Abstract
Integrally convex functions constitute a fundamental function class in discrete convex analysis, including M-convex functions, L-convex functions, and many others. This paper aims at a rather comprehensive survey of recent results on integrally convex functions with some new technical results. Topics covered in this paper include characterizations of integral convex sets and functions, operations on integral convex sets and functions, optimality criteria for minimization with a proximity-scaling algorithm, integral biconjugacy, and the discrete Fenchel duality. While the theory of M-convex and L-convex functions has been built upon fundamental results on matroids and submodular functions, developing the theory of integrally convex functions requires more general and basic tools such as the Fourier–Motzkin elimination.
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1 Introduction
Discrete convex analysis [39, 40] has found applications and connections to a wide variety of disciplines. Early applications include those to combinatorial optimization, matrix theory, and economics, as described in the book [40] and a survey paper [43]. More recently, there have been active interactions with operations research (inventory theory) [4, 5, 60], economics and game theory [44, 53, 59, 63], and algebra (Lorentzian polynomials, in particular) [3]. The reader is referred to [40] for basic concepts and terminology in discrete convex analysis, and to [17, 22, 31,32,33, 35, 45, 46, 58] for recent theoretical and algorithmic developments.
Integrally convex functions, due to Favati–Tardella [13], constitute a fundamental function class in discrete convex analysis. A function \(f: {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}\) is called integrally convex if its local convex extension \(\tilde{f}: {\mathbb {R}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}\) is (globally) convex in the ordinary sense, where \(\tilde{f}\) is defined as the collection of convex extensions of f in each unit hypercube \(\{ x \in {\mathbb {R}}^{n} \mid a_{i} \le x_{i} \le a_{i} + 1 \ (i=1,\ldots , n) \}\) with \(a \in {\mathbb {Z}}^{n}\); see Sect. 3.2 for more precise statements. A subset of \({\mathbb {Z}}^{n}\) is called integrally convex if its indicator function \(\delta _{S}: {\mathbb {Z}}^{n} \rightarrow \{ 0, +\infty \}\) (\(\delta _{S}(x) = 0\) for \(x \in S\) and \(= + \infty \) for \(x \notin S\)) is an integrally convex function. The concept of integral convexity is used in formulating discrete fixed point theorems [25, 26, 28, 68], and designing solution algorithms for discrete systems of nonlinear equations [30, 67]. In game theory the integral concavity of payoff functions guarantees the existence of a pure strategy equilibrium in finite symmetric games [27].
Integrally convex functions serve as a common framework for discrete convex functions. Indeed, separable convex, L-convex, L\(^{\natural }\)-convex, M-convex, M\(^{\natural }\)-convex, L\(^{\natural }_{2}\)-convex, and M\(^{\natural }_{2}\)-convex functions are known to be integrally convex [40]. Multimodular functions [23] are also integrally convex, as pointed out in [42]. Moreover, BS-convex and UJ-convex functions [18] are integrally convex. Discrete midpoint convex functions [38] and directed discrete midpoint convex functions [64] are also integrally convex. The relations among those discrete convexity concepts are investigated in [34, 47]. Figure 1 is an overview of the inclusion relations among the most fundamental classes of discrete convex sets. It is noted that the class of integrally convex sets contains the other classes of discrete convex sets.
Classes of discrete convex sets
In the last several years, significant progress has been made in the theory of integrally convex functions. A proximity theorem for integrally convex functions is established in [37] together with a proximity-scaling algorithm for minimization. Fundamental operations for integrally convex functions such as projection and convolution are investigated in [33, 45, 46]. It is revealed that integer-valued integrally convex functions enjoy integral biconjugacy [50], and a discrete Fenchel-type min-max formula is established for a pair of integer-valued integrally convex and separable convex functions [51]. The present paper aims at a rather comprehensive survey of those recent results on integrally convex functions with some new technical results. While the theory of M-convex and L-convex functions has been built upon fundamental results on matroids and submodular functions, developing the theory of integrally convex functions requires more general and basic tools such as the Fourier–Motzkin elimination.
This paper is organized as follows. In Sect. 2 we review integrally convex sets, with new observations on their polyhedral properties. In Sect. 3 the concept of integrally convex functions is reviewed with emphasis on their characterizations. Section 4 deals with properties related to minimization and minimizers, including a proximity-scaling algorithm. Section 5 is concerned with integral subgradients and biconjugacy, and Sect. 6 with the discrete Fenchel duality.
2 Integrally convex sets
2.1 Hole-free property
Let n be a positive integer and \(N = \{ 1,2,\ldots , n \}\). For a subset I of N, we denote by \(\varvec{1}^{I}\) the characteristic vector of I; the ith component of \(\varvec{1}^{I}\) is equal to 1 or 0 according to whether \(i \in I\) or not. We use a short-hand notation \(\varvec{1}^{i}\) for \(\varvec{1}^{ \{ i \} }\), which is the ith unit vector. The vector with all components equal to 1 is denoted by \(\textbf{1}\), that is, \(\textbf{1}=(1,1,\ldots , 1) = \varvec{1}^{N}\).
For two vectors \(\alpha \in ({\mathbb {R}}\cup \{ -\infty \})^{n}\) and \(\beta \in ({\mathbb {R}}\cup \{ +\infty \})^{n}\) with \(\alpha \le \beta \), we define notation \( [\alpha ,\beta ]_{{\mathbb {R}}} = \{ x \in {\mathbb {R}}^{n} \mid \alpha \le x \le \beta \}\), which represents the set of real vectors between \(\alpha \) and \(\beta \). An integral box will mean a set B of real vectors represented as \(B = [\alpha ,\beta ]_{{\mathbb {R}}}\) for integer vectors \(\alpha \in ({\mathbb {Z}}\cup \{ -\infty \})^{n}\) and \(\beta \in ({\mathbb {Z}}\cup \{ +\infty \})^{n}\) with \(\alpha \le \beta \). The set of integer vectors contained in an integral box will be called a box of integers or an interval of integers. We use notation \([\alpha ,\beta ]_{{\mathbb {Z}}}: = [\alpha ,\beta ]_{{\mathbb {R}}} \cap {\mathbb {Z}}^{n} = \{ x \in {\mathbb {Z}}^{n} \mid \alpha \le x \le \beta \}\) for \(\alpha \in ({\mathbb {Z}}\cup \{ -\infty \})^{n}\) and \(\beta \in ({\mathbb {Z}}\cup \{ +\infty \})^{n}\) with \(\alpha \le \beta \).
For a subset S of \({\mathbb {R}}^{n}\), we denote its convex hull by \(\overline{S}\), which is, by definition, the smallest convex set containing S. As is well known, \(\overline{S}\) coincides with the set of all convex combinations of (finitely many) elements of S. We say that a set \(S \subseteq {\mathbb {Z}}^{n}\) is hole-free if
Since the inclusion \(S \subseteq \overline{S} \cap {\mathbb {Z}}^{n}\) is trivially true for any S, the content of this condition lies in
stating that the integer points contained in the convex hull of S all belong to S itself. A finite set of integer points is hole-free if and only if it is the set of integer points in some integral polytope.
For a set \(S \subseteq {\mathbb {R}}^{n}\) we define its indicator function \(\delta _{S}: {\mathbb {R}}^{n} \rightarrow \{ 0, +\infty \}\) by
Remark 2.1
In a standard textbook [24, Section A.1.3], the convex hull of a subset S of the n-dimensional Euclidean space \({\mathbb {R}}^{n}\) is denoted by \(\textrm{co}\, S\) and the closed convex hull by \(\overline{\textrm{co}}\, S\), where the closed convex hull of S is defined to be the intersection of all closed convex set containing S. It is known that \(\overline{\textrm{co}}\, S\) coincides with the (topological) closure of \(\textrm{co}\, S\), which is expressed as \(\overline{\textrm{co}}\, S = \textrm{cl}(\textrm{co}\, S)\) with the use of notation \(\textrm{cl}\) for closure operation. Using our notation \(\overline{S}\), we have \(\textrm{co}\, S = \overline{S}\) and \(\overline{\textrm{co}}\, S = \textrm{cl}(\overline{S})\). For a finite set S, we have \(\textrm{co}\, S = \overline{\textrm{co}}\, S\). To see the difference of \(\textrm{co}\, S\) and \(\overline{\textrm{co}}\, S\) for an infinite set S, consider \(S = \{ (0,1) \} \cup \{ (k,0) \mid k \in {\mathbb {Z}}\}\). The convex hull is \(\textrm{co}\, S = \overline{S} = \{ (0,1) \} \cup \{ (x_{1},x_{2}) \mid 0 \le x_{2} < 1 \}\) and the closed convex hull is \(\overline{\textrm{co}}\, S = \textrm{cl}(\overline{S}) = \{ (x_{1},x_{2}) \mid 0 \le x_{2} \le 1 \}\). We have \(\overline{S} \cap {\mathbb {Z}}^{2} =S\), which shows that this set S is hole-free, while \(\textrm{cl}(\overline{S}) \cap {\mathbb {Z}}^{2} \ne S\). \(\blacksquare \)
2.2 Definition of integrally convex sets
For \(x \in {\mathbb {R}}^{n}\) the integral neighborhood of x is defined by
It is noted that strict inequality “ < ” is used in this definition and N(x) admits an alternative expression
where, for \(t \in {\mathbb {R}}\) in general, \(\left\lfloor t \right\rfloor \) denotes the largest integer not larger than t (rounding-down to the nearest integer) and 
is the smallest integer not smaller than t (rounding-up to the nearest integer). That is, N(x) consists of all integer vectors z between 
and 
. See Fig. 2 for N(x) when \(n=2\).
Integral neighborhood N(x) of x (\(\bullet \): point of N(x))
For a set \(S \subseteq {\mathbb {Z}}^{n}\) and \(x \in {\mathbb {R}}^{n}\) we call the convex hull of \(S \cap N(x)\) the local convex hull of S around x. A nonempty set \(S \subseteq {\mathbb {Z}}^{n}\) is said to be integrally convex if the union of the local convex hulls \(\overline{S \cap N(x)}\) over \(x \in {\mathbb {R}}^{n}\) is convex [40]. In other words, a set \(S \subseteq {\mathbb {Z}}^{n}\) is called integrally convex if
Since the inclusion \(\overline{S} \supseteq \bigcup _{x} \overline{S \cap N(x)}\) is trivially true for any S, the content of the condition (2.6) lies in
Example 2.1
The concept of integrally convex sets is illustrated by simple examples. The six-point set in Fig. 3(a) is integrally convex. The removal of the middle point (Fig. 3(b)) breaks integral convexity (cf., Proposition 2.2). The four-point set \(S = \{ \hbox {A, B, C, D} \}\) in Fig. 3(c) is not integrally convex, since its convex hull \(\overline{S}\), which is the triangle ACD, does not coincide with the union of the local convex hulls \(\overline{S \cap N(x)}\), which is the union of the line segment AB and the triangle BCD. \(\blacksquare \)
Concept of integrally convex sets
Example 2.2
Obviously, every subset of \(\{ 0, 1\}^{n}\) is integrally convex, and every interval of integers \((\subseteq {\mathbb {Z}}^{n})\) is integrally convex. \(\blacksquare \)
Integral convexity can be defined by seemingly different conditions. Here we mention the following two.
-
Every point x in the convex hull of S is contained in the convex hull of \(S \cap N(x)\), i.e.,
$$\begin{aligned} x \in \overline{S} \ \Longrightarrow x \in \overline{S \cap N(x)}. \end{aligned}$$(2.8) -
For each \(x \in {\mathbb {R}}^{n}\), the intersection of the convex hulls of S and N(x) is equal to the convex hull of the intersection of S and N(x), i.e.,
$$\begin{aligned} \overline{S} \cap \overline{N(x)} = \overline{S \cap N(x)}. \end{aligned}$$(2.9)Since the inclusion \(\overline{S} \cap \overline{N(x)} \supseteq \overline{S \cap N(x)}\) is trivially true for any S and x, the content of this condition lies in
$$\begin{aligned} \overline{S} \cap \overline{N(x)} \subseteq \overline{S \cap N(x)}. \end{aligned}$$(2.10)
The following proposition states the equivalence of the five conditions (2.6) to (2.10). Thus, any one of these conditions characterizes integral convexity of a set of integer points.
Proposition 2.1
For \(S \subseteq \mathbb {Z}^{n}\), the five conditions (2.6) to (2.10) are all equivalent.
Proof
The proof is easy and straightforward, but we include it for completeness. We already mentioned the equivalences [(2.6)\(\Leftrightarrow \)(2.7)] and [(2.9)\(\Leftrightarrow \)(2.10)]. [(2.7)\(\Rightarrow \)(2.8)]: Take any \(x \in \overline{S}\). By (2.7) we obtain \(x \in \overline{S \cap N(y)}\) for some y. This means that there exist integer points \(x^{(1)}, x^{(2)}, \ldots , x^{(m)} \in S \cap N(y)\) and coefficients \(\lambda _{1}, \lambda _{2}, \ldots , \lambda _{m} > 0\) such that \(x = \sum _{k=1}^{m} \lambda _{k} \, x^{(k)}\) and \(\sum _{k=1}^{m} \lambda _{k} = 1\). Since all the points x, \(x^{(1)}, x^{(2)}, \ldots , x^{(m)}\) lie between \(\left\lfloor y \right\rfloor \) and 
, we must have \(x^{(k)} \in N(x)\) for all k. Therefore, \(x \in \overline{S \cap N(x)}\).
[(2.8)\(\Rightarrow \)(2.7)]: Take any \(y \in \overline{S}\). By (2.8) we obtain \(y \in \overline{S \cap N(y)}\), whereas \(\overline{S \cap N(y)} \subseteq \bigcup _{x \in {\mathbb {R}}^{n}} \overline{S \cap N(x)}\). Thus (2.7) is shown.
[(2.8)\(\Rightarrow \)(2.10)]: Take any \(y \in \overline{S} \cap \overline{N(x)}\). By \(y \in \overline{S}\) and (2.8) we obtain \(y \in \overline{S \cap N(y)}\), whereas \(y \in \overline{N(x)}\) implies \(N(y) \subseteq N(x)\). Hence \(y \in \overline{S \cap N(x)}\).
[(2.10)\(\Rightarrow \)(2.8)]: Take any \(x \in \overline{S}\). Since \(x \in \overline{N(x)}\), we have \(x \in \overline{S} \cap \overline{N(x)}\). Then \(x \in \overline{S \cap N(x)}\) by (2.10). \(\square \)
As an application of Proposition 2.1 we give a formal proof to the statement that an integrally convex set is hole-free.
Proposition 2.2
For an integrally convex set S, we have \(S = \overline{S} \cap {\mathbb {Z}}^{n}\).
Proof
It suffices to show \(\overline{S} \cap {\mathbb {Z}}^{n} \subseteq S\) in (2.2). Take any \(x \in \overline{S} \cap {\mathbb {Z}}^{n}\). Since \(N(x) = \{ x \}\), (2.8) shows \(x \in \overline{S \cap \{ x \}}\), which implies \(x \in S\). \(\square \)
The following (new) characterization of an integrally convex set is often useful, which is used indeed in the proof of Theorem 3.2 in Sect. 3.4. While the condition (2.8) refers to all real vectors x, the condition (2.11) below restricts x to the midpoints of two vectors in S.
Theorem 2.1
A nonempty set \(S \subseteq {\mathbb {Z}}^{n}\) is integrally convex if and only if
for every \(y, y' \in S\) with \(\Vert y - y' \Vert _{\infty } \ge 2\).
Proof
The only-if-part is obvious from (2.8), since \((y + y')/2 \in \overline{S}\). To prove the if-part, let \(x \in \overline{S}\). This implies the existence of \(y^{(1)},y^{(2)},\ldots , y^{(m)} \in S\) such that
where \(\sum _{i=1}^{m} \lambda _{i} = 1\) and \(\lambda _{i} > 0 \ (i=1,2,\ldots , m)\). In the following we modify the generating points \(y^{(1)},y^{(2)},\ldots , y^{(m)}\) repeatedly and eventually arrive at an expression of the form (2.12) with the additional condition that \(y^{(i)} \in N(x)\) for all i, showing \(x \in \overline{S \cap N(x)}\). Then the integral convexity of S is established by (2.8).
For each \(j=n,n-1,\ldots , 1\), we look at the j-th component of \(y^{(1)},y^{(2)}, \ldots , y^{(m)}\). Let \(j=n\) and define
If \(\beta _{n} - \alpha _{n} \le 1\), we are done with \(j=n\). Suppose that \(\beta _{n} - \alpha _{n} \ge 2\). By translation and reversal of the n-th coordinate, we may assume \(0 \le x_{n} \le 1\), \(\alpha _{n} \le 0\), and \(\beta _{n} \ge 2\). By renumbering the generators we may assume \(1 \in I_{\min }\) and \(2 \in I_{\max }\), i.e., \(y^{(1)}_{n} = \alpha _{n}\) and \(y^{(2)}_{n} = \beta _{n}\). We have \(\Vert y^{(1)} - y^{(2)} \Vert _{\infty } \ge 2\).
By (2.11) for \((y^{(1)}, y^{(2)})\) we have
with \(\mu _{k} > 0\)\((k=1,2,\ldots , l)\) and \(\sum _{k=1}^{l} \mu _{k} = 1\). With notation \(\lambda = \min (\lambda _{1}, \lambda _{2})\), it follows from (2.12) and (2.14) that
which is another representation of the form (2.12).
With reference to this new representation we define \(\hat{\alpha }_{n}\), \(\hat{\beta }_{n}\), \({\hat{I}}_{\min }\), and \({\hat{I}}_{\max }\), as in (2.13). Since \(\beta _{n} - \alpha _{n} \ge 2\), we have
which implies \(\alpha _{n} + 1 \le z^{(k)}_{n} \le \beta _{n} - 1\) for all k. Hence, \(\alpha _{n} \le \hat{\alpha }_{n}\) and \(\hat{\beta }_{n} \le \beta _{n}\). Moreover, if \((\hat{\alpha }_{n}, \hat{\beta }_{n})=(\alpha _{n},\beta _{n})\), then \(|{\hat{I}}_{\min }| + |{\hat{I}}_{\max }| \le |I_{\min }| + |I_{\max }| - 1\). Therefore, by repeating the above process with \(j=n\), we eventually arrive at a representation of the form of (2.12) with \(\beta _{n} - \alpha _{n} \le 1\).
We next apply the above procedure for the \((n-1)\)-st component. What is crucial here is that the condition \(\beta _{n} - \alpha _{n} \le 1\) is maintained in the modification of the generators via (2.14) for the \((n-1)\)-st component. Indeed, for each k, the inequality \(\alpha _{n} \le z^{(k)}_{n} \le \beta _{n}\) follows from \(\alpha _{n} \le (y^{(1)}_{n} +y^{(2)}_{n})/2 \le \beta _{n}\) and \(z^{(k)} \in N( (y^{(1)} + y^{(2)})/2 )\). Therefore, we can obtain a representation of the form of (2.12) with \(\beta _{n} - \alpha _{n} \le 1\) and \(\beta _{n-1} - \alpha _{n-1} \le 1\), where \(\alpha _{n-1} = \min _{i} y^{(i)}_{n-1}\) and \(\beta _{n-1} = \max _{i} y^{(i)}_{n-1}\).
Then we continue the above process for \(j=n-2,n-3, \ldots ,1\), to finally obtain a representation of the form of (2.12) with \(|y^{(i)}_{j} - y^{(i')}_{j}| \le 1\) for all \(i, i'\) and \(j=1,2,\dots ,n\). This means, in particular, that \(y^{(i)} \in S \cap N(x)\) for all i. \(\square \)
2.3 Polyhedral aspects
A subset of \({\mathbb {R}}^{n}\) is called a polyhedron if it is described by a finite number of linear inequalities. A polyhedron is said to be rational if it is described by a finite number of linear inequalities with rational coefficients. A polyhedron is called an integer polyhedron if \(P=\overline{P \cap {\mathbb {Z}}^{n}}\), i.e., if it coincides with the convex hull of the integer points contained in it, or equivalently, if P is rational and each face of P contains an integer vector. See [56, 57] for terminology about polyhedra. For two vectors \(a, x \in {\mathbb {R}}^{n}\), we use notation \(\langle a, x \rangle = a^{\top } x = \sum _{i=1}^{n} a_{i} x_{i}\).
The convex hull of an integrally convex set is an integer polyhedron (see [50, Sec. 4.1] for a rigorous proof). However, not much is known about the inequality system \(Ax \le b\) to describe an integrally convex set. This is not surprising because every subset of \(\{ 0, 1\}^{n}\) is integrally convex (as noted in Example 2.2), and most of the NP-hard combinatorial optimization problems can be formulated on \(\{ 0, 1\}^{n}\) polytopes.
When \(n = 2\), the following fact is easy to see.
Proposition 2.3
([37]) A set \(S \subseteq \mathbb {Z}^{2}\) is integrally convex if and only if it can be represented as \(S = \{ (x_{1},x_{2}) \in \mathbb {Z}^{2} \mid a^{i} x_{1} + b^{i} x_{2} \le c^{i} \ (i=1,2,\ldots ,m) \}\) for some \(a^{i}, b^{i} \in \{ -1,0,1 \}\) and \(c^{i} \in {\mathbb {Z}}\) \((i=1,2,\ldots ,m)\).
Polyhedral descriptions are known for major subclasses of integrally convex sets. The present knowledge for various kinds of discrete convex sets is summarized in Table 1, which shows the possible forms of the vector a for an inequality \(a^{\top } x \le b\) to describe the convex hull \(\overline{S}\) of a discrete convex set S. It should be clear that each vector a corresponds (essentially) to the normal vector of a face of \(\overline{S}\). Since an M\(_{2}\)-convex (resp., M\(^{\natural }_{2}\)-convex) set is, by definition, the intersection of two M-convex (resp., M\(^{\natural }\)-convex) sets [40], the polyhedral description of an M\(_{2}\)-convex (resp., M\(^{\natural }_{2}\)-convex) set is obtained immediately as the union of the inequality systems for the constituent M-convex (resp., M\(^{\natural }\)-convex) sets.
In all cases listed in Table 1, we have \(a \in \{ -1,0,+1 \}^{n},\,{ thatis},\,{ everycomponentof}a{ belongsto}\{ -1,0,+1 \}{} \). However, this is not the case with a general integrally convex set.
Example 2.3
Let \(S = \{ (1,1,0,0), \ (0,1,1,0), \ (1,0,1,0), \ (0,0,0,1) \},\,{ whichisobviouslyanintegrallyconvexsetsince}S \subseteq \{ 0, 1\}^{4}.{ Becauseallthesepointslieonthehyperplane}x_{1} + x_{2} + x_{3} + 2 x_{4} = 2,\,{ weneedavector}a= \pm (1,1,1,2){ todescribetheconvexhull}\overline{S}{} \). \(\blacksquare \)
The following example illustrates a use of the results in Table 1.
Example 2.4
Consider \(S = \{ x \in {\mathbb {Z}}^{4} \mid x_{1} + x_{2} = x_{3} + x_{4} \}.{ Thissetisdescribedbytwoinequalitiesoftheformof}a^{\top } x \le 0{ with}a = (1,1,-1,-1), (-1,-1,1,1).{ Thisshowsthat}S\) is not L\(^{\natural }\)-convex, because we must have \(a=\varvec{1}^{i} - \varvec{1}^{j}{} { or}\pm \varvec{1}^{i}{} \) for an L\(^{\natural }\)-convex set (see Table 1). M\(^{\natural }\)-convexity of \(S{ isalsodeniedbecause}a=\pm \varvec{1}^{I}{} \) for an M\(^{\natural }\)-convex set. The set \(S\) is, in fact, an L\(_{2}\)-convex set with \(a=\varvec{1}^{J} - \varvec{1}^{I}{} { for}(I,J)=(\{ 1,2 \},\{ 3,4 \}){ and}(I,J)=(\{ 3,4 \},\{ 1,2 \})\). \(\blacksquare \)
Each face of the convex hull \(\overline{S}{} { ofanintegrallyconvexset}S\) induces an integrally convex set.
Proposition 2.4
Let \(S \subseteq {\mathbb {Z}}^{n}{} { beanintegrallyconvexset}.{ Foranyface}F{ of}\overline{S},\,F \cap {\mathbb {Z}}^{n}{} \) is integrally convex.
Proof
Consider inequality descriptions of \(\overline{S}{} { and}F,\,{ say},\,\overline{S} = \{ x \in {\mathbb {R}}^{n} \mid \langle a^{(i)}, x \rangle \le b^{(i)} \ (i \in I) \}{} { and}F = \{ x \in {\mathbb {R}}^{n} \mid \langle a^{(j)}, x \rangle = b^{(j)} \ (j \in J), \ \langle a^{(i)}, x \rangle \le b^{(i)} \ (i \in I {\setminus } J) \}{} { withsomeindexsets}I \supseteq J.{ Since}\overline{S}{} { isanintegerpolyhedron},\,{ wemayassume}\textbf{0}\in F,\,{ whichimplies}b^{(j)} = 0{ for}j \in J.{ Takeany}x \in F.{ Since}x \in \overline{S}{} { and}S{ isintegrallyconvex},\,{ thereexist}y^{(1)}, y^{(2)}, \ldots , y^{(m)} \in S \cap N(x){ andcoefficients}\lambda _{1}, \lambda _{2}, \ldots , \lambda _{m} > 0{ suchthat}x = \sum _{k=1}^{m} \lambda _{k} \, y^{(k)}{} { and}\sum _{k=1}^{m} \lambda _{k} = 1.{ Foreach}j \in J,\,{ wehave}0 = \langle a^{(j)}, x \rangle = \langle a^{(j)}, \sum _{k=1}^{m} \lambda _{k} \, y^{(k)} \rangle = \sum _{k=1}^{m} \lambda _{k} \langle a^{(j)}, y^{(k)} \rangle ,\,{ where}\langle a^{(j)}, y^{(k)} \rangle \le 0{ since}y^{(k)} \in S.{ Therefore}\langle a^{(j)}, y^{(k)} \rangle = 0{ forall}k,\,{ whichimplies}y^{(k)} \in F,\,{ andhence}y^{(k)} \in (F \cap {\mathbb {Z}}^{n}) \cap N(x){ forall}k. \square \)
The edges of \(\overline{S}{} { havearemarkablepropertythatthedirectionofanedgeisgivenbya}\{ -1,0,+1 \}{} \)-vector. This property follows from a basic fact that every edge (line) of \(\overline{S}{} { containsapairoflatticepointsinatranslatedunithypercube},\,{ whosedifferenceisa}\{ -1,0,+1 \}{} \)-vector. In this connection, the following fact is known.
Proposition 2.5
([50, Proposition 1]) Let \(S \subseteq {\mathbb {Z}}^{n}{} { beanintegrallyconvexset}.{ Foranyface}F{ of}\overline{S},\,{ thesmallestaffinesubspacecontaining}F\) is given as \(\{ x + \sum _{k=1}^{h} c_{k} d^{(k)} \mid c_{1}, c_{2}, \ldots , c_{h} \in {\mathbb {R}}\}{} \) for any point \(x{ in}F{ andsomedirectionvectors}d^{(k)} \in \{ -1,0,+1 \}^{n} (k=1,2,\ldots , h)\).
Remark 2.2
( [50]) The property mentioned in Proposition 2.5 does not characterize integral convexity of a set. For example, let \(S = \{ (0,0,0), (1,0,1), (1,1,-1), (2,1,0) \}.{ Theconvexhull}\overline{S}{} { isaparallelogramwithedgedirections}(1,0,1){ and}(1,1,-1),\,{ andhenceisanintegerpolyhedronwiththepropertythatthesmallestaffinesubspacecontainingeachfaceisspannedby}\{ -1,0,+1 \}{} \)-vectors. However, \(S\) is not integrally convex, since (2.8) is violated by \(x = [(1,0,1) + (1,1,-1) ]/2 = (1,1/2,0) \in \overline{S},\,{ forwhich}N(x) = \{ (1,0,0), (1,1,0) \}{} { and}S \cap N(x) = \emptyset \). \(\blacksquare \)
The following result is concerned with a direction of infinity in the discrete setting, to be used in the proof of Proposition 3.3 in Sect. 3.7.
Proposition 2.6
Let \(S \subseteq {\mathbb {Z}}^{n}{} { beanintegrallyconvexset},\,y \in S,\,{ and}d \in {\mathbb {Z}}^{n}.{ If}y + kd \in S{ forallintegers}k \ge 1,\,{ thenforany}z \in S,\,{ wehave}z + kd \in S{ forallintegers}k \ge 1\).
Proof
It suffices to prove that, for any \(x \in S,\,{ wehave}x + d \in S.{ Foraninteger}k \ge 1 ({ tobespecifiedlater}),\,{ consider}u = \frac{k-1}{k} x + \frac{1}{k}(y + kd) = x+d + \frac{1}{k}(y-x),\,{ whichisaconvexcombinationof}x{ and}y + kd.{ Bytaking}k{ largeenough},\,{ wecanassume} \Vert u - (x+d) \Vert _{2} = \Vert y - x \Vert _{2}/k < 1 / \sqrt{n} ,\,{ whichimpliesthat}x+d \in N(u){ and}u \notin \overline{N(u) \setminus \{ x+d \}}.{ Ontheotherhand},\,{ wehave}u \in \overline{S \cap N(u)},\,{ since}u \in \overline{S}{} { and}S{ isintegrallyconvex}.{ Therefore},\,{ wemusthave}x+d \in S. \square \)
2.3.1 Box-integer and box-TDI polyhedra
A polyhedron \(P \subseteq {\mathbb {R}}^{n}{} \) is called box-integer if \(P \cap [ l, u ]_{{\mathbb {R}}} (=P \cap \{ x \in {\mathbb {R}}^{n} \mid l \le x \le u \}){ isanintegerpolyhedronforeachchoiceofintegervectors}l, u \in {\mathbb {Z}}^{n}{} { with}l \le u\) ( [57, Section 5.15]). This concept is closely related (or essentially equivalent) to that of integrally convex sets, as follows.
Proposition 2.7
([46]) If a set \(S \subseteq {\mathbb {Z}}^{n}{} { isintegrallyconvex},\,{ thenitsconvexhull}\overline{S}{} \) is a box-integer polyhedron. Conversely, if \(P\) is a box-integer polyhedron, then \(S = P \cap {\mathbb {Z}}^{n}{} \) is an integrally convex set.
It follows from Proposition 2.7 that a set \(S\) of integer points is integrally convex if and only if it is hole-free and its convex hull \(\overline{S}{} \) is a box-integer polyhedron.
The concept of (box-)total dual integrality has long played a major role in combinatorial optimization [7, 8, 11, 12, 56, 57]. A linear inequality system \(Ax \le b\) is said to be totally dual integral (TDI) if the entries of \(A{ and}b\) are rational numbers and the minimum in the linear programming duality equation
has an integral optimal solution \(y{ foreveryintegralvector}w{ suchthattheminimumisfinite}.{ Alinearinequalitysystem}Ax \le b\) is said to be box-totally dual integral (box-TDI) if the system \([ Ax \le b, d \le x\le c ]\) is TDI for each choice of rational (finite-valued) vectors \(c{ and}d\). It is known [57, Theorem 5.35] that a system \(A x \le b\) is box-TDI if the matrix \(A\) is totally unimodular.
A polyhedron is called a box-TDI polyhedron if it can be described by a box-TDI system. It was pointed out in [8] that every TDI system describing a box-TDI polyhedron is a box-TDI system, which fact indicates that box-TDI is a property of a polyhedron. In this connection it is worth mentioning that every rational polyhedron can be described by a TDI system, showing that TDI is a property of a system of inequalities and not of a polyhedron.
An integral box-TDI polyhedron is box-integer [57, (5.82), p. 83]. Although the converse is not true (see Example 2.5 below), it is possible to characterize a box-TDI polyhedron in terms of box-integrality of its dilations. For a positive integer \(\alpha ,\,{ the}\alpha \)-dilation of a polyhedron \(P=\{x \mid A x \le b \}{} { meansthepolyhedron}\alpha P:=\{x \mid A x \le \alpha b \} = \{x \mid \frac{1}{\alpha }x \in P \}{} \).
Proposition 2.8
( [6, Theorem 2 & Prop. 2]) An integer polyhedron \(P\) is box-TDI if and only if the \(\alpha \)-dilation \(\alpha P\) is box-integer for any positive integer \(\alpha \).
Example 2.5
Here is an example of a \(\{ 0,1 \}{} \)-polyhedron that is not box-TDI. Let \(P (\subseteq {\mathbb {R}}^{4}){ betheconvexhullof}S = \{ (1,1,0,0), \ (0,1,1,0), \ (1,0,1,0), (0,0,0,1) \}{} \) (considered in Example 2.3). Since \(P{ isa}\{ 0,1 \}{} \)-polyhedron, it is obviously box-integer. However, the 2-dilation \(2P\) is not box-integer. To see this we note that
whereas \((1,1,1,1/2) \in [ l, u ]_{{\mathbb {R}}}{} { for}l =(1,1,1,0){ and}u =(1,1,1,1),\,{ and}l \notin 2P{ and}u \notin 2P.{ Wecaneasilyshowthat}(2P) \cap [ l, u ]_{{\mathbb {R}}}{} { consistsof}(1,1,1,1/2){ only}.{ Thus}2P\) is not box-integer, which implies, by Proposition 2.8, that \(P\) is not box-TDI. \(\blacksquare \)
Following [15] we call the set of integral elements of an integral box-TDI polyhedron a discrete box-TDI set, or just a box-TDI set. A box-TDI set is an integrally convex set, but the converse is not true (Example 2.5). That is, box-TDI sets form a proper subclass of integrally convex sets. On the other hand, the major classes of discrete convex sets considered in discrete convex analysis are known to be box-TDI as follows.
Proposition 2.9
([35]) An L\(^{\natural }_{2}\)-convex set is a box-TDI set.
Proposition 2.10
An M\(^{\natural }_{2}\)-convex set is a box-TDI set.
Proposition 2.9 for L\(^{\natural }_{2}\)-convex sets is established recently in [35] and Proposition 2.10 for M\(^{\natural }_{2}\)-convex sets is a reformulation of the fundamental fact about polymatroid intersection [57] in the language of discrete convex analysis. Theses propositions imply, in particular, that L\(_{2}\)-, L\(^{\natural }\)-, L-, M\(_{2}\)-, M\(^{\natural }\)-, M-convex sets are all box-TDI.
The following are examples of a box-TDI set \(S\) that is neither L\(^{\natural }_{2}\)-convex nor M\(^{\natural }_{2}\)-convex. The former consists of \(\{ 0,1 \}{} \)-vectors and the latter arises from a cone.
Example 2.6
Consider \(S = \{ (0,0,0), \ (1,1,0), \ (1,0,1), \ (0,1,1) \}{} \). This set is described by four inequalities
The first inequality, of the form of \(a^{\top } x \le b{ with}a =(1,1,1)\), denies L\(^{\natural }_{2}\)-convexity of \(S,\,{ becausewemusthave}a=\varvec{1}^{J} - \varvec{1}^{I}{} { with}|I|-|J| \in \{ -1,0,1 \}{} \) for an L\(^{\natural }_{2}\)-convex set (see Table 1). In the second inequality we have \(a =(1,-1,-1)\), which denies M\(^{\natural }_{2}\)-convexity of \(S,\,{ because}a=\pm \varvec{1}^{I}{} \) for an M\(^{\natural }_{2}\)-convex set. The set \(S\) is box-TDI, that is, its convex hull \(\overline{S}{} \) is a box-TDI polyhedron, which we can verify on the basis of Proposition 2.8. \(\blacksquare \)
Example 2.7
The set \(S = \{ x \in {\mathbb {Z}}^{2} \mid x_{1} + x_{2} \le 0, x_{1} - x_{2} \le 0 \}{} \) is neither L\(^{\natural }_{2}\)-convex nor M\(^{\natural }_{2}\)-convex, whereas it is box-TDI since the convex hull \(\overline{S}{} \) is a box-TDI polyhedron by Proposition 2.8. \(\blacksquare \)
We can summarize the above argument as
where Examples 2.5, 2.6, and 2.7 demonstrate the strict inclusions (\(\subsetneqq \)). See Fig. 1.
2.4 Basic operations
In this section we show how integral convexity of a set behaves under basic operations. Let \(S{ beasubsetof}{\mathbb {Z}}^{n},\,{ i}.{ e}.,\,S \subseteq {\mathbb {Z}}^{n}{} \).
Origin shift:
For an integer vector \(b \in {\mathbb {Z}}^{n}{} \), the origin shift of \(S{ by}b{ meansaset}T{ definedby}T = \{ x - b \mid x \in S \}{} \). The origin shift of an integrally convex set is an integrally convex set.
Inversion of coordinates:
The independent coordinate inversion of \(S{ meansaset}T\) defined by
with an arbitrary choice of \(\tau _{i} \in \{ +1, -1 \} (i=1,2,\ldots ,n)\). The independent coordinate inversion of an integrally convex set is an integrally convex set. This is a nice property of integral convexity, not shared by L\(^{\natural }\)-, L\(^{\natural }_{2}\)-, M\(^{\natural }\), or M\(^{\natural }_{2}\)-convexity.
Permutation of coordinates:
For a permutation \(\sigma { of}(1,2,\ldots ,n)\), the permutation of \(S{ by}\sigma { meansaset}T\) defined by
The permutation of an integrally convex set is an integrally convex set.
Remark 2.3
Integral convexity is not preserved under a transformation by a (totally) unimodular matrix. For example, \(S=\{ (0,0), (1,0), (1,1) \}{} { isintegrallyconvexand}A = {\left[ \begin{array}{cc} 1 &{} 1 \\ 0 &{} 1 \\ \end{array} \right] }{} { istotallyunimodular}.{ However},\,\{ Ax \mid x \in S \} = \{ (0,0), (1,0), (2,1) \}{} \) is not integrally convex. \(\blacksquare \)
Scaling: For a positive integer \(\alpha \), the scaling of \(S{ by}\alpha { meansaset}T\) defined by
Note that the same scaling factor \(\alpha { isusedforallcoordinates}.{ If}\alpha = 2,\,{ forexample},\,{ thisoperationamountstoconsideringthesetofevenpointscontainedin}S\). The scaling of an integrally convex set is not necessarily integrally convex (Example 2.8 below). However, when \(n = 2,\,{ integralconvexityadmitsthescalingoperation}.{ Thatis},\,{ if}S \subseteq {\mathbb {Z}}^{2}{} { isintegrallyconvex},\,{ then}T = \{ y \in {\mathbb {Z}}^{2} \mid \alpha y \in S \}{} \) is integrally convex ( [37, Proposition 3.1]).
Example 2.8
( [37, Example 3.1]) This example shows that integral convexity is not preserved under scaling. Let \(S{ beasubsetof}{\mathbb {Z}}^{3}{} \) defined by
which is an integrally convex set (Fig. 4, left). With the scaling factor \(\alpha =2,\,{ however},\,{ thescaledset} T = \{ y \in {\mathbb {Z}}^{3} \mid 2 y \in S \} = \{ (0,0,0), (1,0,0), (1,0,1), (2,1,1) \}{} \) is not integrally convex (Fig. 4, right). \(\blacksquare \)
Dilation: The dilation operation for a polyhedron (described in Sect. 2.3) is another kind of scaling operation. An adaptation of this operation to a hole-free discrete set \(S \subseteq {\mathbb {Z}}^{n},\,{ wemaycalltheset}T' = (\alpha \overline{S}) \cap {\mathbb {Z}}^{n}{} { the}\alpha \)-dilation of \(S,\,{ where}\alpha \) is a positive integer. Note that the scaling in (2.15) can be expressed as \(T = (\frac{1}{\alpha } \overline{S}) \cap {\mathbb {Z}}^{n}{} { when}S\) is hole-free.
The dilation operation does not always preserve integral convexity. Indeed, Example 2.5 shows that the \(2\)-dilation of an integrally convex set is not necessarily integrally convex.
Remark 2.4
Failure of dilation operation is rather exceptional for discrete convex sets. Indeed, all kinds of discrete convexity (box, L-, L\(^{\natural }\)-, L\(_{2}\)-, L\(^{\natural }_{2}\)-, M-, M\(^{\natural }\)-, M\(_{2}\)-, M\(^{\natural }_{2}\)-convexity, and multimodularity) listed in Table 1 are preserved under the dilation operation. In contrast, the scaling operation in (2.15) preserves L-convexity and its relatives (box, L-, L\(^{\natural }\)-, L\(_{2}\)-, L\(^{\natural }_{2}\)-convexity, and multimodularity), and not M-convexity and its relatives (M-, M\(^{\natural }\)-, M\(_{2}\)-, M\(^{\natural }_{2}\)-convexity). \(\blacksquare \)
Restriction:
For a set \(S \subseteq {\mathbb {Z}}^{N}{} { andasubset}U{ oftheindexset}N = \{ 1,2,\ldots , n \}{} \), the restriction of \(S{ to}U{ isasubset}T{ of}{\mathbb {Z}}^{U}{} \) defined by
where \(\textbf{0}_{N \setminus U}{} { denotesthezerovectorin}{\mathbb {Z}}^{N \setminus U}.{ Thenotation}(y,\textbf{0}_{N \setminus U}){ meansthevectorin}{\mathbb {Z}}^{N}{} { whose}i{ thcomponentisequalto}y_{i}{} { for}i \in U{ andto}0{ for}i \in N {\setminus } U\). The restriction of an integrally convex set is integrally convex (if the resulting set is nonempty).
Projection:
For a set \(S \subseteq {\mathbb {Z}}^{N}{} { andasubset}U{ oftheindexset}N = \{ 1,2,\ldots , n \}{} \), the projection of \(S{ to}U{ isasubset}T{ of}{\mathbb {Z}}^{U}{} \) defined by
where the notation \((y,z){ meansthevectorin}{\mathbb {Z}}^{N}{} { whose}i{ thcomponentisequalto}y_{i}{} { for}i \in U{ andto}z_{i}{} { for}i \in N \setminus U\). The projection of an integrally convex set is integrally convex ( [33, Theorem 3.1]).
Splitting:
Suppose that we are given a family \(\{ U_{1}, U_{2}, \dots , U_{n} \}{} { ofdisjointnonemptysetsindexedby}N = \{ 1, 2, \dots , n\}.{ Let}m_{i}= |U_{i}|{ for}i=1,2,\ldots , n{ anddefine}m= \sum _{i=1}^{n} m_{i},\,{ where}m \ge n.{ Foreach}i \in N{ wedefinean}m_{i}{} \)-dimensional vector \(y_{[i]} = ( y_{j} \mid j \in U_{i} ){ andexpress}y \in {\mathbb {Z}}^{m}{} { as}y = (y_{[1]}, y_{[2]}, \dots , y_{[n]}).{ Foraset}S \subseteq {\mathbb {Z}}^{n},\,{ thesubsetof}{\mathbb {Z}}^{m}{} \) defined by
is called the splitting of \(S{ by}\{ U_{1}, U_{2}, \dots , U_{n} \},\,{ where}y_{[i]}(U_{i}) = \sum \{ y_{j} \mid j \in U_{i} \}.{ Forexample},\,T = \{ (y_{1}, y_{2}, y_{3}) \in {\mathbb {Z}}^{3} \mid (y_{1}, y_{2}+ y_{3}) \in S \}{} { isasplittingof}S \subseteq {\mathbb {Z}}^{2}{} { for}U_{1} = \{ 1 \}{} { and}U_{2} = \{ 2,3 \},\,{ where}n=2{ and}m=3\). The splitting of an integrally convex set is integrally convex ( [46, Proposition 3.4]).
Aggregation:
Let \({\mathcal {P}} = \{ N_{1}, N_{2}, \dots , N_{m} \}{} { beapartitionof}N = \{ 1,2, \ldots , n \}{} { intodisjointnonemptysubsets}\,:\,N = N_{1} \cup N_{2} \cup \dots \cup N_{m}{} { and}N_{i} \cap N_{j} = \emptyset { for}i \not = j.{ Foraset}S \subseteq {\mathbb {Z}}^{N}{} { thesubsetof}{\mathbb {Z}}^{m},\,{ where}m \le n\), defined by
is called the aggregation of \(S{ by}{\mathcal {P}}.{ Forexample},\,T = \{ (y_{1}, y_{2}) \in {\mathbb {Z}}^{2} \mid y_{1} = x_{1}, y_{2} = x_{2} + x_{3} \hbox { for some } (x_{1}, x_{2}, x_{3}) \in S \}{} { isanaggregationof}S \subseteq {\mathbb {Z}}^{3}{} { for}N_{1} = \{ 1 \}{} { and}N_{2} = \{ 2,3 \},\,{ where}n=3{ and}m=2\). The aggregation of an integrally convex set is not necessarily integrally convex.
Example 2.9
( [46, Example 3.4]) Set \(S = \{ (0,0,1,0), (0,0,0,1), (1,1,1,0), (1,1,0,1) \}{} { isanintegrallyconvexset}.{ Forthepartitionof}N = \{ 1,2,3,4 \}{} { into}N_{1} = \{ 1,3 \}{} { and}N_{2} = \{ 2,4 \},\,{ theaggregationof}S{ by}\{ N_{1}, N_{2} \}{} { isgivenby}T = \{ (1,0), (0,1), (2,1), (1,2) \}{} \), which is not integrally convex. \(\blacksquare \)
Intersection:
The intersection \(S_{1} \cap S_{2}{} { ofintegrallyconvexsets}S_{1},\,S_{2} \subseteq {\mathbb {Z}}^{n}{} \) is not necessarily integrally convex (Example 2.10 below). However, it is obviously true (almost from definition) that the intersection of an integrally convex set with a box of integers is integrally convex.
Example 2.10
( [47, Example 4.4]) The intersection of two integrally convex sets is not necessarily integrally convex. Let \(S_{1} = \{(0, 0, 0), (0, 1, 1), (1, 1, 0), (1, 2, 1)\}{} { and}S_{2} = \{(0, 0, 0), (0, 1, 0), (1, 1, 1), (1, 2, 1)\},\,{ forwhich}S_{1} \cap S_{2}= \{(0, 0, 0), (1, 2, 1)\}.{ Thesets}S_{1}{} { and}S_{2}{} { areintegrallyconvex},\,{ whereas}S_{1} \cap S_{2}{} \) is not. \(\blacksquare \)
Minkowski sum:
The Minkowski sum of two sets \(S_{1},\,S_{2} \subseteq {\mathbb {Z}}^{n}{} { meansthesubsetof}{\mathbb {Z}}^{n}{} \) defined by
The Minkowski sum of integrally convex sets is not necessarily integrally convex (Example 2.11 below). However, the Minkowski sum of an integrally convex set with a box of integers is integrally convex ( [33, Theorem 4.1]).
Example 2.11
( [40, Example 3.15]) The Minkowski sum of \(S_{1} = \{ (0,0), (1,1) \}{} { and}S_{2} = \{ (1,0), (0,1) \}{} { isequalto}S_{1}+S_{2} = \{ (1,0), (0,1), (2,1), (1,2) \}{} \), which has a “hole” at \((1,1),\,{ i}.{ e}.,\,(1,1) \in \overline{S_{1}+S_{2}}{} { and}(1,1) \not \in S_{1}+S_{2}{} \). \(\blacksquare \)
Failure of convexity in Minkowski sum (2.18)
Remark 2.5
The Minkowski sum is often a source of difficulty in a discrete setting, because
is not always true (Example 2.11). In other words, the equality (2.18), if true, captures a certain essence of discrete convexity. The property (2.18) is called “convexity in Minkowski sum” in [40, Section 3.3]. We sometimes call (2.17) the discrete (or integral) Minkowski sum of \(S_{1}{} { and}S_{2}{} \) to emphasize discreteness. \(\blacksquare \)
Remark 2.6
The Minkowski sum plays a central role in discrete convex analysis. The Minkowski sum of two (or more) M\(^{\natural }{} \)-convex sets is M\(^{\natural }{} \)-convex. The Minkowski sum of two L\(^{\natural }{} \)-convex sets is not necessarily L\(^{\natural }{} \)-convex, but it is integrally convex. The Minkowski sum of three L\(^{\natural }\)-convex sets is no longer integrally convex. For example ([47, Example 4.12]), \(S_{1} = \{(0, 0, 0), (1, 1, 0)\},\,S_{2} = \{(0, 0, 0), (0, 1, 1)\},\,{ and}S_{3} = \{(0, 0, 0), (1, 0, 1)\}{} \) are L\(^{\natural }\)-convex sets, and their Minkowski sum \(S = S_{1} + S_{2} + S_{3}{} \) is given as
which is not integrally convex, since \((1,1,1) \in \overline{S}{} { and}(1,1,1) \not \in S\). \(\blacksquare \)
The following theorem is a discrete analogue of a well-known decomposition of a polyhedron into a bounded part and a conic part (recession cone or characteristic cone) [56, Theorem 8.5]. An integrally convex set is called conic if its convex hull is a cone.
Theorem 2.2
([52]) Every integrally convex set \(S{ canberepresentedasa}({ discrete}){ Minkowskisumofaboundedintegrallyconvexset}Q{ andaconicintegrallyconvexset}C,\,{ thatis},\,S = Q + C\).
3 Integrally convex functions
3.1 Convex extension
For a function \(g: {\mathbb {R}}^{n} \rightarrow {\mathbb {R}}\cup \{ -\infty , +\infty \}{} { ingeneral},\,\mathrm{dom\,}g:= \{ x \in {\mathbb {R}}^{n} \mid -\infty< g(x) < +\infty \}{} \) is called the effective domain of \(g.{ Inthissectionwealwaysassumethat}f: {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} { and}\mathrm{dom\,}f \ne \emptyset ,\,{ thatis},\,f{ isafunctiondefinedon}{\mathbb {Z}}^{n}{} { takingvaluesin}{\mathbb {R}}\cup \{ +\infty \}{} { and}\mathrm{dom\,}f = \{ x \in {\mathbb {Z}}^{n} \mid f(x) < +\infty \}{} { isnonempty}.{ Wesaythat}f\) is convex-extensible if there exists a convex function \(g: {\mathbb {R}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} { satisfying}g(x) = f(x){ forall}x \in {\mathbb {Z}}^{n}.{ When}n=1,\,f: {\mathbb {Z}}\rightarrow {\mathbb {R}}\cup \{ +\infty \}{} \) is convex-extensible if and only if \(\mathrm{dom\,}f{ isanintervalofintegersand}f(k-1) + f(k+1) \ge 2 f(k){ forall}k \in {\mathbb {Z}}.{ Inthiscase},\,{ aconvexextensionof}f\) is given by the piecewise-linear function whose graph consists of line segments connecting \((k,f(k)){ and}(k+1,f(k+1)){ forall}k \in {\mathbb {Z}}.{ Wesaythatafunction}g: {\mathbb {R}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} \) minorizes \(f{ if}g(x) \le f(x){ forall}x \in {\mathbb {Z}}^{n}.{ Inthissectionwealwaysassumethat}f{ isminorizedbysomeaffinefunction}g(x) = \langle p, x \rangle + \alpha ,\,{ where}p \in {\mathbb {R}}^{n},\,\alpha \in {\mathbb {R}},\,{ and}\langle p, x \rangle := \sum _{i=1}^{n} p_{i} x_{i}{} { denotestheinnerproduct}({ ordualitypairing},\,{ tobemoreprecise}){ of}p=(p_{1}, p_{2}, \ldots , p_{n}){ and}x=(x_{1}, x_{2}, \ldots , x_{n})\). Note that every convex-extensible function is minorized by an affine function.
The convexification of \(f,\,{ tobedenotedby}\check{f}{} \), is defined as
where \(\Lambda { denotesthesetofcoefficientsforconvexcombinationsindexedby}y\in {\mathbb {Z}}^{n}{} \):
It is known [24, Section B.2.5] that \(\check{f}{} { isaconvexfunctionandthat}\check{f}{} { coincideswiththepointwisesupremumofallconvexfunctionsminorizing}f\), that is,
Therefore, \(f\) is convex-extensible if and only if \(\check{f}(x) = f(x){ forall}x \in {\mathbb {Z}}^{n}{} \).
The convex envelope of \(f,\,{ tobedenotedby}\overline{f},\,{ isdefinedasthepointwisesupremumofallaffinefunctionsminorizing}f\), that is,
This function \(\overline{f}{} { isaclosedconvexfunctionandcoincideswiththepointwisesupremumofallclosedconvexfunctionsminorizing}f\), that is,
In this paper we often refer to the condition
as the convex-extensibility of \(f,\,{ althoughthisconditionisslightlystrongerthanthecondition}f = \check{f}\,|_{{\mathbb {Z}}^{n}}{} { mentionedabove}.{ Accordingly},\,{ weoftenreferto}\overline{f}{} \) as the convex extension of \(f\) if (3.3) is the case.
Example 3.1
The quadratic function \(f(x)=x^{2}{} { definedfor}x \in {\mathbb {Z}}\) is convex-extensible, where \(g(x)=x^{2} (x \in {\mathbb {R}}){ isanobviousconvexextensionof}f.{ Theconvexenvelope}\overline{f}{} \) in (3.2) is a piecewise-linear function given by
It is noted that \(\overline{f}(x) = x^{2}{} { forintegers}x{ and}\overline{f}(x) > x^{2}{} \) for non-integral \(x;\,{ forexample},\,\overline{f}(1/2) = 1/2 > 1/4.{ Theconvexification}\check{f}{} \) in (3.1) coincides with \(\overline{f}{} \). \(\blacksquare \)
Remark 3.1
In a standard textbook [24, Section B.2.5], (3.1) is called the convex hull of \(f{ anddenotedby}\textrm{co} f\), whereas (3.2) is called the closed convex hull of \(f{ anddenotedby}\overline{\textrm{co}}\, f{ or}\textrm{cl}(\textrm{co} f).{ Usingournotationwehave}\textrm{co} f = \check{f}{} { and}\overline{\textrm{co}}\, f = \overline{f}{} \). \(\blacksquare \)
Remark 3.2
We have \(\overline{f}(x) \le \check{f}(x){ forall}x \in {\mathbb {R}}^{n}{} \), and the equality may fail in general (Example 3.2 below). However, when \(\mathrm{dom\,}f{ isbounded},\,{ wehave}\overline{f}(x) = \check{f}(x){ forall}x \in {\mathbb {R}}^{n}{} \). The proofs are as follows. In (3.2) we have \(\langle p, y \rangle + \alpha \le f(y){ foreach}y.{ Byusing}\lambda \in \Lambda { satisfying}\sum _{y} \lambda _{y} y = x\), we obtain
from which \(\overline{f}(x) = \sup _{(p, \alpha )} \{ \langle p, x \rangle + \alpha \} \le \inf _{\lambda }\{ \sum _{y} \lambda _{y} f(y) \} = \check{f}(x).{ When}\mathrm{dom\,}f{ isbounded},\,\mathrm{dom\,}f{ isafiniteset}.{ Foreach}x \in {\mathbb {R}}^{n}{} \), consider a pair of (mutually dual) linear programs:
where \((p, \alpha ) \in {\mathbb {R}}^{n} \times {\mathbb {R}}{ and}(\lambda _{y} \mid y \in \mathrm{dom\,}f){ arethevariablesof}({ P}){ and}({ D}),\,{ respectively}.{ Theoptimalvaluesof}({ P}){ and}({ D}){ areequalto}\overline{f}(x){ and}\check{f}(x),\,{ respectively}.{ Problem}({ P}){ isfeasible}({ e}.{ g}.,\,{ take}p=0{ andasufficientlysmall}\alpha ).{ ByLPduality},\,({ P}){ and}({ D}){ havethesame}({ finiteorinfinite}){ optimalvalues},\,{ thatis},\,\overline{f}(x) = \check{f}(x).{ Notethat}({ D}){ isfeasibleifandonlyif}x \in \overline{\mathrm{dom\,}f}{} \), in which case the optimal values are finite. \(\blacksquare \)
Example 3.2
Let \(f: {\mathbb {Z}}^{2} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} { betheindicatorfunction}\delta _{S}{} { oftheset}S\) considered in Remark 2.1. For \(x= (x_{1},1){ with}x_{1} \ne 0,\,{ wehave}x \in \textrm{cl}(\overline{S}) \setminus \overline{S}.{ Hence}0= \overline{f}(x)< \check{f}(x) = +\infty \). \(\blacksquare \)
Remark 3.3
For a set \(S \subseteq {\mathbb {Z}}^{n},\,{ theconvexificationoftheindicatorfunction}\delta _{S}{} { coincideswiththeindicatorfunctionofitsconvexhull}\overline{S},\,{ thatis},\,\check{\delta }_{S} = \delta _{\overline{S}}.{ Aset}S \subseteq {\mathbb {Z}}^{n}{} \) is hole-free if and only if the indicator function \(\delta _{S}{} \) is convex-extensible. \(\blacksquare \)
3.2 Definition of integrally convex functions
Recall the notation \(N(x){ fortheintegralneighborhoodof}x \in {\mathbb {R}}^{n}{} \) (cf., (2.4), Fig. 2). For a function \(f: \mathbb {Z}^{n} \rightarrow \mathbb {R} \cup \{ +\infty \}{} \), the local convex extension \(\tilde{f}: {\mathbb {R}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} { of}f{ isdefinedastheunionofallconvexextensions}({ convexifications}){ of}f{ on}N(x)\). That is,
where \(\Lambda (x){ denotesthesetofcoefficientsforconvexcombinationsindexedby}N(x)\):
It follows from this definition that, for each \(x \in {\mathbb {R}}^{n},\,{ thefunction}{\tilde{f}}{} { restrictedto}\overline{N(x)}{} { isaconvexfunction}.{ Ingeneral},\,{ wehave}{\tilde{f}} (x) \ge \check{f}(x) \ge \overline{f}(x){ forall}x \in {\mathbb {R}}^{n},\,{ where}\check{f}{} { and}\overline{f}{} \) are defined by (3.1) and (3.2), respectively.
We say that a function \(f\) is integrally convex if its local convex extension \({\tilde{f}}{} { is}({ globally}){ convexontheentirespace}{\mathbb {R}}^{n}.{ Inthiscase},\,{\tilde{f}}{} { isaconvexfunctionsatisfying}{\tilde{f}} (x) = f(x){ forall}x \in {\mathbb {Z}}^{n},\,{ whichmeansthat}f\) is convex-extensible. Moreover, \({\tilde{f}}{} { coincideswith}\check{f}{} { and}\overline{f}{} \), that is,
In particular, we have \(\mathrm{dom\,}{\tilde{f}} = \overline{\mathrm{dom\,}f}.{ Since}{\tilde{f}} (x) = f(x){ for}x \in {\mathbb {Z}}^{n}{} \), (3.6) implies
Proposition 3.1
-
(1)
The effective domain of an integrally convex function is integrally convex.
-
(2)
A set \(S \subseteq {\mathbb {Z}}^{n}{} { isintegrallyconvexifandonlyifitsindicatorfunction}\delta _{S}{} \) is integrally convex.
The following is an example of a convex-extensible function that is not integrally convex.
Example 3.3
Let \(f: {\mathbb {Z}}^{2} \rightarrow {\mathbb {R}}{ bedefinedby}f(x_{1}, x_{2})= | 2 x_{1} - x_{2} |{ forall}(x_{1}, x_{2}) \in {\mathbb {Z}}^{2}{} \). Obviously, this function is convex-extensible and the convex envelope is given by \(\overline{f}(x_{1}, x_{2})= | 2 x_{1} - x_{2} |{ forall}(x_{1}, x_{2}) \in {\mathbb {R}}^{2}.{ For}y=(1/2,1){ wehave}N(y)=\{ (0,1), (1,1) \}{} { andthelocalconvexextension}{\tilde{f}}{} { of}f{ around}y\) is given by
On the other hand, \(y = (1/2,1){ isthemidpointof}u=(0,0){ and}v=(1,2){ with}{\tilde{f}}(u)= f(0,0) = 0{ and}{\tilde{f}}(v)= f(1,2) = 0.{ Thisshowsthatthefunction}{\tilde{f}}{} { isnotconvex},\,{ and}f{ isnotintegrallyconvex}.{ Alsonotethat}0=\overline{f}(1/2,1) \not = {\tilde{f}}(1/2,1)=1\).
\(\blacksquare \)
Integrally convex functions in two variables (\(n = 2){ canbedefinedbysimpleinequalityconditionswithoutreferringtothelocalconvexextension}\tilde{f}{} \) (see Theorem 3.3 in Sect. 3.4).
Remark 3.4
The concept of integrally convex functions is introduced in [13] for functions defined on a box of integers. The extension to functions with general integrally convex effective domains is straightforward, which is found in [40]. \(\blacksquare \)
3.3 Examples
Three classes of integrally convex functions are given below.
Example 3.4
A function \(\Phi : {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} { in}x=(x_{1}, x_{2}, \ldots ,x_{n}) \in {\mathbb {Z}}^{n}{} \) is called separable convex if it can be represented as
with univariate discrete convex functions \(\varphi _{i}: {\mathbb {Z}}\rightarrow {\mathbb {R}}\cup \{ +\infty \},\,{ whichmeans},\,{ bydefinition},\,{ that}\mathrm{dom\,}\varphi _{i}{} \) is an interval of integers and
A separable convex function is integrally convex (actually, both L\(^{\natural }\)- and M\(^{\natural }\)-convex). \(\blacksquare \)
Example 3.5
A symmetric matrix \(Q =(q_{ij})\) that satisfies the condition
is called a diagonally dominant matrix (with nonnegative diagonals). If \(Q\) is diagonally dominant in the sense of (3.10), then \(f(x) = x^{\top } Q x\) is integrally convex [13, Proposition 4.5]. The converse is also true if \(n \le 2\) [13, Remark 4.3]. Recently it has been shown in [64, Theorem 9] that the diagonally dominance (3.10) of \(Q{ isequivalenttothedirecteddiscretemidpointconvexityof}f(x) = x^{\top } Q x\); see [64] for details. \(\blacksquare \)
Example 3.6
A function \(f: {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} \) is called 2-separable convex if it can be expressed as the sum of univariate convex, diff-convex, and sum-convex functions, i.e., if
where \(\varphi _{i}, \varphi _{ij}, \psi _{ij}: \mathbb {Z} \rightarrow \mathbb {R} \cup \{ +\infty \} (i, j =1,2,\ldots ,n; \ i \not = j)\) are univariate convex functions. A 2-separable convex function is known to be integrally convex [64, Theorem 4], whereas it is L\(^{\natural }\)-convex if \(\psi _{ij} \equiv 0{ forall}(i,j){ with}i \ne j.{ Aquadraticfunction}f(x) = x^{\top } Q x{ with}Q\) satisfying (3.10) is an example of a 2-separable convex function. \(\blacksquare \)
In addition to the above, almost all kinds of discrete convex functions treated in discrete convex analysis are integrally convex. It is known that separable convex, L-convex, L\(^{\natural }\)-convex, M-convex, M\(^{\natural }\)-convex, L\(^{\natural }_{2}\)-convex, and M\(^{\natural }_{2}\)-convex functions are integrally convex [40]. Multimodular functions [23] are also integrally convex [42]. Moreover, BS-convex and UJ-convex functions [18] are integrally convex.
3.4 Characterizations
In this section we give two characterizations of integrally convex functions in terms of an inequality of the form
where \(\tilde{f}{} { denotesthelocalconvexextensionof}f\) defined by (3.4). By the definition of \(\tilde{f}{} \), the inequality (3.11) above is true for \((x,y){ with}\Vert x - y \Vert _{\infty } \le 1{ foranyfunction}f: {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}.{ If}f\) is integrally convex, the inequality (3.11) holds for any \((x,y)\), as follows.
Proposition 3.2
If \(f\) is integrally convex, then (3.11) holds for every \(x, y \in \mathrm{dom\,}f\).
Proof
The function \(\tilde{f}{} { isconvexbyintegralconvexityof}f\), and hence
where the equalities \(\tilde{f}(x)=f(x){ and}\tilde{f}(y)=f(y)\) by (3.7) are used. \(\square \)
Integral convexity of a function can be characterized by a local condition under the assumption that the effective domain is an integrally convex set.
Theorem 3.1
([13, 37]) Let \(f: \mathbb {Z}^{n} \rightarrow \mathbb {R} \cup \{ +\infty \}{} \) be a function with an integrally convex effective domain. Then the following properties are equivalent.
-
(a)
\(f\) is integrally convex.
-
(b)
Inequality (3.11) holds for every \(x, y \in \mathrm{dom\,}f{ with}\Vert x - y \Vert _{\infty } =2\).
Proof
[(a) \(\Rightarrow \) (b)]: This is shown in Proposition 3.2.
[(b) \(\Rightarrow \) (a)]: (The proof given in [37, Appendix A] is sketched here.) For an integer vector \({a} \in {\mathbb {Z}}^{n},\,{ defineabox}B \subseteq {\mathbb {R}}^{n}{} \) of size two by
It can be shown ( [37, Lemma A.1]) that, if \(\mathrm{dom\,}f\) is integrally convex and the condition (b) is satisfied, then \(\tilde{f}{} { isconvexon}B \cap \overline{\mathrm{dom\,}f}.{ Fixarbitrary}x, y \in \overline{\mathrm{dom\,}f},\,{ anddenoteby}L{ the}({ closed}){ linesegmentconnecting}x{ and}y.{ Weshowthat}\tilde{f}{} { isconvexon}L.{ Considertheboxes}B\) of the form of (3.12) that intersect \(L.{ Thereexistsafinitenumberofsuchboxes},\,{ say},\,B_{1},B_{2}, \ldots , B_{m},\,{ and}L{ iscoveredbythelinesegments}L_{j} = L \cap B_{j} (j=1,2,\ldots , m).{ Thatis},\, L = \bigcup _{j=1}^{m} L_{j}.{ Foreachpoint}z \in L {\setminus } \{ x, y \},\,{ thereexistssome}L_{j}{} { thatcontains}z{ initsinterior},\,{ and}\tilde{f}{} { isconvexon}L_{j}{} \) by the above-mentioned fact. Hence \(\tilde{f}{} { isconvexon}L\) (cf. [66, Lemma 2]). This implies the convexity of \(\tilde{f},\,{ thatis},\,{ theintegralconvexityof}f. \square \)
The second characterization of integral convexity of a function is free from the assumption on the effective domain, but is not a local condition as it refers to all pairs \((x, y){ with}\Vert x - y \Vert _{\infty } \ge 2\).
Theorem 3.2
([38, Theorem A.1]) Let \(f: \mathbb {Z}^{n} \rightarrow \mathbb {R} \cup \{ +\infty \}{} { beafunctionwith}\mathrm{dom\,}f \ne \emptyset \). Then the following properties are equivalent.
-
(a)
\(f\) is integrally convex.
-
(b)
Inequality (3.11) holds for every \(x, y \in \mathrm{dom\,}f{ with}\Vert x - y \Vert _{\infty } \ge 2\).
Proof
[(a) \(\Rightarrow \) (b)]: This is shown in Proposition 3.2.
[(b) \(\Rightarrow \) (a)]: By Theorem 3.1, it suffices to show that \(\mathrm{dom\,}f\) is an integrally convex set, which follows from Theorem 2.1 applied to \(S=\mathrm{dom\,}f\). Note that the condition (2.11) in Theorem 2.1 holds by the assumption (b). \(\square \)
Remark 3.5
Theorem 3.1 originates in [13, Proposition 3.3], which shows the equivalence of (a) and (b) when the effective domain is a box of integers, while their equivalence for a general integrally convex effective domain is proved in [37, Appendix A]. Theorem 3.2 is given in [38, Theorem A.1] with a direct proof without using Theorem 3.1, while here we have given an alternative proof that relies on Theorem 3.1 via Theorem 2.1. \(\blacksquare \)
Integrally convex functions in two variables (\(n = 2){ canbecharacterizedbysimpleinequalityconditionsasfollows}.{ Weusenotation}f_{z}(x):=f(z+x)\).
Theorem 3.3
A function \(f: \mathbb {Z}^{2} \rightarrow \mathbb {R} \cup \{ +\infty \}{} \) is integrally convex if and only if its effective domain is an integrally convex set and the following five inequalities
are satisfied by both \(g(x_{1},x_{2})=f_{z}(x_{1},x_{2}){ and}g(x_{1},x_{2})=f_{z}(x_{2},x_{1}){ forany}z \in \mathrm{dom\,}f\).
Proof
This follows immediately from Theorem 3.1, since when \(n=2{ and}\Vert x - y \Vert _{\infty }= 2\), (3.11) is equivalent to
For example, if \(x=z{ and}y = z +(2,1),\,{ then}{} { and}\left\lfloor (x+y)/2 \right\rfloor = z+ (1,0)\), and (3.13) gives the first inequality for \(g(x_{1},x_{2})=f_{z}(x_{1},x_{2}). \square \)
Remark 3.6
For a function \(g: \mathbb {Z}^{2} \rightarrow \mathbb {R} \cup \{ +\infty \}{} \) (in general), an inequality of the form
is called the (basic) parallelogram inequality in [37]. It is shown in [37, Proposition 3.3] that for any integrally convex function \(f{ intwovariablesandapoint}z \in \mathrm{dom\,}f,\,{ thefunction}g(x)=f_{z}(x)\) satisfies the inequality (3.14). Note that (3.14) with \((a,b)=(1,1)\) coincides with the first inequality in Theorem 3.3. Furthermore, the inequality (3.14) holds also for \(g(x_{1},x_{2})=f_{z}(x_{2},x_{1}), f_{z}(x_{1},-x_{2}),\,{ and}f_{z}(-x_{2},x_{1})\), as integral convexity is preserved under such coordinate inversions (cf., (3.17), (3.18)). \(\blacksquare \)
In this section we have given three theorems (Theorems 3.1, 3.2, and 3.3) to characterize integrally convex functions. In Sect. 4.2 we give two additional theorems (Theorems 4.3 and 4.4). Their logical dependence (in our presentation) is illustrated in Fig. 6.
Characterizations of integrally convex functions
3.5 Simplicial divisions
As is well known ( [17, Section 16.3], [40, Section 7.7]), the convex extension of an L\(^{\natural }{} \)-convex function can be constructed in a systematic manner using a regular simplicial division (the Freudenthal simplicial division) of unit hypercubes. This is a generalization of the Lovász extension for a submodular set function. In addition, the concepts of BS-convex and UJ-convex functions are introduced on the basis of other regular simplicial divisions in [18].
By definition, an integrally convex function \(f\) is convex-extensible, and its convex envelope \(\overline{f}{} { canbeconstructedlocallywithineachunithypercube},\,{ since}\overline{f}{} { coincideswiththelocalconvexextension}\tilde{f}.{ However},\,{ generalintegrallyconvexfunctionsarenotassociatedwitharegularsimplicialdivision}.{ Indeed},\,{ thefollowingconstructionshowsthat},\,{ when}n=2,\,{ anarbitrarytriangulationcanarisefromanintegrallyconvexfunction}.{ Considertherectangulardomain}[\textbf{0}, a]_{{\mathbb {R}}},\,{ where}a = (a_{1}, a_{2}){ withpositiveintegers}a_{1}{} { and}a_{2},\,{ andassumethatwearegivenanarbitrarytriangulationofeachunitsquareinthedomain}[\textbf{0}, a]_{{\mathbb {R}}}{} \) such as the one in Fig. 7(a). We can construct an integrally convex function \(f{ suchthattheconvexenvelope}\overline{f}{} \) corresponds to the given triangulation.
A given triangulation and the corresponding function \(g\)
According to the given triangulation, we classify the unit squares into two types, type M and type L. We say that a unit square is of type M (resp., type L) if it has a diagonal line segment on \(x_{1} + x_{2} = c ({ resp}.,\,x_{1} - x_{2} = c){ forsome}c\); see Fig. 7(b). For each \(x = (x_{1}, x_{2}) \in [\textbf{0}, a]_{{\mathbb {Z}}},\,{ wedenotethenumberofunitsquaresoftypeM}({ resp}.,\,{ typeL}){ containedinthedomain}[\textbf{0}, x]_{{\mathbb {R}}}{} { by}g(x; \textrm{M}) ({ resp}.,\,g(x; \textrm{L})),\,{ anddefine}g(x):= g(x; \textrm{M}) - g(x; \textrm{L}).{ For}x=(2,2)\) in Fig. 7(b), for example, we have \(g(x; \textrm{M})=3{ and}g(x; \textrm{L}) = 1,\,{ andtherefore}g(2,2) = g(x; \textrm{M}) - g(x; \textrm{L}) = 3-1 = 2.{ Finally},\,{ wedefineafunction}f{ on}[\textbf{0}, a]_{{\mathbb {Z}}}{} \) by
with a positive constant \(A.{ If}A \ge a_{1} + a_{2},\,{ thisfunction}f\) is integrally convex and the associated triangulation of each unit square coincides with the given one (proved in Remark 3.7). It is noted that, while \(f{ isintegrallyconvex},\,g\) itself may not be integrally convex. For example, in Fig. 7(b), we have
(cf., Theorem 3.3).
Remark 3.7
First, we prove the integral convexity of \(f\) in (3.15) by showing that
holds for every \(x, y \in [\textbf{0}, a]_{{\mathbb {Z}}}{} { with}\Vert x - y \Vert _{\infty } =2.{ Bysymmetrybetween}x{ and}y\) and that between coordinate axes, we have five cases to consider (cf., Theorem 3.3): (i) \(y=x+(2,1),\,({ ii}) y=x+(2,-1),\,({ iii}) y=x+(2,0),\,({ iv}) y=x+(2,2),\,{ and}({ v}) y=x+(2,-2).{ Let}h(x):= {x_{1}}^{2} + {x_{2}}^{2}{} \), for which we have
On the other hand, we have
in either case. Therefore, if \(A \ge a_{1} + a_{2}{} \), the inequality in (3.16) holds.
Next, we observe that the function \(f = A h + g{ inducesatriangulationofthespecifiedtypewithineachunitsquare}.{ Considerasquare}[x, y]_{{\mathbb {R}}}{} { with}y=x+\textbf{1}{ and}x, y \in [\textbf{0}, a]_{{\mathbb {Z}}}.{ Let}u:= (x_{1}+1,x_{2}){ and} v:= (x_{1},x_{2}+1).{ Then}h(x) + h(y) - h(u) - h(v) = 0,\,{ while}g(x) + g(y) - g(u) - g(v){ isequalto}+1{ and}-1{ accordingtowhetherthesquare}[x, y]_{{\mathbb {R}}}{} { isoftypeMorL}.{ Thisshowsthatthetriangulationof}[x, y]_{{\mathbb {R}}}{} { inducedby}f\) coincides with the given one. \(\blacksquare \)
Next we give an example of a simplicial division associated with an integrally convex function in three variables.
Simplicial division associated with an integrally convex function
Example 3.7
Consider \(S = [ (0,0,0), (2,1,1) ]_{{\mathbb {Z}}} =\{ x \in {\mathbb {Z}}^{3} \mid 0 \le x_{1} \le 2, \ 0 \le x_{i} \le 1 \ (i=2,3) \}{} \) (see Fig. 8), and define \(f: {\mathbb {Z}}^{3} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} { with}\mathrm{dom\,}f = S\) by
and \(f(x_{1},0,0) = f(x_{1},1,1),\,f(x_{1},1,0) = f(x_{1},0,1){ for}x_{1}=0,1,2\). By Theorem 3.1 we can verify that \(f{ isintegrallyconvex}.{ Forexample},\,{ for}x= (0,0,0){ and}y= (2,1,1)\), we have
in (3.11). The simplicial division of \(\overline{S} = [ (0,0,0), (2,1,1) ]_{{\mathbb {R}}}{} { fortheconvexextensionof}f{ issymmetricwithrespecttotheplane}x_{1}=1.{ Theleftcube}[ (0,0,0), (1,1,1) ]_{{\mathbb {R}}}{} { isdecomposedintofivesimplices};\,{ oneofthemhasverticesat}(0,0,0),\,(1,1,0),\,(1,0,1),\,(0,1,1){ andhasvolume}1/3 ({ drawninboldline}),\,{ whereastheotherfoursimpliciesarecongruenttothestandardsimplex},\,{ havingvolume}1/6.{ Therightcube}[ (1,0,0), (2,1,1) ]_{{\mathbb {R}}}{} { isdecomposedsimilarlyintofivesimplices};\,{ oneofthemhasverticesat}(2,0,0),\,(1,1,0),\,(1,0,1),\,(2,1,1),\,{ andhasvolume}1/3,\,{ whereastheotherfoursimpliciesarecongruenttothestandardsimplex},\,{ havingvolume}1/6.{ Thusthesimplicesarenotuniforminvolume},\,{ whereastheyhavethesamevolume}(=1/6\)) for an L\(^{\natural }\)-convex function. It is added that this function \(f\) is neither L\(^{\natural }_{2}\)-convex nor M\(^{\natural }_{2}\)-convex, since \(\arg \min f = \{ (0,0,0), \ (1,1,0), \ (1,0,1), \ (0,1,1) \}{} \) is neither L\(^{\natural }_{2}\)-convex nor M\(^{\natural }_{2}\)-convex (as discussed in Example 2.6). \(\blacksquare \)
3.6 Basic operations
In this section we show how integral convexity of a function behaves under basic operations. Let \(f{ beafunctionon}{\mathbb {Z}}^{n},\,{ i}.{ e}.,\,f: {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} \).
Origin shift:
For an integer vector \(b \in {\mathbb {Z}}^{n}{} \), the origin shift of \(f{ by}b{ meansafunction}g{ on}{\mathbb {Z}}^{n}{} { definedby}g(y) = f(y-b)\). The origin shift of an integrally convex function is an integrally convex function.
Inversion of coordinates:
The independent coordinate inversion of \(f{ meansafunction}g{ on}{\mathbb {Z}}^{n}{} \) defined by
with an arbitrary choice of \(\tau _{i} \in \{ +1, -1 \} (i=1,2,\ldots ,n)\). The independent coordinate inversion of an integrally convex function is an integrally convex function. This is a nice property of integral convexity, not shared by L\(^{\natural }\)-, L\(^{\natural }_{2}\)-, M\(^{\natural }\)-, or M\(^{\natural }_{2}\)-convexity.
Permutation of coordinates:
For a permutation \(\sigma { of}(1,2,\ldots ,n)\), the permutation of \(f{ by}\sigma { meansafunction}g{ on}{\mathbb {Z}}^{n}{} \) defined by
The permutation of an integrally convex function is integrally convex.
Variable-scaling:
For a positive integer \(\alpha \), the variable-scaling (or scaling for short) of \(f{ by}\alpha { meansafunction}g{ on}{\mathbb {Z}}^{n}{} \) defined by
Note that the same scaling factor \(\alpha { isusedforallcoordinates}.{ If}\alpha = 2\), for example, this operation amounts to considering the function values at even points. The scaling operation is used effectively in minimization algorithms (see Sect. 4). The scaling of an integrally convex function is not necessarily integrally convex. The indicator function \(f=\delta _{S}{} { oftheintegrallyconvexset}S \subseteq {\mathbb {Z}}^{3}{} \) in Example 2.8 is such an example. Another example of a function on the integer box \([(0,0,0), (4,2,2)]_{{\mathbb {Z}}}{} \) can be found in [37, Example 3.1]. In the case of \(n = 2,\,{ integralconvexityadmitsthescalingoperation}.{ Thatis},\,{ if}f: {\mathbb {Z}}^{2} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} { isintegrallyconvex},\,{ then}g: {\mathbb {Z}}^{2} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} \) is integrally convex ( [37, Theorem 3.2]).
As another kind of scaling, the dilation operation can be defined for a function \(f: {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} \) if it is convex-extensible. For any positive integer \(\alpha ,\,{ the}\alpha \) -dilation of \(f{ isdefinedasafunction}g{ on}{\mathbb {Z}}^{n}{} \) given by
where \(\overline{f}{} { denotestheconvexenvelopeof}f.{ Thedilationofanintegrallyconvexfunctionisnotnecessarilyintegrallyconvex}.{ Forexample},\,{ theindicatorfunction}f=\delta _{S}{} { oftheintegrallyconvexset}S \subseteq {\mathbb {Z}}^{4}{} \) in Example 2.5 is an integrally convex function for which the \(2\)-dilation is not integrally convex.
Remark 3.8
Although integral convexity is not compatible with the dilation operation, other kinds of discrete convexity such as L-, L\(^{\natural }\)-, L\(_{2}\)-, L\(^{\natural }_{2}\)-, M-, M\(^{\natural }\)-, M\(_{2}\)-, M\(^{\natural }_{2}\)-convexity, and multimodularity are preserved under dilation. In contrast, the scaling operation in (3.19) preserves L-convexity and its relatives (box, L-, L\(^{\natural }\)-, L\(_{2}\)-, L\(^{\natural }_{2}\)-convexity, and multimodularity), and not M-convexity and its relatives (M-, M\(^{\natural }\)-, M\(_{2}\)-, M\(^{\natural }_{2}\)-convexity).
\(\blacksquare \)
Value-scaling:
For a function \(f: {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} { andanonnegativefactor}a \ge 0\), the value-scaling of \(f{ by}a{ meansafunction}g: {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} { definedby}g(y) = a f(y){ for}y \in {\mathbb {Z}}^{n}.{ Wemayalsointroduceanadditiveconstant}b \in {\mathbb {R}}{ andalinearfunction}\langle c, y \rangle = \sum _{i=1}^{n} c_{i} y_{i},\,{ where}c \in {\mathbb {R}}^{n}{} \), to obtain
The operation (3.21) preserves integral convexity of a function.
Restriction:
For a function \(f: {\mathbb {Z}}^{N} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} { andasubset}U{ oftheindexset}N = \{ 1,2,\ldots , n \}{} \), the restriction of \(f{ to}U{ isafunction}g: {\mathbb {Z}}^{U} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} \) defined by
where \(\textbf{0}_{N \setminus U}{} { denotesthezerovectorin}{\mathbb {Z}}^{N \setminus U}.{ Thenotation}(y,\textbf{0}_{N \setminus U}){ meansthevectorwhose}i{ thcomponentisequalto}y_{i}{} { for}i \in U{ andto}0{ for}i \in N \setminus U\). The restriction of an integrally convex function is integrally convex (if the effective domain of the resulting function is nonempty).
Projection:
For a function \(f: {\mathbb {Z}}^{N} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} { andasubset}U{ oftheindexset}N = \{ 1,2,\ldots , n \}{} \), the projection of \(f{ to}U{ isafunction}g: {\mathbb {Z}}^{U} \rightarrow {\mathbb {R}}\cup \{ -\infty , +\infty \}{} \) defined by
where the notation \((y,z){ meansthevectorwhose}i{ thcomponentisequalto}y_{i}{} { for}i \in U{ andto}z_{i}{} { for}i \in N {\setminus } U\). The projection is also called partial minimization. The resulting function \(g\) is referred to as the marginal function of \(f\) in [24]. The projection of an integrally convex function is integrally convex ( [33, Theorem 3.1]) if \(g > -\infty ,\,{ orelsewehave}g \equiv -\infty \) (see Proposition 3.4 in Sect. 3.7).
Splitting:
Suppose that we are given a family \(\{ U_{1}, U_{2}, \dots , U_{n} \}{} { ofdisjointnonemptysetsindexedby}N = \{ 1, 2, \dots , n\}.{ Let}m_{i}= |U_{i}|{ for}i=1,2,\ldots , n{ anddefine}m= \sum _{i=1}^{n} m_{i},\,{ where}m \ge n.{ Foreach}i \in N{ wedefinean}m_{i}{} \)-dimensional vector \(y_{[i]} = ( y_{j} \mid j \in U_{i} ){ andexpress}y \in {\mathbb {Z}}^{m}{} { as}y = (y_{[1]}, y_{[2]}, \dots , y_{[n]}).{ Forafunction}f: {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{+\infty \}{} \), the splitting of \(f{ by}\{ U_{1}, U_{2}, \dots , U_{n} \}{} { isdefinedasafunction}g: {\mathbb {Z}}^{m} \rightarrow {\mathbb {R}}\cup \{+\infty \}{} \) given by
where \(y_{[i]}(U_{i}) = \sum \{ y_{j} \mid j \in U_{i} \}.{ Forexample},\,g(y_{1}, y_{2}, y_{3}) = f(y_{1}, y_{2}+ y_{3}){ isasplittingof}f: {\mathbb {Z}}^{2} \rightarrow {\mathbb {R}}\cup \{+\infty \}{} { for}U_{1} = \{ 1 \}{} { and}U_{2} = \{ 2,3 \},\,{ where}n=2{ and}m=3\). The splitting of an integrally convex function is integrally convex ( [46, Proposition 4.4]).
Aggregation:
Let \({\mathcal {P}} = \{ N_{1}, N_{2}, \dots , N_{m} \}{} { beapartitionof}N = \{ 1,2, \ldots , n \}{} { intodisjointnonemptysubsets}\,:\,N = N_{1} \cup N_{2} \cup \dots \cup N_{m}{} { and}N_{i} \cap N_{j} = \emptyset { for}i \not = j.{ Wehave}m \le n.{ Forafunction}f: {\mathbb {Z}}^{N} \rightarrow {\mathbb {R}}\cup \{+\infty \}{} \), the aggregation of \(f{ withrespectto}{\mathcal {P}}{} { isdefinedasafunction}g: {\mathbb {Z}}^{m} \rightarrow {\mathbb {R}}\cup \{+\infty , -\infty \}{} \) given by
For example, \(g(y_{1}, y_{2}) = \inf \{ f(x_{1}, x_{2}, x_{3}) \mid x_{1} = y_{1}, \ x_{2} + x_{3} = y_{2} \}{} { isanaggregationof}f: {\mathbb {Z}}^{3} \rightarrow {\mathbb {R}}\cup \{+\infty \}{} { for}N_{1} = \{ 1 \}{} { and}N_{2} = \{ 2,3 \},\,{ where}n=3{ and}m=2\). The aggregation of an integrally convex function is not necessarily integrally convex (Example 2.9).
Direct sum:
The direct sum of two functions \(f_{1}: {\mathbb {Z}}^{n_{1}} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} { and}f_{2}: {\mathbb {Z}}^{n_{2}} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} { isafunction}f_{1} \oplus f_{2}: {\mathbb {Z}}^{n_{1}+n_{2}} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} \) defined as
The direct sum of two integrally convex functions is integrally convex.
Addition:
The sum of two functions \(f_{1}, f_{2}: {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} \) is defined by
For two sets \(S_{1},\,S_{2} \subseteq {\mathbb {Z}}^{n},\,{ thesumoftheirindicatorfunctions}\delta _{S_{1}}{} { and}\delta _{S_{2}}{} { coincideswiththeindicatorfunctionoftheirintersection}S_{1} \cap S_{2},\,{ thatis},\,\delta _{S_{1}} + \delta _{S_{2}} = \delta _{S_{1} \cap S_{2}}{} \). The sum of integrally convex functions is not necessarily integrally convex (Example 2.10). However, the sum of an integrally convex function with a separable convex function
is integrally convex.
Convolution:
The (infimal) convolution of two functions \(f_{1}, f_{2}: {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} \) is defined by
where it is assumed that the infimum is bounded from below (i.e., \((f_{1} \Box \,f_{2})(x) > -\infty { forevery}x \in {\mathbb {Z}}^{n}).{ Fortwosets}S_{1},\,S_{2} \subseteq {\mathbb {Z}}^{n},\,{ theconvolutionoftheirindicatorfunctions}\delta _{S_{1}}{} { and}\delta _{S_{2}}{} { coincideswiththeindicatorfunctionoftheirMinkowskisum}S_{1}+S_{2} = \{ y + z \mid y \in S_{1}, z \in S_{2} \},\,{ thatis},\,\delta _{S_{1}} \Box \,\delta _{S_{2}} = \delta _{S_{1}+S_{2}}{} \). The convolution of integrally convex functions is not necessarily integrally convex (Example 2.11). The convolution of an integrally convex function and a separable convex function is integrally convex [33, Theorem 4.2] (also [45, Proposition 4.17]).
Remark 3.9
The convolution operation plays a central role in discrete convex analysis. The convolution of two (or more) M\(^{\natural }{} \)-convex functions is M\(^{\natural }{} \)-convex. The convolution of two L\(^{\natural }{} \)-convex functions is not necessarily L\(^{\natural }{} \)-convex, but it is integrally convex. The convolution of three L\(^{\natural }\)-convex functions is no longer integrally convex (Remark 2.6). \(\blacksquare \)
3.7 Technical supplement
This section is a technical supplement concerning the projection operation defined in (3.22). We first consider a direction \(d{ inwhichthefunctionvaluedivergesto}-\infty \).
Proposition 3.3
Let \(f: {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} { beanintegrallyconvexfunction},\,y \in \mathrm{dom\,}f,\,{ and}d \in {\mathbb {Z}}^{n}.{ If}\displaystyle \lim _{k \rightarrow \infty } f(y + k d)= -\infty ,\,{ thenforany}z \in \mathrm{dom\,}f,\,{ wehave}\displaystyle \lim _{k \rightarrow \infty } f(z + k d)= -\infty \).
Proof
Let \(S = \mathrm{dom\,}f.{ Foreach}x \in S,\,g_{x}(k) = f(x + k d){ isaconvexfunctionin}k \in {\mathbb {Z}}\), which follows from the convex-extensibility of \(f.{ Let}T{ denotethesetof}x \in S{ forwhich}\displaystyle \lim _{k \rightarrow \infty } g_{x}(k) = -\infty .{ Wewanttoshowthat}T = \emptyset { or}T=S.{ Toprovethisbycontradiction},\,{ assumethatboth}T{ and}S {\setminus } T{ arenonempty}.{ Consider}y \in T{ and}z \in S {\setminus } T{ thatminimize}C \Vert y-z \Vert _{\infty } + \Vert y-z \Vert _{1},\,{ where}C \gg 1.{ Fromintegralconvexityof}S,\,{ wecaneasilyshowthat}\Vert y-z \Vert _{\infty } =1{ and}S \cap N \big (\frac{y + z}{2} \big ) = \{ y,z \}.{ Moreover},\,{ for}y^{(k)} = y + k d{ and}z^{(k)} = z + k d\), we have
(Proof of (3.26): Since \(g_{y}{} { isconvex},\,y \in S,\,{ and}\displaystyle \lim _{k \rightarrow \infty } g_{y}(k) = -\infty ,\,{ wehave}y^{(k)} = y + k d \in S{ forall}k\). This implies, by Proposition 2.6, that \(z^{(k)} = z + k d \in S{ forall}k.{ Wealsohave}C \Vert y^{(k)}-z^{(k)} \Vert _{\infty } + \Vert y^{(k)}-z^{(k)} \Vert _{1} = C \Vert y-z \Vert _{\infty } + \Vert y-z \Vert _{1}{} \).) Since \(f\) is integrally convex, we have
by Proposition 3.2, where the left-hand side can be expressed as
by \(y^{(k+1)} + z^{(k-1)} = y^{(k)} + z^{(k)} \) and (3.26). Therefore, we have
By adding these inequalities for \(k=1,2,\ldots , {{\hat{k}}}{} \), we obtain
By letting \({{\hat{k}}} \rightarrow \infty \) we obtain a contradiction, since the right-hand side tends to \(-\infty \) while the left-hand side does not. \(\square \)
Using the above proposition we can show that the projection \(g(y) = \inf \{ f(y,z) \mid z \in {\mathbb {Z}}^{N {\setminus } U} \}{} \), defined in (3.22), is away from the value of \(-\infty { unlessitisidenticallyequalto}-\infty \).
Proposition 3.4
Let \(f: {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} { beanintegrallyconvexfunction},\,{ and}g{ betheprojectionof}f{ to}U.{ If}g(y^{0}) = -\infty { forsome}y^{0} \in {\mathbb {Z}}^{U},\,{ then}g(y) = -\infty { forall}y \in {\mathbb {Z}}^{U}{} \).
Proof
First suppose \(|N {\setminus } U|=1{ with}N {\setminus } U = \{ v \},\,{ andassume}g(y^{0}) = -\infty { forsome}y^{0} \in {\mathbb {Z}}^{U}.{ Then}x^{0}:= (y^{0}, z^{0}) \in \mathrm{dom\,}f{ forsome}z^{0} \in {\mathbb {Z}},\,{ and}\displaystyle \lim _{k \rightarrow \infty } f(x^{0} + k d)= -\infty { for}d=\varvec{1}^{v}{} { or}d=-\varvec{1}^{v}{} \). This implies, by Proposition 3.3, that \(\displaystyle \lim _{k \rightarrow \infty } f(x + k d)= -\infty { forall}x \in \mathrm{dom\,}f,\,{ whichshows}g \equiv -\infty .{ Nextweconsiderthecasewhere}|N {\setminus } U| \ge 2.{ Let}N {\setminus } U =\{ v_{1}, v_{2}, \ldots , v_{r} \}{} { with}2 \le r < n,\,{ and}g^{(k)}{} { denotetheprojectionof}f{ to}U \cup \{ v_{k+1}, v_{k+2}, \ldots , v_{r} \}{} { for}k =0,1,\ldots , r.{ Then}g^{(0)}=f,\,g^{(r)}=g,\,{ and}g^{(k)}{} { istheprojectionof}g^{(k-1)}{} { for}k =1,2,\ldots , r.{ Bytheargumentforthecasewith}|N {\setminus } U|=1,\,{ wehave}g^{(1)} > -\infty { or}g^{(1)} \equiv -\infty .{ Inthelattercaseweobtain}g^{(k)} \equiv -\infty { for}k=1,2,\ldots ,r.{ Intheformercase},\,g^{(1)}{} \) is an integrally convex function by [33, Theorem 3.1], as already mentioned in Sect. 3.6. This allows us to apply the same argument to \(g^{(1)}{} { toobtainthat}g^{(2)}> -\infty { or}g^{(2)} \equiv -\infty .{ Continuingthiswaywearriveattheconclusionthat}g > -\infty { or}g \equiv -\infty . \square \)
4 Minimization and minimizers
4.1 Optimality conditions
The global minimum of an integrally convex function can be characterized by a local condition.
Theorem 4.1
( [13, Proposition 3.1]; see also [40, Theorem 3.21]) Let \(f: \mathbb {Z}^{n} \rightarrow \mathbb {R} \cup \{ +\infty \}{} { beanintegrallyconvexfunctionand}x^{*} \in \mathrm{dom\,}f.{ Then}x^{*}{} { isaminimizerof}f{ ifandonlyif}f(x^{*}) \le f(x^{*} + d){ forall}d \in \{ -1, 0, +1 \}^{n}{} \).
A more general form of this local optimality criterion is known as “box-barrier property” in Theorem 4.2 below (see Fig. 9). A special case of Theorem 4.2 with \({{\hat{x}}} = x^{*},\,p= x^{*} - \varvec{1},\,{ and}q= x^{*} + \varvec{1}{} \) coincides with Theorem 4.1 above.
Theorem 4.2
(Box-barrier property [37, Theorem 2.6]) Let \(f: \mathbb {Z}^{n} \rightarrow \mathbb {R} \cup \{ +\infty \}{} { beanintegrallyconvexfunction},\,{ and}p \in (\mathbb {Z} \cup \{ -\infty \})^{n}{} { and}q \in (\mathbb {Z} \cup \{ +\infty \})^{n},\,{ where}p \le q\). Define
and \(W = \bigcup _{i=1}^{n} (W_{i}^{+} \cup W_{i}^{-}).{ Let}{{\hat{x}}} \in S \cap \mathrm{dom\,}f.{ If}f({{\hat{x}}}) \le f(y){ forall}y \in W,\,{ then}f({{\hat{x}}}) \le f(z){ forall}z \in {\mathbb {Z}}^{n} {\setminus } S\).
Proof
(The proof of [37] is described here.) Let \(U_{i}^{+}{} { and}U_{i}^{-}{} { denotetheconvexhullsof}W_{i}^{+}{} { ans}W_{i}^{-},\,{ respectively},\,{ anddefine}U = \bigcup _{i=1}^{n} (U_{i}^{+} \cup U_{i}^{-}).{ Then}W = U \cap \mathbb {Z}^{n}.{ Forapoint}z \in {\mathbb {Z}}^{n} \setminus S,\,{ thelinesegmentconnecting}{{\hat{x}}}{} { and}z{ intersects}U{ atapoint},\,{ say},\,u \in {\mathbb {R}}^{n}.{ Thenitsintegralneighborhood}N(u){ iscontainedin}W.{ Sincethelocalconvexextension}\tilde{f}(u){ isaconvexcombinationofthe}f(y)\)’s with \(y \in N(u),\,{ and}f(y) \ge f({{\hat{x}}}){ forevery}y \in W,\,{ wehave}\tilde{f}(u) \ge f({{\hat{x}}}).{ Ontheotherhand},\,{ itfollowsfromtheconvexityof}\tilde{f}{} { that}\tilde{f}(u) \le (1 - \lambda ) f({{\hat{x}}}) + \lambda f(z){ forsome}\lambda { with}0 < \lambda \le 1.{ Hence}f({{\hat{x}}}) \le \tilde{f}(u) \le (1 - \lambda ) f({{\hat{x}}}) + \lambda f(z),\,{ andtherefore},\,f({{\hat{x}}}) \le f(z). \square \)
Box barrier property
Remark 4.1
The optimality criterion in Theorem 4.1 is certainly local, but not satisfactory from the computational complexity viewpoint. We need \(O(3^{n})\) function evaluations to verify the local optimality condition. \(\blacksquare \)
4.2 Minimizer sets
It is (almost) always the case that if a function is equipped with some kind of discrete convexity, then the set of its minimizers is equipped with the discrete convexity of the same kind. This is indeed the case with an integrally convex function.
Proposition 4.1
Let \(f: {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} \) be an integrally convex function. If it attains a (finite) minimum, the set of its minimizers is integrally convex.
Proof
Let \(\alpha { denotetheminimumvalueof}f,\,{ and}S:=\arg \min f.{ Takeany}x \in \overline{S}.{ Wehave}\alpha = \overline{f}(x) = \tilde{f}(x).{ Bythedefinitionof}\tilde{f}{} \) in (3.4), this implies \(x \in \overline{S \cap N(x)}{} \). Thus (2.8) holds, showing the integral convexity of \(S. \square \)
In the following we discuss how integral convexity of a function can be characterized in terms of the integral convexity of the minimizer sets. For a function \(f: {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} { andavector}p \in {\mathbb {R}}^{n},\,f[-p]\) will denote the function defined by
We use notation
for the set of the minimizers of \(f[-p]\).
Theorem 4.3
Let \(f: {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} { andassumethattheconvexenvelope}\overline{f}{} \) is a polyhedral convex function and
Then \(f{ isintegrallyconvexifandonlyif}\arg \min f[-p]{ isanintegrallyconvexsetforeach}p \in {\mathbb {R}}^{n}{} { forwhich}f[-p]\) attains a (finite) minimum.
Proof
The only-if-part is immediate from Proposition 4.1, since if \(f{ isintegrallyconvex},\,{ sois}f[-p]{ forany}p\). We prove the if-part by using Theorem 3.2. Take any \(x,y \in \mathrm{dom\,}f{ with}\Vert x - y \Vert _{\infty } \ge 2,\,{ andlet}u:=(x+y)/2\). Our goal is to show
in (3.11), where \(\tilde{f}{} { isthelocalconvexextensionof}f.{ Themidpoint}u{ belongstotheconvexhull}\overline{\mathrm{dom\,}f}{} { of}\mathrm{dom\,}f,\,{ whichimpliesthat}\overline{f}(u){ isfiniteand}u \in \arg \min \overline{f}[-p]{ forsome}p.{ Let}S_{p}:= \arg \min f[-p]{ forthis}p,\,{ where}S_{p} \subseteq {\mathbb {Z}}^{n}.{ Since}\arg \min \overline{f}[-p] = \overline{\arg \min f[-p]} = \overline{S_{p}},\,{ wehave}u \in \overline{S_{p}}.{ Bytheassumedintegralconvexityof}S_{p},\,{ thisimplies}u \in \overline{S_{p} \cap N(u)}{} \) (see (2.8)). Therefore, there exist \(z^{(1)}, z^{(2)}, \ldots , z^{(m)} \in S_{p} \cap N(u){ aswellaspositivenumbers}\lambda _{1}, \lambda _{2}, \ldots , \lambda _{m}{} { with}\sum _{i=1}^{m} \lambda _{i} = 1\) such that
Since each \(z^{(i)} (\in S_{p}){ isaminimizerof}f[-p]\), we have
that is,
For the linear parts in this expression we have
and therefore, (4.6) is equivalent to
Combining this with the expression of \({\tilde{f}}(u)\) in (4.5), we obtain (4.4). This completes the proof of Theorem 4.3. \(\square \)
The next theorem gives a similar characterization of integrally convex functions under a different assumption.
Theorem 4.4
([40, Theorem 3.29]) Let \(f: {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} { beafunctionwithaboundednonemptyeffectivedomain}.{ Then}f{ isintegrallyconvexifandonlyif}\arg \min f[-p]{ isanintegrallyconvexsetforeach}p \in {\mathbb {R}}^{n}{} \).
Proof
The only-if-part is immediate from Proposition 4.1, since if \(f{ isintegrallyconvex},\,{ sois}f[-p]{ forany}p\). To prove the if-part by Theorem 4.3, define \(S_{p}:= \arg \min f[-p],\,{ whichisintegrallyconvexbyassumption}.{ Notethat}S_{p}{} { isnonemptyforevery}p{ bytheassumedboundednessof}\mathrm{dom\,}f.{ Theboundednessof}\mathrm{dom\,}f{ alsoimpliesthattheconvexenvelope}\overline{f}{} { isapolyhedralconvexfunction}.{ Theintegralconvexityof}S_{p}{} { impliesthat}S_{p}{} \) is hole-free (\(S_{p} = \overline{S_{p}} \cap {\mathbb {Z}}^{n}),\,{ fromwhichweobtain}\overline{f}(x) = f(x) (x \in {\mathbb {Z}}^{n})\) in (4.3). Then the integral convexity of \(f\) follows from Theorem 4.3. \(\square \)
Remark 4.2
The characterization of integral convexity of \(f{ by}\arg \min f[-p]\) originates in [40, Theorem 3.29], which is stated as Theorem 4.4 above. In this paper, we have given an alternative proof to this theorem by first establishing Theorem 4.3 that employs the assumption of convex-extensibility. See also Fig. 6 in Sect. 3.4. \(\blacksquare \)
Remark 4.3
Theorems 4.3 and 4.4 impose the assumption of convex-extensibility of \(f{ orboundednessof}\mathrm{dom\,}f.{ Suchanassumptionseemsinevitable}.{ Considerafunction}f: {\mathbb {Z}}\rightarrow {\mathbb {R}}{ definedby} f(x)= \left\{ \begin{array}{ll} 0 &{} ( x = 0 ) , \\ 1 &{} ( x \not = 0 ) . \end{array} \right. { Then}\arg \min f[-p]{ isequalto}\{ 0 \}{} { ortheemptysetforeach}p \in {\mathbb {R}}.{ However},\,{ thisfunctionisnotintegrallyconvex}.{ Notethat}f\) is not convex-extensible nor \(\mathrm{dom\,}f\) is bounded. \(\blacksquare \)
4.3 Proximity theorems
The proximity-scaling approach is a fundamental technique in designing efficient algorithms for discrete or combinatorial optimization. For a function \(f: \mathbb {Z}^{n} \rightarrow \mathbb {R} \cup \{ +\infty \}{} { inintegervariablesandapositiveinteger}\alpha \in \mathbb {Z}_{++},\,{ calledascalingunit},\,{ the}\alpha \)-scaling of \(f{ meansthefunction}f^{\alpha }{} { definedby}f^{\alpha }(x) = f(\alpha x) (x \in \mathbb {Z}^{n})\) (cf., (3.19)). A proximity theorem is a result guaranteeing that a (local) minimum of the scaled function \(f^{\alpha }{} { is}({ geometrically}){ closetoaminimizeroftheoriginalfunction}f.{ Moreprecisely},\,{ wesaythat}x^{\alpha } \in \mathrm{dom\,}f{ isan}\alpha \)-local minimizer of \(f ({ or}\alpha \)-local minimal for \(f){ if}f(x^{\alpha }) \le f(x^{\alpha }+ \alpha d){ forall}d \in \{ -1,0, +1 \}^{n},\,{ andaproximitytheoremgivesabound}B(n,\alpha ){ suchthatforany}\alpha \)-local minimizer \(x^{\alpha }{} { of}f,\,{ thereexistsaminimizer}x^{*}{} { of}f{ satisfying} \Vert x^{\alpha } - x^{*}\Vert _{\infty } \le B(n,\alpha ).{ Thescaledfunction}f^{\alpha }{} { isexpectedtobesimplerandhenceeasiertominimize},\,{ whereasthequalityoftheobtainedminimizerof}f^{\alpha }{} { asanapproximationtotheminimizerof}f\) is guaranteed by a proximity theorem. The proximity-scaling approach consists in applying this idea for a decreasing sequence of \(\alpha ,\,{ oftenbyhalvingthescaleunit}\alpha \).
In discrete convex analysis the following proximity theorems are known for L\(^{\natural }\)-convex and M\(^{\natural }\)-convex functions.
Theorem 4.5
([29]; [40, Theorem 7.18]) Suppose that \(f\) is an L\(^{\natural }\)-convex function, \(\alpha \in \mathbb {Z}_{++},\,{ and}x^{\alpha } \in \mathrm{dom\,}f.{ If}f(x^{\alpha }) \le f(x^{\alpha } + \alpha d){ forall}d \in \{ 0, 1 \}^{n} \cup \{ 0, -1 \}^{n},\,{ thenthereexistsaminimizer}x^{*}{} { of}f{ satisfying}\Vert x^{\alpha } - x^{*} \Vert _{\infty } \le n (\alpha -1)\).
Theorem 4.6
([36]; [40, Theorem 6.37]) Suppose that \(f\) is an M\(^{\natural }\)-convex function, \(\alpha \in \mathbb {Z}_{++},\,{ and}x^{\alpha } \in \mathrm{dom\,}f.{ If}f(x^{\alpha }) \le f(x^{\alpha } + \alpha d){ forall}d \in \{ \varvec{1}^{i}, -\varvec{1}^{i} \ (1 \le i \le n) \} \cup \{ \varvec{1}^{i} - \varvec{1}^{j} \ (i \not = j) \} ,\,{ thenthereexistsaminimizer}x^{*}{} { of}f{ satisfying}\Vert x^{\alpha } - x^{*} \Vert _{\infty } \le n (\alpha -1)\).
The proximity bounds in Theorems 4.5 and 4.6 are known to be tight [37, Examples 4.2 and 4.3]. It is noteworthy that we have the same proximity bound \(B(n,\alpha ) = n (\alpha -1)\) for L\(^{\natural }\)-convex and M\(^{\natural }\)-convex functions, and that \(B(n,\alpha ){ islinearin}n.{ Forintegrallyconvexfunctionswith}n \ge 3,\,{ thisbound}n (\alpha -1)\) is no longer valid, which is demonstrated by Examples 4.4 and 4.5 in [37]. More specifically, the latter example shows a quadratic lower bound \(B(n,\alpha ) \ge (n-2)^{2}(\alpha -1)/4\) for an integrally convex function arising from bipartite graphs.
The following is a proximity theorem for integrally convex functions.
Theorem 4.7
([37, Theorem 5.1]) Let \(f: \mathbb {Z}^{n} \rightarrow \mathbb {R} \cup \{ +\infty \}{} { beanintegrallyconvexfunction},\,\alpha \in \mathbb {Z}_{++},\,{ and}x^{\alpha } \in \mathrm{dom\,}f.{ If}f(x^{\alpha }) \le f(x^{\alpha }+ \alpha d){ forall}d \in \{ -1,0, +1 \}^{n},\,{ then}\arg \min f \not = \emptyset { andthereexists}x^{*} \in \arg \min f\) with
where \(\beta _{n}{} \) is defined by
The proximity bound \(\beta _{n}{} \) satisfies
The bound \(\beta _{n} (\alpha - 1)\) in (4.7) is superexponential in \(n\), as seen from (4.9). The numerical values of \(\beta _{n}{} \) are as follows:
While the proof of Theorem 4.7 for general \(n\) is quite long [37], its special case for \(n=2\) admits an alternative method based on the box-barrier property (Theorem 4.2) and the parallelogram inequality (Remark 3.6). See the proof of [37, Theorem 4.1] for the detail.
Finally we mention that proximity theorems are also available for L\(^{\natural }_{2}\)-convex and M\(^{\natural }_{2}\)-convex functions [49]. The proximity bound for L\(^{\natural }_{2}\)-convex functions is linear in \(n\) [49, Theorem 6] and that for M\(^{\natural }_{2}\)-convex functions is quadratic in \(n\) [49, Theorem 10].
4.4 Scaling algorithm
In spite of the fact that integral convexity is not preserved under variable-scaling, it is possible to design a scaling algorithm for minimizing an integrally convex function with a bounded effective domain.
In the following we briefly describe the algorithm of [37]. The algorithm is justified by the proximity bound in Theorem 4.7 and the optimality criterion in Theorem 4.1. Let \(K_{\infty }{} { denotethe}\ell _{\infty }{} \)-size of the effective domain of \(f\), i.e.,
Initially, the scaling unit \(\alpha { issetto}2^{\lceil \log _{2} K_{\infty } \rceil } \approx K_{\infty }.{ InStep}~{ S}1{ ofthealgorithm},\,{ thefunction}{{\hat{f}}}(y){ istherestrictionof}f(x + \alpha y)\) to the set
which is a box of integers. Then a local minimizer \(y^{*}{} { of}{{\hat{f}}}(y){ isfoundtoupdate}x{ to}x+ \alpha y^{*}.{ Alocalminimizerof}{{\hat{f}}}(y){ canbefound},\,{ e}.{ g}.,\,{ byanydescentmethod}({ thesteepestdescentmethod},\,{ inparticular}),\,{ although}{{\hat{f}}}(y)\) is not necessarily integrally convex.
In the final phase with \(\alpha = 1,\,{{\hat{f}}}{} \) is an integrally convex function, and hence, by Theorem 4.1, the local minimizer in Step S1 is a global minimizer of \({{\hat{f}}}{} \). Furthermore, it can be shown, with the use of Theorem 4.7, that this point is a global minimizer of \(f.{ Thecomplexityofthealgorithmisasfollows}.{ Thenumberofiterationsinthedescentmethodisboundedby}|Y| \le (4\beta _{n})^{n}.{ Foreach}z,\,{ the}3^{n}{} { neighboringpointsareexaminedtofindadescentdirectionorverifyitslocalminimality}.{ ThusStep}~{ S}1{ canbedonewithatmost}(12\beta _{n})^{n}{} { functionevaluations}.{ Thenumberofscalingphasesis}\log _{2} K_{\infty }.{ Therefore},\,{ thetotalnumberoffunctionevaluationsinthealgorithmisboundedby}(12\beta _{n})^{n}\log _{2} K_{\infty }.{ Forafixed}n,\,{ thisgivesapolynomialbound}O(\log _{2} K_{\infty })\) in the problem size. It is emphasized in [37, Remark 6.2] that the linear dependence of \(B(n,\alpha )= \beta _{n} (\alpha - 1){ on}\alpha { iscriticalforthecomplexity}O(\log _{2} K_{\infty }).{ Finally},\,{ wementionthatnoalgorithmcanminimizeeveryintegrallyconvexfunctionintimepolynomialin}n,\,{ sinceanyfunctionontheunitcube}\{ 0,1 \}^{n}{} \) is integrally convex.
5 Subgradient and biconjugacy
5.1 Subgradient
In convex analysis [2, 24, 55], the subdifferential of a convex function \(g: {\mathbb {R}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} { at}x \in \mathrm{dom\,}g\) is defined by
and an element \(p{ of}\partial g(x)\) is called a subgradient of \(g{ at}x.{ Analogously},\,{ forafunction}f: {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \},\,{ thesubdifferentialof}f{ at}x \in \mathrm{dom\,}f\) is defined as
and an element \(p{ of}\partial f(x){ iscalledasubgradientof}f{ at}x\). An alternative expression
is often convenient, where \(f[-p](x) = f(x) - \langle p, x \rangle \) defined in (4.1). If \(f\) is convex-extensible in the sense of \(f = \overline{f}\,|_{{\mathbb {Z}}^{n}}{} \) in (3.3), where \(\overline{f}{} { istheconvexenvelopeof}f\) defined in (3.2), then \(\partial f(x) = \partial \overline{f}(x){ foreach}x \in {\mathbb {Z}}^{n}.{ When}f: {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} { isintegrallyconvex},\,\partial f(x){ isnonemptyfor}x \in \mathrm{dom\,}f,\,{ since}f = \overline{f}\,|_{{\mathbb {Z}}^{n}}{} \) by (3.7). Furthermore, we can rewrite (5.2) by making use of Theorem 4.1 for the minimality of an integrally convex function. Namely, by Theorem 4.1 applied to \(f[-p],\,{ wemayrestrict}y\) in (5.2) to the form of \(y=x + d{ with}d \in \{-1,0,+1\}^n\), to obtain
This expression shows, in particular, that \(\partial f(x){ isapolyhedrondescribedbyinequalitieswithcoefficientstakenfrom}\{ -1, 0, +1 \}.{ Todiscussanintegralitypropertyof}\partial f(x)\) in Sect. 5.2, it is useful to investigate the projections of \(\partial f(x){ alongcoordinateaxes}.{ Let}P:= \partial f(x){ fornotationalsimplicity},\,{ andforeach}l = 1,2,\ldots ,n,\,{ let}[P]_{l}{} { denotetheprojectionof}P{ tothespaceof}(p_{l},p_{l+1},\ldots ,p_n).{ Inequalitysystemstodescribetheprojections}[P]_{l}{} { for}l = 1,2,\ldots ,n\) can be obtained by applying the Fourier–Motzkin elimination procedure [48, 56] to the system of inequalities in (5.4), where the variable \(p_{1}{} { iseliminatedfirst},\,{ andthen}p_{2}, p_{3}, \ldots ,\,{ tofinallyobtainaninequalityin}p_{n}{} { only}.{ Byvirtueoftheintegralconvexityof}f\), a drastic simplification occurs in this elimination process. The inequalities that are generated in the elimination process are actually redundant and need not be added to the current system of inequalities, which is a crucial observation made in [50]. Thus we obtain the following theorem.
Theorem 5.1
([50]) Let \(f: {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} { beanintegrallyconvexfunctionand}x \in \mathrm{dom\,}f.{ Then}\partial f(x){ isanonemptypolyhedron},\,{ andforeach}l = 1,2,\ldots ,n,\,{ theprojection}[\partial f(x)]_{l}{} { of}\partial f(x){ tothespaceof}(p_{l},p_{l+1},\ldots ,p_n)\) is given by
While the Fourier–Motzkin elimination for the proof of Theorem 5.1 depends on the linear ordering of \(N = \{ 1,2,\ldots , n \}{} \), it is possible to formulate the obtained identity (5.5) without referring to the ordering of \(N\). This is stated below as a corollary.
Corollary 5.1
Let \(J{ beanynonemptysubsetof}N = \{ 1,2,\ldots , n \}{} \). Under the same assumption as in Theorem 5.1, the projection of \(\partial f(x){ tothespaceof}(p_{j} \mid j \in J){ isgivenby} \{ (p_{j} \mid j \in J) \mid \sum _{j \in J} d_{j} p_{j} \le f(x+d) - f(x) \ \hbox {for all} \ d \in \{ -1,0,+1 \}^{n} \ \hbox {with} \ d_{j} = 0 \ (j \in N {\setminus } J) \}{} \).
5.2 Integral subgradient
For an integer-valued function \(f: {\mathbb {Z}}^{n} \rightarrow {\mathbb {Z}}\cup \{ +\infty \},\,{ wearenaturallyinterestedinintegralvectorsin}\partial f(x).{ Anintegervector}p{ belongingto}\partial f(x)\) is called an integral subgradient and the condition
is sometimes referred to as the integral subdifferentiability of \(f{ at}x\).
Integral subdifferentiability of integer-valued integrally convex functions is established recently by the present authors [50].
Theorem 5.2
([50, Theorem 3]) Let \(f: {\mathbb {Z}}^{n} \rightarrow {\mathbb {Z}}\cup \{ +\infty \}{} \) be an integer-valued integrally convex function. For every \(x \in \mathrm{dom\,}f,\,{ wehave}\partial f(x) \cap {\mathbb {Z}}^{n} \ne \emptyset \).
Proof
The proof is based on Theorem 5.1 concerning projections of \(\partial f(x)\). While the reader is referred to [50] for the formal proof, we indicate the basic idea here. By (5.5) for \(l=n\), we have
which is an interval \([ \alpha _{n}, \beta _{n} ]_{{\mathbb {R}}}{} { with}\alpha _{n} = f(x) - f(x- \varvec{1}^{n} ){ and}\beta _{n}= f(x+ \varvec{1}^{n} ) - f(x).{ Wehave}\alpha _{n} \le \beta _{n}{} { since}\partial f(x) \ne \emptyset ,\,{ while}\alpha _{n}, \beta _{n} \in {\mathbb {Z}}{ since}f\) is integer-valued. Therefore, the interval \([ \alpha _{n}, \beta _{n} ]_{{\mathbb {R}}}{} { containsaninteger},\,{ say},\,p_{n}^{*}{} \). Next, by (5.5) for \(l=n-1,\,{ theset} \{ p_{n-1} \mid p \in \partial f(x), p_{n}= p_{n}^{*} \}{} \) is described by six inequalities
This implies that \(\{ p_{n-1} \mid p \in \partial f(x), p_{n}= p_{n}^{*} \}{} { isanonemptyinterval},\,{ say},\,[ \alpha _{n-1}, \beta _{n-1} ]_{{\mathbb {R}}}{} { with}\alpha _{n-1}, \beta _{n-1} \in {\mathbb {Z}},\,{ whichcontainsaninteger},\,{ say},\,p_{n-1}^{*}.{ Thismeansthatthereexists}p \in \partial f(x){ suchthat}p_{n}= p_{n}^{*} \in {\mathbb {Z}}{ and}p_{n-1}= p_{n-1}^{*} \in {\mathbb {Z}}.{ Continuinginthisway}({ with}l=n-2, n-3, \ldots , 2,1),\,{ wecanconstruct}p^{*} \in \partial f(x) \cap {\mathbb {Z}}^{n}. \square \)
Some supplementary facts concerning Theorem 5.2 are shown below.
-
Theorem 5.2 states that \(\partial f(x) \cap {\mathbb {Z}}^{n} \ne \emptyset ,\,{ butitdoesnotclaimastrongerstatementthat}\partial f(x){ isanintegerpolyhedron}.{ Indeed},\,\partial f(x){ isnotnecessarilyanintegerpolyhedron}.{ Forexample},\,{ let}f: \mathbb {Z}^3 \rightarrow \mathbb {Z} \cup \{+\infty \}{} { bedefinedby}f(0,0,0)=0{ and}f(1,1,0)=f(0,1,1)=f(1,0,1)=1{ with}\mathrm{dom\,}f = \{ (0,0,0), (1,1,0), (0,1,1), (1,0,1) \}.{ This}f{ isintegrallyconvexand}\partial f(\textbf{0}) = \{ p \in {\mathbb {R}}^{3} \mid p_{1} + p_{2} \le 1, p_{2} + p_{3} \le 1, p_{1} + p_{3} \le 1 \}{} \) is not an integer polyhedron, having a non-integral vertex at \(p=(1/2, 1/2, 1/2)\). See [50, Remark 4] for details. In the special cases where \(f\) is L\(^{\natural }\)-convex, M\(^{\natural }\)-convex, L\(^{\natural }_{2}\)-convex, or M\(^{\natural }_{2}\)-convex, the subdifferential \(\partial f(x)\) is known [40] to be an integer polyhedron.
-
If \(\partial f(x){ isbounded},\,\partial f(x){ hasanintegralvertex},\,{ althoughnoteveryvertexof}\partial f(x){ isintegral}.{ Forexample},\,{ let} D = \{ x \in \{ -1,0,+1 \}^{3} \mid |x_{1}| + |x_{2}| + |x_{3}| \le 2 \}{} { anddefine}f{ with}\mathrm{dom\,}f = D{ by}f(\textbf{0}) = 0{ and}f(x) = 1 (x \in D {\setminus } \{ \textbf{0}\}).{ This}f\) is an integer-valued integrally convex function and \(\partial f(\textbf{0})\) is a bounded polyhedron that has eight non-integral vertices \((\pm 1/2, \pm 1/2, \pm 1/2)\) (with arbitrary combinations of double-signs) and six integral vertices \((\pm 1, 0, 0),\,(0, \pm 1, 0),\,{ and}(0,0, \pm 1)\). See [50, Remark 7] for details.
-
Integral subdifferentiability is not guaranteed without the assumption of integral convexity. Consider \(D = \{ (0,0,0), \pm (1,1,0), \pm (0,1,1), \pm (1,0,1) \}{} { anddefine}f: \mathbb {Z}^3 \rightarrow \mathbb {Z} \cup \{+\infty \}{} { with}\mathrm{dom\,}f = D{ by}f(x_{1},x_{2},x_{3}) = (x_{1}+x_{2}+x_{3})/2 .{ Thisfunctionisnotintegrallyconvex},\,\partial f(\textbf{0}) = \{ (1/2, 1/2, 1/2) \},\,{ and}\partial f(\textbf{0}) \cap {\mathbb {Z}}^{3} = \emptyset \). See [39, Example 1.1] or [50, Example 1] for details.
The integral subdifferentiability formulated in Theorem 5.2 can be strengthened with an additional box condition. This stronger form plays the key role in the proof of the Fenchel-type min-max duality theorem (Theorem 6.1) discussed in Sect. 6.
Recall that an integral box means a set \(B{ ofrealvectorsrepresentedas}B = \{ p \in {\mathbb {R}}^{n} \mid \alpha \le p \le \beta \}{} { forintegervectors}\alpha \in ({\mathbb {Z}}\cup \{ -\infty \})^{n}{} { and}\beta \in ({\mathbb {Z}}\cup \{ +\infty \})^{n}{} { satisfying}\alpha \le \beta \). The following theorem states that
which may be referred to as box-integral subdifferentiability.
Theorem 5.3
([51, Theorem 1.2]) Let \(f: {\mathbb {Z}}^{n} \rightarrow {\mathbb {Z}}\cup \{ +\infty \}{} \) be an integer-valued integrally convex function, \(x \in \mathrm{dom\,}f,\,{ and}B{ beanintegralbox}.{ If}\partial f(x) \cap B{ isnonempty},\,{ then}\partial f(x) \cap B{ isapolyhedroncontaininganintegervector}.{ If},\,{ inaddition},\,\partial f(x) \cap B{ isbounded},\,{ then}\partial f(x) \cap B\) has an integral vertex.
We briefly describe how Theorem 5.3 has been proved in [51]. Recall that Theorem 5.2 for integral subdifferentiability (without a box) is proved from a hierarchical system of inequalities (Theorem 5.1) to describe the projection \([\partial f(x)]_{l}{} { of}\partial f(x){ tothespaceof}(p_{l},p_{l+1},\ldots ,p_n){ for}l = 1,2,\ldots ,n\), where Theorem 5.1 itself is proved by means of the Fourier–Motzkin elimination. This approach is extended in [51] to prove Theorem 5.3. Namely, Theorem 4.3 of [51] gives a hierarchical system of inequalities to describe the projection \([\partial f(x) \cap B]_{l}{} { of}\partial f(x) \cap B{ tothespaceof}(p_{l},p_{l+1},\ldots ,p_n){ for}l = 1,2,\ldots ,n\). The proof of this theorem is based on the Fourier–Motzkin elimination. Once inequalities for the projections \([\partial f(x) \cap B]_{l}{} \) are obtained, the derivation of box-integral subdifferentiability (Theorem 5.3) is almost the same as that of integral subdifferentiability (Theorem 5.2) from Theorem 5.1. Finally we mention that alternative proofs of Theorems 5.2 and 5.3 can be found in [19] and [20], respectively.
5.3 Biconjugacy
For an integer-valued function \(f: {\mathbb {Z}}^{n} \rightarrow {\mathbb {Z}}\cup \{ +\infty \}{} { with}\mathrm{dom\,}f \ne \emptyset ,\,{ wedefine}f^{\bullet }: {\mathbb {Z}}^{n} \rightarrow {\mathbb {Z}}\cup \{ +\infty \}{} \) by
called the integral conjugate of \(f.{ Forany}x, p \in {\mathbb {Z}}^{n}{} \) we have
which is a discrete analogue of Fenchel’s inequality [24, (1.1.3), p. 211] or the Fenchel–Young inequality [2, Proposition 3.3.4]. When \(x \in \mathrm{dom\,}f\), we have
Note the asymmetric roles of \(f{ and}f^{\bullet }{} \) in (5.10).
The integral conjugate \(f^{\bullet }{} \) of an integer-valued function \(f\) is also an integer-valued function defined on \({\mathbb {Z}}^{n}{} \). So we can apply the transformation (5.8) to \(f^{\bullet }{} { toobtain}f^{\bullet \bullet } = (f^{\bullet })^{\bullet }{} \), which is called the integral biconjugate of \(f\). It follows from (5.9) and (5.10) that, for each \(x \in \mathrm{dom\,}f\) we have
See [39, Lemma 4.1] or [50, Lemma 1] for the proof. We say that \(f\) enjoys integral biconjugacy if
Example 5.1
Let \(D = \{ (0,0,0), \pm (1,1,0), \pm (0,1,1), \pm (1,0,1) \}{} { andconsiderthefunction}f(x_{1},x_{2},x_{3}) = (x_{1}+x_{2}+x_{3})/2 { on}\mathrm{dom\,}f = D\). (This is the function used in Sect. 5.2 as an example of an integer-valued function lacking in integral subdifferentiability.) According to the definition (5.8), the integral conjugate \(f^{\bullet }{} \) is given by
For \(x=\textbf{0}{ wehave}f(x)= 0\), while (5.13) shows \(f^{\bullet }(p) \ge 1{ foreveryintegervector}p \in {\mathbb {Z}}^{3}.{ Thereforewehavestrictinequality}f(x) + f^{\bullet }(p) > \langle p, x \rangle { for}x=\textbf{0}{ andevery}p \in {\mathbb {Z}}^{3}{} \). This is consistent with (5.10) since \(\partial f(\textbf{0}) = \{ (1/2, 1/2, 1/2) \}{} { andhence}\partial f(\textbf{0}) \cap {\mathbb {Z}}^{3} = \emptyset .{ Fortheintegralbiconjugate}f^{\bullet \bullet }(x) = \sup \{ \langle p, x \rangle - f^{\bullet }(p) \mid p \in {\mathbb {Z}}^3 \}{} \) we have
Therefore we have \(f^{\bullet \bullet }(\textbf{0}) \ne f(\textbf{0})\). This is consistent with (5.11) since \(\partial f(\textbf{0}) \cap {\mathbb {Z}}^{3} = \emptyset \). \(\blacksquare \)
We now assume that \(f: {\mathbb {Z}}^{n} \rightarrow {\mathbb {Z}}\cup \{ +\infty \}{} \) is an integer-valued integrally convex function. The integral conjugate \(f^{\bullet }{} \) is not necessarily integrally convex ( [47, Example 4.15], [50, Remark 2.3]). Nevertheless, the integral biconjugate \(f^{\bullet \bullet }{} { coincideswith}f\) itself, that is, integral biconjugacy holds for an integer-valued integrally convex function. This theorem is established by the present authors [50] based on the integral subdifferentiability \(\partial f(x) \cap {\mathbb {Z}}^{n} \ne \emptyset \) given in Theorem 5.2; see Remark 5.1 below for some technical aspects.
Theorem 5.4
([50, Theorem 4]) For any integer-valued integrally convex function \(f: {\mathbb {Z}}^{n} \rightarrow {\mathbb {Z}}\cup \{ +\infty \}{} { with}\mathrm{dom\,}f \ne \emptyset ,\,{ wehave}f^{\bullet \bullet }(x) =f(x){ forall}x \in {\mathbb {Z}}^{n}{} \).
As special cases of Theorem 5.4 we obtain integral biconjugacy for L-convex, L\(^{\natural }\)-convex, M-convex, M\(^{\natural }\)-convex, L\(^{\natural }_{2}\)-convex, and M\(^{\natural }_{2}\)-convex functions given in [40, Theorems 8.12, 8.36, 8.46], and that for BS-convex and UJ-convex functions given in [50, Corollary 2].
Remark 5.1
There is a subtle gap between integral subdifferentiability in (5.11) and integral biconjugacy in (5.12) for a general integer-valued function \(f: {\mathbb {Z}}^{n} \rightarrow {\mathbb {Z}}\cup \{ +\infty \} ({ whichisnotnecessarilyintegrallyconvex}).{ Whilethelatterimposesthecondition}f^{\bullet \bullet }(x) = f(x){ onall}x \in {\mathbb {Z}}^{n},\,{ theformerrefersto}x{ in}\mathrm{dom\,}f\) only. This means that integral subdifferentiability may possibly be weaker than integral biconjugacy, and this is indeed the case in general (see [39, Remark 4.1] or [50, Remark 6]). However, it is known [39, Lemma 4.2] that integral subdifferentiability does imply integral biconjugacy under the technical conditions that \(\mathrm{dom\,}f = \textrm{cl}(\overline{\mathrm{dom\,}f}) \cap \mathbb {Z}^{n}{} { and}\textrm{cl}(\overline{\mathrm{dom\,}f})\) is rationally-polyhedral, where \(\textrm{cl}(\overline{\mathrm{dom\,}f}){ denotestheclosureoftheconvexhull}({ orclosedconvexhull}){ of}\mathrm{dom\,}f\); see Remark 2.1 for this notation. \(\blacksquare \)
Remark 5.2
In convex analysis [2, 24, 55], the conjugate function of \(g: {\mathbb {R}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} { with}\mathrm{dom\,}g \ne \emptyset { isdefinedtobeafunction}g^{\bullet {\mathbb {R}}}: {\mathbb {R}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} \) given by
where a (non-standard) notation \(g^{\bullet {\mathbb {R}}}{} \) is introduced for discussion here. The biconjugate of \(g{ isdefinedas}(g^{\bullet {\mathbb {R}}})^{\bullet {\mathbb {R}}}{} \) by using the transformation (5.14) twice. We have biconjugacy \((g^{\bullet {\mathbb {R}}})^{\bullet {\mathbb {R}}} = g{ forclosedconvexfunctions}g\). For a real-valued function \(f: {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} \) in discrete variables, we may also define
Then the convex envelope \(\overline{f}{} { coincideswith}(f^{\bullet {\mathbb {R}}})^{\bullet {\mathbb {R}}}{} \), which denotes the function obtained by applying (5.15) to \(f\) and then (5.14) to \(g = f^{\bullet {\mathbb {R}}}.{ Therefore},\,{ if}f\) is convex-extensible in the sense of \(f = \overline{f}\,|_{{\mathbb {Z}}^{n}}{} \) in (3.3), we have \((f^{\bullet {\mathbb {R}}})^{\bullet {\mathbb {R}}} \, |_{{\mathbb {Z}}^{n}} = f,\,{ whichisakindofbiconjugacy}.{ If}f\) is integer-valued, we can naturally consider \((f^{\bullet })^{\bullet }{} \) using (5.8) twice as well as \((f^{\bullet {\mathbb {R}}})^{\bullet {\mathbb {R}}}{} \) using (5.15) and then (5.14). It is most important to recognize that for any \(f{ wehave}(f^{\bullet })^{\bullet }(x)\le (f^{\bullet {\mathbb {R}}})^{\bullet {\mathbb {R}}}(x){ for}x \in {\mathbb {Z}}^{n}{} { andthattheequalitymayfailevenwhen}f = \overline{f}\, |_{{\mathbb {Z}}^{n}}.{ Asanexample},\,{ considerthefunction}f(x_{1},x_{2},x_{3}) = (x_{1}+x_{2}+x_{3})/2 \) in Example 5.1. The convex envelope \(\overline{f}{} { isgivenby}\overline{f}(x_{1},x_{2},x_{3}) = (x_{1}+x_{2}+x_{3})/2 { ontheconvexhullof}D,\,{ andtherefore}f = \overline{f}\, |_{{\mathbb {Z}}^{n}}{} \) holds. Similarly to (5.13) we have
and hence
where the infimum over \(p \in {\mathbb {R}}^3{ isattainedby}p = (1/2, 1/2, 1/2).{ Therefore}(f^{\bullet {\mathbb {R}}})^{\bullet {\mathbb {R}}}(\textbf{0}) = f(\textbf{0}),\,{ whereas} (f^{\bullet })^{\bullet }(\textbf{0}) < f(\textbf{0})\) as we have seen in Example 5.1. Thus, integral biconjugacy \(f = (f^{\bullet })^{\bullet }{} \) in (5.12) is much more intricate than the equality \(f = (f^{\bullet {\mathbb {R}}})^{\bullet {\mathbb {R}}}\, |_{{\mathbb {Z}}^{n}}{} \). \(\blacksquare \)
5.4 Discrete DC programming
A discrete analogue of the theory of DC functions (difference of two convex functions), or discrete DC programming, has been proposed in [31] using L\(^{\natural }\)-convex and M\(^{\natural }\)-convex functions. As already noted in [31, Remark 4.7], such theory of discrete DC functions can be developed for functions that satisfy integral biconjugacy and integral subdifferentiability. It is pointed out in [50] that Theorems 5.2 and 5.4 for integrally convex functions enable us to extend the theory of discrete DC functions to integrally convex functions. In particular, an analogue of the Toland–Singer duality [61, 65] can be established for integrally convex functions as follows.
Theorem 5.5
( [50, Theorem 5]) Let \(g, h: \mathbb {Z}^{n} \rightarrow \mathbb {Z} \cup \{+\infty \}{} \) be integer-valued integrally convex functions. Then
As mentioned already in [50], the assumption of integral convexity of \(g\) is not needed for (5.16) to be true. That is, (5.16) holds for any \(g: \mathbb {Z}^{n} \rightarrow \mathbb {Z} \cup \{+\infty \}{} { aslongas}h: \mathbb {Z}^{n} \rightarrow \mathbb {Z} \cup \{+\infty \}{} \) is integrally convex.
6 Discrete Fenchel duality
6.1 General framework of Fenchel duality
The Fenchel duality is one of the expressions of the duality principle in the form of a min-max relation between a pair of convex and concave functions \((f,g)\) and their conjugate functions. As is well known, the existence of such min-max formula guarantees the existence of a certificate of optimality for the problem of minimizing \(f - g{ over}{\mathbb {R}}^{n}{} { or}{\mathbb {Z}}^{n}.{ Firstwerecalltheframeworkforfunctionsincontinuousvariables}.{ For}f: {\mathbb {R}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} { with}\mathrm{dom\,}f \ne \emptyset ,\,{ thefunction}f^{\bullet }: {\mathbb {R}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} \) defined by
is called the conjugate (or convex conjugate) of \(f.{ For}g: {\mathbb {R}}^{n} \rightarrow {\mathbb {R}}\cup \{ -\infty \}{} { with}\mathrm{dom\,}g \ne \emptyset ,\,{ thefunction}g^{\circ }: {\mathbb {R}}^{n} \rightarrow {\mathbb {R}}\cup \{ -\infty \}{} \) defined by
is called the concave conjugate of \(g.{ Wehave}g^{\circ }(p) = -f^{\bullet }(-p){ if}g(x) = -f(x)\). It follows from the definitions that
for any \(x \in {\mathbb {R}}^{n}{} { and}p \in {\mathbb {R}}^{n}{} \). The relation (6.3) is called weak duality.
The Fenchel duality theorem says that a min-max formula
holds for convex and concave functions \(f: {\mathbb {R}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} { and}g: {\mathbb {R}}^{n} \rightarrow {\mathbb {R}}\cup \{ -\infty \}{} \) satisfying certain regularity conditions. The relation (6.4) is called strong duality in contrast to weak duality. See Bauschke–Combettes [1, Section 15.2], Borwein–Lewis [2, Theorem 3.3.5], Hiriart-Urruty–Lemaréchal [24, (2.3.2), p. 228], Rockafellar [55, Theorem 31.1], Stoer–Witzgall [62, Corollary 5.1.4] for precise statements.
We now turn to functions in discrete variables. For any functions \(f: {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} { and}g: {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{ -\infty \}{} \) we define
where \(\mathrm{dom\,}f \ne \emptyset { and}\mathrm{dom\,}g \ne \emptyset \) are assumed. In this case, the generic form of the Fenchel duality reads:
which is expected to be true when \(f{ and}g{ areequippedwithcertaindiscreteconvexityandconcavity},\,{ respectively}.{ Moreover},\,{ when}f{ and}g\) are integer-valued (\(f: {\mathbb {Z}}^{n} \rightarrow {\mathbb {Z}}\cup \{ +\infty \}{} { and}g: {\mathbb {Z}}^{n} \rightarrow {\mathbb {Z}}\cup \{ -\infty \}{} \)), we are particularly interested in the dual problem with an integer vector, that is,
To relate the discrete case to the continuous case, it is convenient to consider the convex envelope \(\overline{f}{} { of}f{ andtheconcaveenvelope}\overline{g}{} { of}g,\,{ where}\overline{g}:= -\overline{(-g)},\,{ thatis},\,\overline{g}{} { isdefinedtobethenegativeoftheconvexenvelopeof}-g\). By the definitions of convex and concave envelopes and conjugate functions we have
as well as weak dualities
Thus we have the following chain of inequalities:
with notations
for the optimal values of the problems, where \(\textrm{P}(\cdot )\) stands for “Primal problem” and \(\textrm{D}(\cdot )\) for “Dual problem”.
The desired min-max relations (6.7) and (6.8) can be written as \(\textrm{P}({\mathbb {Z}}) = \textrm{D}({\mathbb {R}}){ and}\textrm{P}({\mathbb {Z}}) = \textrm{D}({\mathbb {Z}}),\,{ respectively}.{ Theinequalitybetween}\overline{\textrm{P}}({\mathbb {R}}){ and}\overline{\textrm{D}}({\mathbb {R}})\) in the middle of (6.11) becomes an equality if the Fenchel duality (6.4) holds for \((\overline{f}, \overline{g})\). The first and the last inequality in (6.11) express possible discrepancy between discrete and continuous cases, and we are naturally concerned with when these inequalities turn into equalities. The concept of subdifferential plays the essential role here.
For the subdifferential \(\partial f(x)\) defined in (5.2) we observe that
holds for any \(p \in {\mathbb {R}}^{n}{} { and}x \in \mathrm{dom\,}f\). Similarly, we have
for any \(p \in {\mathbb {R}}^{n}{} { and}x \in \mathrm{dom\,}g,\,{ where}\partial ' g(x)\) means the concave version of the subdifferential defined by
Suppose that the infimum \(\textrm{P}({\mathbb {Z}}){ isattainedbysome}x^{*} \in {\mathbb {Z}}^{n}{} \). It follows from (6.9), (6.12), and (6.13) that \(\textrm{P}({\mathbb {Z}}) = \textrm{D}({\mathbb {R}}){ andthesupremum}\textrm{D}({\mathbb {R}}){ isattainedby}p^{*} \in {\mathbb {R}}^{n}{} { ifandonlyif}p^{*} \in \partial f(x^{*}) \cap \partial ' g(x^{*}).{ Therefore},\,\textrm{P}({\mathbb {Z}}) = \textrm{D}({\mathbb {R}})\) if
Furthermore, if
then we have \(\textrm{P}({\mathbb {Z}}) = \textrm{D}({\mathbb {Z}})\). Thus (6.15) and (6.16), respectively, imply the Fenchel-type min-max formulas (6.7) for real-valued functions and (6.8) for integer-valued functions. It is noted that this implication does not presuppose the Fenchel duality \(\overline{\textrm{P}}({\mathbb {R}}) = \overline{\textrm{D}}({\mathbb {R}}){ for}(\overline{f}, \overline{g}).{ When}\overline{\textrm{P}}({\mathbb {R}}) = \overline{\textrm{D}}({\mathbb {R}})\) is known to hold, the min-max relation \(\textrm{P}({\mathbb {Z}}) = \textrm{D}({\mathbb {R}})\) for real-valued functions follows, by (6.11), from \(\textrm{P}({\mathbb {Z}}) = \overline{\textrm{P}}({\mathbb {R}}),\,{ wherethelattercondition}\textrm{P}({\mathbb {Z}}) = \overline{\textrm{P}}({\mathbb {R}})\) holds if
It is emphasized that (6.17) does not follow from the individual convex or concave-extensibility of \(f{ and}g\). See Example 6.1 in Sect. 6.2.
6.2 Fenchel duality for integrally convex functions
The following two examples show that the min-max formula (6.7) or (6.8) is not necessarily true when \(f{ and}g{ areintegrallyconvexandconcavefunctions}. ({ Naturally},\,{ function}g\) is called integrally concave if \(-g\) is integrally convex.)
Example 6.1
( [43, Example 5.6]) Let \(f, g: {\mathbb {Z}}^{2} \rightarrow {\mathbb {Z}}\) be defined as
The function \(f\) is integrally convex (actually M\(^{\natural }\)-convex) and \(g\) is integrally concave (actually L\(^{\natural }\)-concave). We have
Thus the min-max identity (6.8) as well as (6.7) fails because of the primal integrality gap \(\textrm{P}({\mathbb {Z}}) > \overline{\textrm{P}}({\mathbb {R}}).{ Indeed},\,{ thecondition}\overline{f - g} = \overline{f}- \overline{g}{} \) in (6.17) fails as follows. Let \(h:= f - g.{ Since}h(0,0) = h(1,0) = h(0,1) = h(1,1) = 0,\,{ wehave}\overline{h}(1/2,1/2) = 0,\,{ whereas}\overline{f}(1/2,1/2) - \overline{g}(1/2,1/2) = 0 -1 = -1.{ Thus}\overline{f - g} \ne \overline{f}- \overline{g}{} \). \(\blacksquare \)
Example 6.2
( [43, Example 5.7]) Let \(f, g: {\mathbb {Z}}^{2} \rightarrow {\mathbb {Z}}\) be defined as
The function \(f\) is integrally convex (actually M\(^{\natural }\)-convex) and \(g\) is integrally concave (actually L\(^{\natural }\)-concave). We have
Although the min-max identity (6.7) with real-valued \(p\) holds, the formula (6.8) with integer-valued \(p{ failsbecauseofthedualintegralitygap}\textrm{D}({\mathbb {R}}) > \textrm{D}({\mathbb {Z}}).{ Theoptimalvalue}\min _{{\mathbb {Z}}}\{ f - g \} = 0{ isattainedby}x^{*} = \textbf{0},\,{ atwhichthesubdifferentialsaregivenby}\partial f(\textbf{0}) = \{ (p_{1}, p_{2}) \mid 0 \le p_{1}=p_{2} \le 1 \}{} { and}\partial ' g(\textbf{0}) = \{ (p_{1}, p_{2}) \mid p_{1}+p_{2}= 1, 0 \le p_{1} \le 1 \}.{ Wehave}\partial f(\textbf{0}) \cap \partial ' g(\textbf{0}) = \{ (1/2, 1/2)\}{} { and}\partial f(\textbf{0}) \cap \partial ' g(\textbf{0}) \cap {\mathbb {Z}}^{2} = \emptyset \), which shows the failure of (6.16). \(\blacksquare \)
As the min-max formula (6.7) is not true (in general) when \(f{ and}-g{ areintegrallyconvex},\,{ wearemotivatedtorestrict}g{ toasubclassofintegrallyconcavefunctions},\,{ whileallowing}f{ tobeageneralintegrallyconvexfunction}.{ However},\,{ thepossibilityof}g\) being M\(^{\natural }\)-concave or L\(^{\natural }\)-concave is denied by the above examples. That is, we cannot hope for the combination of (integrally convex, M\(^{\natural }\)-convex) nor (integrally convex, L\(^{\natural }\)-convex) for \((f,-g)\). Furthermore, since a function in two variables is M\(^{\natural }\)-convex if and only if it is multimodular [32, Remark 2.2], the possibility of the combination of (integrally convex, multimodular) for \((f,-g)\) is also denied. Thus we are motivated to consider the combination of (integrally convex, separable convex).
In the following we address the Fenchel-type min-max formula for a pair of an integrally convex function and a separable concave function. A function \(\Psi : {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{ -\infty \}{} { in}x=(x_{1}, x_{2}, \ldots ,x_{n}) \in {\mathbb {Z}}^{n}{} \) is called separable concave if it can be represented as
with univariate discrete concave functions \(\psi _{i}: {\mathbb {Z}}\rightarrow {\mathbb {R}}\cup \{ -\infty \},\,{ whichmeans},\,{ bydefinition},\,{ that}\mathrm{dom\,}\psi _{i}{} \) is an interval of integers and
The concave conjugate of \(\Psi { isdenotedby}\Psi ^{\circ }{} \), that is,
This is a separable concave function represented as
where
It is often possible to obtain an explicit form of the (integral) conjugate function of an (integer-valued) separable convex (or concave) function; see [14, 15].
The following is the Fenchel-type min-max formula for a pair of an integrally convex function and a separable concave function. The case of integer-valued functions, which is more interesting, is given in [51, Theorem 1.1], while we include here the case of real-valued functions for completeness.
Theorem 6.1
Let \(f: {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} { beanintegrallyconvexfunctionand}\Psi : {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{ -\infty \}{} { aseparableconcavefunction}.{ Assumethat}\mathrm{dom\,}f \cap \mathrm{dom\,}\Psi \ne \emptyset { and}\inf \{ f(x) - \Psi (x) \mid x \in {\mathbb {Z}}^{n} \}{} { isattainedbysome}x^{*}{} \). Then
and the supremum in (6.23) is attained by some \(p^{*} \in {\mathbb {R}}^{n}.{ If},\,{ inaddition},\,f{ and}\Psi \) are integer-valued, then
and the supremum in (6.24) is attained by some \(p^{*} \in {\mathbb {Z}}^{n}{} \).
Proof
The proof of the real-valued case (6.23) consists in showing (6.15) for \((f,g)= (f,\Psi )\), which is proved in Sect. 6.3. The integral case (6.24) has been proved in [51, Theorem 1.1] by showing (6.16) for \((f,g)= (f,\Psi )\) as a consequence of box-integral subdifferentiability described in Theorem 5.3. \(\square \)
Remark 6.1
In the integer-valued case, the existence of \(x^{*}{} \) attaining the infimum in (6.24) is guaranteed if (and only if) the set \(\{ f(x) - \Psi (x) \mid x \in {\mathbb {Z}}^{n} \}{} \) of function values is bounded from below. It is known [51, Lemma 3.2] that if the supremum on the right-hand side of (6.24) is finite, then the infimum on the left-hand side is also finite. \(\blacksquare \)
Theorem 6.1 implies a min-max theorem for separable convex minimization on a box-integer polyhedron. The case of integer-valued functions, which is more interesting, is stated in [51, Theorem 3.1]. We define notation \(\mu _{P}(p) = \inf \{ \langle p, x \rangle \mid x\in P \}{} { forapolyhedron}P\).
Theorem 6.2
Let \(P \ (\subseteq {\mathbb {R}}^{n})\) be a nonempty box-integer polyhedron, and \(\Phi : {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} { aseparableconvexfunction}.{ Assumethat}\inf \{ \Phi (x) \mid x \in P \cap {\mathbb {Z}}^{n} \}{} { is}({ finiteand}){ attainedbysome}x^{*}{} \). Then
and the supremum is attained by some \(p^{*} \in {\mathbb {R}}^{n}.{ If},\,{ inaddition},\,\Phi \) is integer-valued, then
and the supremum is attained by some \(p^{*} \in {\mathbb {Z}}^{n}{} \).
Proof
Denote the indicator function of \(P \cap {\mathbb {Z}}^{n}{} { by}\delta \), which is an integer-valued integrally convex function because \(P\) is a box-integer polyhedron. Then the statements follow from Theorem 6.1 for \(f=\delta { and}\Psi = -\Phi . \square \)
This theorem generalizes a recent result of Frank–Murota [15, Theorem 3.4], which asserts the min-max formula (6.26) for integer-valued \(\Phi { when}P\) is an integral box-TDI polyhedron.
6.3 Proof of (6.23) for real-valued functions (Theorem 6.1)
In this section we prove the min-max formula (6.23) for real-valued functions in Theorem 6.1. Let \(x^{*}{} { denoteanelementof}\mathrm{dom\,}f \cap \mathrm{dom\,}\Psi \ (\subseteq {\mathbb {Z}}^{n})\) that attains the infimum in (6.23). According to the general framework described in Sect. 6.1, it suffices to show \(\partial f(x^{*}) \cap \partial ' \Psi (x^{*}) \ne \emptyset .{ Since}\overline{f - \Psi } = \overline{f} - \overline{\Psi }{} \) by Proposition 2.1 of [51], we have
which implies that \(x^{*}{} { isalsoaminimizerof}\overline{f} - \overline{\Psi }{} { over}{\mathbb {R}}^{n}.{ Define}\alpha := \inf \{ f(x) - \Psi (x) \mid x \in {\mathbb {Z}}^{n} \} = f(x^{*}) - \Psi (x^{*}){ and}f_{1}(x):= f(x) - \alpha { for}x \in {\mathbb {Z}}^{n}.{ Then}f_{1}{} { isanintegrallyconvexfunctionsatisfying}\overline{f_{1}} \ge \overline{\Psi }{} { and}f_{1}(x^{*}) = \Psi (x^{*}).{ Thedesirednonemptiness}\partial f(x^{*}) \cap \partial ' \Psi (x^{*}) \ne \emptyset \) follows from Proposition 6.1 below applied to \((f,g)=(f_{1},\Psi )\).
Proposition 6.1
Let \(f: {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{ +\infty \}{} { and}g: {\mathbb {Z}}^{n} \rightarrow {\mathbb {R}}\cup \{ -\infty \}{} { beintegrallyconvexandconcavefunctions},\,{ respectively},\,{ and}x^{*} \in \mathrm{dom\,}f \cap \mathrm{dom\,}g.{ If}\overline{f} \ge \overline{g}{} { on}{\mathbb {R}}^{n}{} { and}f(x^{*}) = g(x^{*}),\,{ then}\partial f(x^{*}) \cap \partial ' g(x^{*}) \ne \emptyset \).
Proof
We may assume that \(x^{*} = \textbf{0}{ and}f(\textbf{0}) = g(\textbf{0}) = 0\). By (5.4) we have
where \(\{ -1,0,+1 \}^{n} \cap \mathrm{dom\,}f{ isrepresentedas}\{ d^{(i)} \mid i \in I \}{} { and}\{ -1,0,+1 \}^{n} \cap \mathrm{dom\,}g{ as}\{ {{\hat{d}}}^{(j)} \mid j \in J \}{} \). By the Farkas lemma (or linear programming duality) [56], there exists \(p \in \partial f(\textbf{0}) \cap \partial ' g(\textbf{0})\) if and only if
for any \(u_{i} \ge 0~\textrm{and}\,v_{j} \ge 0\) satisfying \(\sum _{i \in I} u_{i} d^{(i)} - \sum _{j \in J} v_{j} {{\hat{d}}}^{(j)} = \textbf{0}. {{ Let}}\,{{\hat{x}}}:= \sum _{i \in I} u_{i} d^{(i)} = \sum _{j \in J} v_{j} {{\hat{d}}}^{(j)},\,U:= \sum _{i \in I} u_{i}, {{ and}} V:= \sum _{j \in J} v_{j}.{{ Byhomogeneitywemayassume}} U + V \le 1.{{ Then}} {{\hat{x}}} \in \mathrm {dom\,}\overline{f} \cap \mathrm {dom\,}\overline{g}.{{ If}} U > 0\), it follows from the convexity of \(\overline{f}{} {{ aswellas}} \overline{f}(\textbf{0})=0{{ and}} U \le 1\) that
The resulting inequality \( \sum _{i \in I} u_{i} f(d^{(i)}) \ge \overline{f}({{\hat{x}}}){ isalsotruewhen}U=0.{ Similarly},\,{ weobtain} \sum _{j \in J} v_{j} g({{\hat{d}}}^{(j)}) \le \overline{g}({{\hat{x}}}),\,{ whereas}\overline{f}({{\hat{x}}}) - \overline{g}({{\hat{x}}}) \ge 0{ bytheassumption}\overline{f} \ge \overline{g}{} \). Therefore, (6.29) holds. \(\square \)
6.4 Connection to min-max theorems on bisubmodular functions
Let \(N = \{ 1,2,\ldots , n \}{} { anddenoteby}3^{N}{} { thesetofallpairs}(X,Y){ ofdisjointsubsets}X, Y{ of}N,\,{ thatis},\,3^{N} = \{ (X,Y) \mid X, Y \subseteq N, \ X \cap Y = \emptyset \}.{ Afunction}f: 3^{N} \rightarrow {\mathbb {R}}\) is called bisubmodular if
holds for all \((X_{1}, Y_{1}), (X_{2}, Y_{2}) \in 3^{N}.{ Inthefollowingweassume}f(\emptyset ,\emptyset ) =0\). The associated bisubmodular polyhedron is defined by
which, in turn, determines \(f\) by
If \(f\) is integer-valued, \(P(f)\) is an integral polyhedron. The reader is referred to [17, Section 3.5(b)] and [18] for bisubmodular functions and polyhedra.
In a study of \(b\)-matching degree-sequence polyhedra, Cunningham–Green-Krótki [9] obtained a min-max formula for the maximum component sum \(z(N) = \sum _{i \in N} z_{i}{} { of}z \in P(f)\) upper-bounded by a given vector \(w\).
Theorem 6.3
( [9, Theorem 4.6]) Let \(f: 3^{N} \rightarrow {\mathbb {R}}{ beabisubmodularfunctionwith}f(\emptyset ,\emptyset ) =0,\,{ and}w \in {\mathbb {R}}^{n}.{ Ifthereexists}z \in P(f){ with}z \le w\), then
Moreover, if \(f{ and}w\) are integer-valued, then there exists an integral vector \(z\) that attains the maximum on the left-hand side of (6.31).
The min-max formula (6.31) can be extended to a box constraint (with both upper and lower bounds on \(z\)). This extension is given in (6.32) below. Although this formula is not explicit in Fujishige–Patkar [21], it can be derived without difficulty from the results of [21]; see Remark 6.2.
Theorem 6.4
([21]) Let \(f: 3^{N} \rightarrow {\mathbb {R}}{ beabisubmodularfunctionwith}f(\emptyset ,\emptyset ) =0,\,{ and}\alpha { and}\beta { berealvectorswith}\alpha \le \beta .{ Ifthereexists}z \in P(f){ with}\alpha \le z \le \beta ,\,{ then},\,{ foreach}(A,B) \in 3^{N}{} \), we have
Moreover, if \(f,\,\alpha ,\,{ and}\beta \) are integer-valued, then there exists an integral vector \(z\) that attains the maximum on the left-hand side of (6.32).
Theorem 6.4 can be derived from Theorem 6.1 as follows. Let \({{\hat{f}}}{} { denotetheconvexextensionofthegivenbisubmodularfunction}f: 3^{N} \rightarrow {\mathbb {R}}\), as defined by Qi [54, Eqn (5)] as a generalization of the Lovász extension of a submodular function. This function \({{\hat{f}}}: {\mathbb {R}}^{n} \rightarrow {\mathbb {R}}{ isapositivelyhomogeneousconvexfunctionanditisanextensionof}f{ inthesensethat}{{\hat{f}}} (\varvec{1}^{X}-\varvec{1}^{Y}) = f(X,Y){ forall}(X,Y) \in 3^{N}.{ Itfollowsfromthepositivehomogeneityof}{{\hat{f}}}{} \) and Lemma 11 of [54] that
This implies, by Theorem 3.1, that the function \({{\hat{f}}}{} { restrictedto}{\mathbb {Z}}^{n}{} { isanintegrallyconvexfunction}.{ Inthefollowingwedenotetherestrictionof}{{\hat{f}}}{} { to}{\mathbb {Z}}^{n}{} { alsoby}{{\hat{f}}}.{ Fix}(A,B) \in 3^{N}{} { andlet}C = N \setminus (A \cup B).{ Wedefineaseparableconcavefunction}\Psi (x) = \sum _{i \in N} \psi _{i}(x_{i}){ with}\psi _{i}: {\mathbb {Z}}\rightarrow {\mathbb {R}}{ givenasfollows}\,:\,{ For}i \in A\),
For \(i \in B\),
For \(i \in C\),
We apply Theorem 6.1 to the integrally convex function \({{\hat{f}}}{} { andtheseparableconcavefunction}\Psi \). For these functions the min-max formula (6.23) reads
where (6.24) gives integrality of \(p\) in the integer-valued case. In the following we show
to obtain “\(\max = \min \)” in (6.32).
The proof of (6.37) consists of showing two equations
We first show (6.40) while postponing the proof of (6.39). On identifying a vector \(x \in \{-1, 0, +1 \}^{n}{} { with}(X,Y) \in 3^{N}{} { by}x = \varvec{1}^{X}-\varvec{1}^{Y},\,{ wehave}{{\hat{f}}}(x) = {{\hat{f}}} (\varvec{1}^{X}-\varvec{1}^{Y}) = f(X,Y)\) and
which is easily verified from (6.33)–(6.35). Hence follows (6.40).
Next we turn to (6.38) for the maximum in (6.36). The conjugate function \({{\hat{f}}}^{\bullet }{} { isequaltotheindicatorfunctionof}P(f).{ Thatis},\,{{\hat{f}}}^{\bullet }(p){ isequalto}0{ if}p \in P(f),\,{ and}+\infty { otherwise}.{ Theconcaveconjugate}\Psi ^{\circ }{} \) is given by
for \(p \in {\mathbb {R}}^{n}.{ Indeed},\,\mathrm{dom\,}\psi ^{\circ }_{i} = [\alpha _{i}, \beta _{i}]_{{\mathbb {R}}}{} { forall}i \in N{ and},\,{ for}l \in [\alpha _{i}, \beta _{i}]_{{\mathbb {R}}}{} \), we have
from (6.33), (6.34), and (6.35). Therefore, we have
where the variable \(p{ correspondsto}z\) in (6.32).
It remains to show (6.39). The function \({{\hat{f}}} - \overline{\Psi }{} \) is a polyhedral convex function and is bounded from below since the value of (6.41) is finite. Therefore, \({{\hat{f}}} - \Psi { hasaminimizer}.{ Let}{{\hat{x}}} \in {\mathbb {Z}}^{n}{} { beaminimizerof}{{\hat{f}}} - \Psi { with}\Vert {{\hat{x}}} \Vert _{\infty }{} { minimum}.{ Toprovebycontradiction},\,{ assume}\Vert {{\hat{x}}} \Vert _{\infty } \ge 2.{ Define} U = \{ i \in N \mid {{\hat{x}}}_{i} = \Vert {{\hat{x}}} \Vert _{\infty }\}{} { and}V = \{ i \in N \mid {{\hat{x}}}_{i} = -\Vert {{\hat{x}}} \Vert _{\infty }\},\,{ andlet}d = \varvec{1}^{U} - \varvec{1}^{V}{} \). By (6.33)–(6.35), each \(\psi _{i}{} { isalinear}({ affine}){ functiononeachoftheintervals}(-\infty ,-1]{ and}[+1,+\infty ).{ Combiningthiswiththefundamentalpropertyoftheextension}{{\hat{f}}},\,{ weseethatthereexistsavector}q \in {\mathbb {R}}^{n}{} \) for which
holds, where the double-sign corresponds. Since \({{\hat{x}}}{} { isaminimizerof}{{\hat{f}}} - \Psi ,\,{ wemusthave}f(U,V) - \langle q, d \rangle = 0.{ Thisimplies},\,{ however},\,{ that}{{\hat{x}}} - d{ isalsoaminimizerof}{{\hat{f}}} - \Psi ,\,{ whereaswehave}\Vert {{\hat{x}}} - d\Vert _{\infty } < \Vert {{\hat{x}}}\Vert _{\infty }{} \), a contradiction. We have thus completed the derivation of Theorem 6.4 from Theorem 6.1.
Remark 6.2
The min-max formula (6.32) can be derived from the results of [21] as follows. Given \(\alpha , \beta \in {\mathbb {R}}^{n}{} { with}\alpha \le \beta ,\,{ wecanconsiderabisubmodularfunction}w_{\alpha \beta }{} { definedby}w_{\alpha \beta }(X,Y) = \beta (X) - \alpha (Y){ fordisjointsubsets}X{ and}Y.{ Theconvolutionof}f{ with}w = w_{\alpha \beta }{} \) is defined (and denoted) as
This function is bisubmodular [21, Theorem 3.2]. By (6.30) applied to \(f \circ w\), we obtain
On the other hand, Theorem 3.3 of [21] shows
By substituting this expression into \(P(f \circ w)\) on the right-hand side of (6.43) we obtain
The combination of (6.42) and (6.45) gives the desired equality (6.32).
Finally we mention that the paper [21] considers a more general setting where \(\alpha \in ({\mathbb {R}}\cup \{ -\infty \})^{n},\,\beta \in ({\mathbb {R}}\cup \{ +\infty \})^{n},\,{ and}f{ isabisubmodularfunctiondefinedonasubset}{\mathcal {F}}{} { of}3^{N}{} \) such that
The min-max formula (6.32) remains true in this general case. \(\blacksquare \)
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Acknowledgements
The authors are thankful to Satoko Moriguchi and Fabio Tardella for recent joint work on integrally convex functions, and to Satoru Fujishige for a suggestive comment that led to Sect. 6.4. Detailed comments from the referees were helpful to improve the paper. This work was supported by JSPS/MEXT KAKENHI JP20K11697, JP20H00609, and JP21H04979.
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Murota, K., Tamura, A. Recent progress on integrally convex functions. Japan J. Indust. Appl. Math. 40, 1445–1499 (2023). https://doi.org/10.1007/s13160-023-00589-4
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DOI: https://doi.org/10.1007/s13160-023-00589-4
Keywords
- Discrete convex analysis
- Integrally convex function
- Box-total dual integrality
- Fenchel duality
- Integral subgradient
- Minimization