Note on minimization of quasi M$^{\natural }$-convex functions

Abstract

For a class of discrete quasi convex functions called semi-strictly quasi M$^{\natural }$-convex functions, we investigate fundamental issues relating to minimization, such as optimality condition by local optimality, minimizer cut property, geodesic property, and proximity property. Emphasis is put on comparisons with (usual) M$^{\natural }$-convex functions. The same optimality condition and a weaker form of the minimizer cut property hold for semi-strictly quasi M$^{\natural }$-convex functions, while geodesic property and proximity property fail.

1 Introduction

In this paper, we deal with a class of discrete quasi convex functions called semi-strictly quasi M$^{\natural }$-convex functions [2] (see also [10]). The concept of semi-strictly quasi M$^{\natural }$-convex function is introduced as a “quasi convex” version of M$^{\natural }$-convex function [9], which is a major concept in the theory of discrete convex analysis introduced as a variant of M-convex function [5, 6, 8]. Application of (semi-strictly) quasi M$^{\natural }$-convex functions can be found in mathematical economics [2, 11] and operations research [1].

An M-convex function is defined as a function $f:\mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ satisfying a certain exchange axiom (see Sect. 4.2), which implies that the effective domain ${\textrm{dom}\,}f = \{x \in \mathbb {Z}^n \mid f(x) < + \infty \}$ is contained in a hyperplane of the form $\sum _{i=1}^n x(i)= r$ for some $r \in \mathbb {Z}$. Due to this fact, it is natural to consider the projection of an M-convex function to the $(n-1)$-dimensional space along a coordinate axis, which is called an M$^{\natural }$-convex function. A nontrivial argument shows that an M$^{\natural }$-convex function $f:\mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ is characterized by the following exchange axiom:

(M$^{\natural }$-EXC) $\forall x, y \in {\textrm{dom}\,}f$, $\forall i \in \textrm{supp}^{+}(x - y)$, $\exists j \in \textrm{supp}^{-}(x - y)\cup \{0\}$ such that

$$\begin{aligned} f(x) + f(y) \ge f(x - \chi _{i} + \chi _{j}) + f(y + \chi _{i} - \chi _{j}), \end{aligned}$$

(1.1)

where $N = \{1,2,\ldots , n\}$, $\chi _i \in \{0, 1\}^n$ is the characteristic vector of $i \in N$, $\chi _0 = 0$, and

$$\begin{aligned}&\textrm{supp}^{+}(x-y) = \{i \in N \mid x(i) > y(i)\}, \qquad \textrm{supp}^{-}(x-y) = \{j \in N \mid x(j) < y(j)\}. \end{aligned}$$

The inequality (1.1) implies that at least one of the following three conditions holds:

$$\begin{aligned}&f(x - \chi _i + \chi _j)< f(x), \end{aligned}$$

(1.2)

$$\begin{aligned}&f(y + \chi _i - \chi _j) < f(y), \end{aligned}$$

(1.3)

$$\begin{aligned}&f(x - \chi _i + \chi _j) = f(x) \text{ and } f(y + \chi _i - \chi _j) = f(y). \end{aligned}$$

(1.4)

Using this, a semi-strictly quasi M$^{\natural }$-convex function (s.s. quasi M$^{\natural }$-convex function, for short) is defined as follows: $f:\mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ is called an s.s. quasi M$^{\natural }$-convex function if it satisfies the following exchange axiom:

(SSQM$^{\natural }$) $\forall x, y \in {\textrm{dom}\,}f$, $\forall i \in \textrm{supp}^{+}(x - y)$, $\exists j \in \textrm{supp}^{-}(x - y)\cup \{0\}$ satisfying at least one of the conditions (1.2), (1.3), and (1.4).

The main aim of this paper is to investigate fundamental issues relating to minimization of an s.s. quasi M$^{\natural }$-convex function. It is known that minimizers of an M-convex function have various nice properties (to be described in Sect. A.1 of Appendix) such as

$\bullet$ optimality condition by local optimality,

$\bullet$ minimizer cut property,

$\bullet$ geodesic property,

$\bullet$ proximity property.

The definition of M$^{\natural }$-convex function implies that these properties of M-convex functions are inherited by M$^{\natural }$-convex functions, as shown in Sect. 2. In this paper, we examine which of the above properties are satisfied by s.s. quasi M$^{\natural }$-convex functions. For each of the properties, if it holds for s.s. quasi M$^{\natural }$-convex functions, we describe the precise statement of the property in question and give its proof; otherwise, we provide an example to show the failure of the property.

It is added that there is a notion called “s.s. quasi M-convex function” [8, 10], which corresponds directly to M-convexity. Although M-convex and M$^{\natural }$-convex functions are known to be essentially equivalent, it turns out that their quasi-convex versions, namely, s.s. quasi M-convexity and s.s. quasi M$^{\natural }$-convexity, are significantly different. We also discuss such subtle points in Sect. 4.2.

2 Properties on minimization of quasi M$^{\natural }$-convex functions

2.1 Optimality condition by local optimality

In this section we consider an optimality condition for minimization in terms of local optimality and also a minimization algorithm based on the optimality condition. Before dealing with quasi M$^{\natural }$-convex functions, we describe the existing results for M$^{\natural }$-convex functions.

A minimizer $x^*$ of an M$^{\natural }$-convex function f can be characterized by the local minimality within the neighborhood consisting of vectors $y \in \mathbb {Z}^n$ with $\Vert y - x^*\Vert _1 \le 2$.

Theorem 2.1

(cf.  [5, Theorem 2.4], [7, Theorem 2.2]) Let $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ be an M$^{\natural }$-convex function. A vector $x^* \in {\textrm{dom}\,}f$ is a minimizer of f if and only if

$$\begin{aligned} f(x^* - \chi _i + \chi _j) \ge f(x^*) \qquad (i, j \in N\cup \{0\}). \end{aligned}$$

(2.1)

Theorem 2.1 makes it possible to apply the following steepest descent algorithm to find a minimizer of an M$^{\natural }$-convex function.

Algorithm BasicSteepestDescent

Step 0: Let $x_0 \in {\textrm{dom}\,}f$ be an arbitrarily chosen initial vector. Set $x:=x_0$.

Step 1: If $f(x - \chi _{i} + \chi _{j}) \ge f(x)$ for every $i, j \in N\cup \{0\}$, then output x and stop.

Step 2: Find $i, j \in N\cup \{0\}$ that minimize $f(x - \chi _{i} + \chi _{j})$.

Step 3: Set $x:= x - \chi _{i} + \chi _{j}$ and go to Step 1.

Corollary 2.2

(cf.  [12]) For an M$^{\natural }$-convex function $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ with $\arg \min f\ne \emptyset$, Algorithm BasicSteepestDescent finds a minimizer of f in a finite number of iterations.

The optimality condition for M$^{\natural }$-convex functions in Theorem 2.1 can be generalized to s.s. quasi M$^{\natural }$-convex functions.

Theorem 2.3

Let $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ be a function satisfying (SSQM$^{\natural }$). A vector $x^* \in {\textrm{dom}\,}f$ is a minimizer of f if and only if the condition (2.1) holds.

The “only if” part of the theorem is easy to see. The “if” part is implied immediately by the following lemma.

Lemma 2.4

Let $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ be a function satisfying (SSQM$^{\natural }$). For $x, y \in {\textrm{dom}\,}f$, if $f(x) > f(y)$, then there exist some $i\in \textrm{supp}^{+}(x-y)\cup \{0\}$ and $j\in \textrm{supp}^{-}(x-y)\cup \{0\}$ satisfying $f(x)>f(x-\chi _i + \chi _j)$.

Proof

Putting

$$\begin{aligned} \alpha = \sum _{i\in \textrm{supp}^{+}(x-y)}|x(i)-y(i)|, \qquad \beta = \sum _{j\in \textrm{supp}^{-}(x-y)}|x(j)-y(j)|, \end{aligned}$$

we prove the lemma by induction on the pair of values $(\alpha , \beta )$. If $\alpha \le 1$ and $\beta \le 1$, then $y = x - \chi _i + \chi _j$ for some $i,j \in N\cup \{0\}$, and therefore the claim holds immediately.

Suppose $\alpha \ge 2$ and let $i \in \textrm{supp}^{+}(x-y)$. By (SSQM$^{\natural }$) applied to x, y, and i, there exists some $j \in \textrm{supp}^{-}(x-y)\cup \{0\}$ satisfying $f(x - \chi _i + \chi _j) < f(x)$ or $f(y + \chi _i - \chi _j) \le f(y)$ (or both). In the former case, we are done. In the latter case, we can apply the induction hypothesis to x and $y'=y + \chi _i - \chi _j$ to obtain some $i'\in \textrm{supp}^{+}(x-y')\cup \{0\} \subseteq \textrm{supp}^{+}(x-y)\cup \{0\}$ and $j'\in \textrm{supp}^{-}(x-y')\cup \{0\}\subseteq \textrm{supp}^{-}(x-y)\cup \{0\}$ satisfying $f(x)>f(x-\chi _{i'} + \chi _{j'})$. The proof for the case $\beta \ge 2$ is similar. $\square$

It follows from Theorem 2.3 that we can also apply the steepest descent algorithm to find a minimizer of an s.s. quasi M$^{\natural }$-convex function.

Corollary 2.5

For a function $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ with $\arg \min f \ne \emptyset$ satisfying (SSQM$^{\natural }$), Algorithm BasicSteepestDescent finds a minimizer of f in a finite number of iterations.

2.2 Minimizer cut property

The minimizer cut property, originally shown for M-convex functions [12, Theorem 2.2] (see also Theorem A.2 in Appendix), states that a separating hyperplane between a given vector x and some minimizer can be found by using the steepest descent direction at x (i.e., vector $- \chi _{i} + \chi _{j}$ with $i, j \in N$ that minimizes $f(x - \chi _{i} + \chi _{j})$). By rewriting the minimizer cut property for M-convex functions based on the relationship between M-convexity and M$^{\natural }$-convexity, we obtain the following minimizer cut property for M$^{\natural }$-convex functions, where $y(N)= \sum _{i \in N} y(i)$ for $y \in \mathbb {Z}^n$.

Theorem 2.6

(cf.  [12, Theorem 2.2]) Let $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ be an M$^{\natural }$-convex function with $\arg \min f \ne \emptyset$, and $x \in {\textrm{dom}\,}f$ be a vector with $x \not \in \arg \min f$. For a pair (i, j) of distinct elements in $N\cup \{0\}$ minimizing the value $f(x - \chi _i + \chi _j)$, there exists some minimizer $x^*$ of f satisfying

$$\begin{aligned} {\left\{ \begin{array}{ll} x^*(i) \le x(i)-1, \ x^*(j) \ge x(j)+1 &{} (\text{ if } i,j \in N),\\ x^*(i) \le x(i)-1, \ x^*(N) \le x(N) -1 &{} (\text{ if } i \in N,\ j =0),\\ x^*(j) \ge x(j)+1, \ x^*(N) \ge x(N) +1 &{} (\text{ if } i =0,\ j \in N). \end{array}\right. } \end{aligned}$$

(2.2)

Other variants of the minimizer cut property of M$^{\natural }$-convex functions are given in Sect. 4.1, which capture the technical core of Theorem 2.6.

Using Theorem 2.6 we can provide an upper bound for the number of iterations in the following variant of the steepest descent algorithm, where ${\textrm{dom}\,}f$ is assumed to be bounded, and the integer interval $[\ell , u] = \{x \in \mathbb {Z}^n \mid \ell \le x \le u\}$ always contains a minimizer of f.

Algorithm ModifiedSteepestDescent

Step 0::

Let $x_0 \in {\textrm{dom}\,}f$ be an arbitrarily chosen initial vector. Set $x:=x_0$.

Let $\ell , u \in \mathbb {Z}^n$ be vectors such that ${\textrm{dom}\,}f \subseteq [\ell , u]$.

Step 1::

If $f(x - \chi _{i} + \chi _{j}) \ge f(x)$ for every $i, j \in N\cup \{0\}$ with $x - \chi _{i} + \chi _{j} \in [\ell , u]$,

then output x and stop.

Step 2::

Find $i, j \in N\cup \{0\}$ with $x - \chi _{i} + \chi _{j} \in [\ell , u]$ that minimize $f(x - \chi _{i} + \chi _{j})$.

Step 3::

Set $x:= x - \chi _{i} + \chi _{j}$, $u(i):=x(i)- 1$ if $i \in N$, and $\ell (j):=x(j)+1$ if $j \in N$.

Go to Step 1.

We define the L$_\infty$-diameter of a bounded set $S \subseteq \mathbb {Z}^n$ by

$$\begin{aligned} L_\infty (S) = \max \{\Vert x-y\Vert _\infty \mid x, y \in S\}. \end{aligned}$$

Corollary 2.7

(cf.  [12, Section 2]) For an M$^{\natural }$-convex function $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ with a bounded effective domain, Algorithm ModifiedSteepestDescent finds a minimizer of f in O(nL) iterations with $L =L_\infty ({\textrm{dom}\,}f)$.

While the number of iterations in the algorithm ModifiedSteepestDescent is proportional to the L$_\infty$-diameter of ${\textrm{dom}\,}f$, the domain reduction approach in [12] (see Sect. 3.2; see also [8, Section 10.1.3]), combined with the minimizer cut property, makes it possible to speed up the computation of a minimizer.

Corollary 2.8

(cf.  [12, Theorem 3.2]) Let $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ be an M$^{\natural }$-convex function with a bounded effective domain, and suppose that a function evaluation oracle for f and a vector in ${\textrm{dom}\,}f$ are available. Then, a minimizer of f can be obtained in $O(n^4 (\log L)^2)$ time with $L =L_\infty ({\textrm{dom}\,}f)$.

Note that faster polynomial-time algorithms based on the scaling technique are also available for M$^{\natural }$-convex function minimization [13, 15].

An s.s. quasi M$^{\natural }$-convex function satisfies the following weaker statement than Theorem 2.6. To be specific, the inequality $x^*(N) \le x(N) -1$ in the second case ($i \in N, \, j =0$) of (2.2) is missing in (2.3) below, and the inequality $x^*(N) \ge x(N) +1$ in the third case ($i =0,\, j \in N$) of (2.2) is missing in (2.3); an example illustrating this difference is given later in Example 2.12.

Theorem 2.9

Let $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ be a function with (SSQM$^{\natural }$) satisfying $\arg \min f \ne \emptyset$, and $x \in {\textrm{dom}\,}f$ be a vector with $x \not \in \arg \min f$. For a pair (i, j) of distinct elements in $N\cup \{0\}$ minimizing the value $f(x - \chi _i + \chi _j)$, there exists some minimizer $x^*$ of f satisfying

$$\begin{aligned} \left\{ \begin{array}{ll} x^*(i) \le x(i)-1, \ x^*(j) \ge x(j)+1 &{} (\text{ if } i,j \in N),\\ x^*(i) \le x(i)-1 &{} (\text{ if } i \in N,\ j =0),\\ x^*(j) \ge x(j)+1 &{} (\text{ if } i =0,\ j \in N). \end{array} \right. \end{aligned}$$

(2.3)

A proof of this theorem is given in Sect. 3.1.

Using Theorem 2.9 we can obtain the same upper bound for the number of iterations in the algorithm ModifiedSteepestDescent applied to s.s. quasi M$^{\natural }$-convex functions.

Corollary 2.10

For a function $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ with a bounded effective domain satisfying (SSQM$^{\natural }$), Algorithm ModifiedSteepestDescent finds a minimizer of f in O(nL) iterations with $L =L_\infty ({\textrm{dom}\,}f)$.

A combination of Theorem 2.9 with the domain reduction approach (described in Sect. 3.2) makes it possible to find a minimizer of an s.s. quasi M$^{\natural }$-convex function in time polynomial in n and $\log L_\infty ({\textrm{dom}\,}f)$, provided that ${\textrm{dom}\,}f$ is an M$^{\natural }$-convex set. Note that the effective domain of an s.s. quasi M$^{\natural }$-convex function is not necessarily an M$^{\natural }$-convex set (see (3.11) in Sect. 3.2 for the definition of M$^{\natural }$-convex set).

Corollary 2.11

Let $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ be a function satisfying (SSQM$^{\natural }$), and suppose that the effective domain of f is bounded and M$^{\natural }$-convex. Also, suppose that a function evaluation oracle for f and a vector in ${\textrm{dom}\,}f$ are available. Then, a minimizer of f can be obtained in $O(n^4 (\log L)^2)$ time with $L =L_\infty ({\textrm{dom}\,}f)$.

A proof of this corollary is given in Sect. 3.2.

Example 2.12

This example shows that the statement of Theorem 2.6, stronger than Theorem 2.9, is not true for s.s. quasi M$^{\natural }$-convex functions. Consider the function $f: \mathbb {Z}^3 \rightarrow \mathbb {R}\cup \{ +\infty \}$ defined by

$$\begin{aligned}&f(2,1,0)=f(2,0,1) = 0,{} & {} f(1,1,0)=f(1,0,1) = 1,\\&f(0,1,1)=f(0,0,2) = 2,{} & {} f(1,1,1)=f(1,0,2) = 3,\\&f(0,1,2)=4,{} & {} f(x_1, x_2, x_3) = + \infty \text{ otherwise } \end{aligned}$$

(see Fig. 1). This function f satisfies (SSQM$^{\natural }$) and has two minimizers $y^* = (2,1,0)$ and $y^{**}=(2,0,1)$ (denoted by ${\bigcirc }$ in Fig. 1). For $x =(0,1,2)$, the pair $(i,j) = (2, 0)$ is a valid choice in Theorem 2.9, since we have

Fig. 1

Values of function f in Example 2.12

$$\begin{aligned}&x - \chi _i + \chi _j = (0,0,2), \\&f(x - \chi _i + \chi _j) = 2 = \min _{i',j'\in N\cup \{0\}} f(x - \chi _{i'} + \chi _{j'}). \end{aligned}$$

However, neither of the two minimizers $y^* = (2,1,0)$ and $y^{**}=(2,0,1)$ satisfies the inequality $x^*(N) \le x(N)-1$ in (2.2), while $y^{**}=(2,0,1)$ satisfies the inequality $x^{*}(i) \le x(i)-1 = 0$ in (2.3). $\square$

2.3 Geodesic property

Geodesic property of a function $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ means that whenever a vector $x \in {\textrm{dom}\,}f$ moves to a local minimizer $x' \in N(x)$ in an appropriately defined neighborhood N(x) of x, the distance $\Vert x^* - x\Vert$ to a nearest minimizer $x^*$ from the current solution x decreases by $\Vert x' - x\Vert$, where $\Vert \cdot \Vert$ is an appropriately chosen norm. It is known that M-convex functions have the geodesic property with respect to the L$_1$-norm [14, Corollary 4.2], [3, Theorem 2.4]. We first point out that M$^{\natural }$-convex functions do not enjoy the geodesic property with respect to the L$_1$-norm.

For a function $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ and a vector $x \in {\textrm{dom}\,}f$, we define

$$\begin{aligned} \mu (x)&= \min \{\Vert x^* - x \Vert _1 \mid x^* \in \arg \min f\}, \end{aligned}$$

(2.4)

$$\begin{aligned} M(x)&= \{x^* \in \mathbb {Z}^n \mid x^* \in \arg \min f, \ \Vert x^* - x \Vert _1 = \mu (x) \}. \end{aligned}$$

(2.5)

The following is an expected plausible statement of the geodesic property for M$^{\natural }$-convex functions with respect to the L$_1$-norm.

Statement A:    Let $x \in {\textrm{dom}\,}f$ be a vector that is not a minimizer of f. Also, let (i, j) be a pair of distinct elements in $N\cup \{0\}$ minimizing the value $f(x - \chi _i + \chi _j)$, and define

$$\begin{aligned} {M}'= {\left\{ \begin{array}{ll} \{x^* \in M(x) \mid x^*(i) \le x(i)-1,\ x^*(j) \ge x(j)+1 \} &{} (\text{ if } i, j \in N),\\ \{x^* \in M(x) \mid x^*(i) \le x(i)-1 \} &{} (\text{ if } i \in N,\ j=0),\\ \{x^* \in M(x) \mid x^*(j) \ge x(j)+1 \} &{} (\text{ if } i=0,\ j \in N). \end{array}\right. } \end{aligned}$$

• (i) There exists some $x^* \in M(x)$ that is contained in ${M}'$; we have ${M}' \ne \emptyset$, in particular.

• (ii) It holds that

  $$\begin{aligned} \mu (x - \chi _i + \chi _j)&= {\left\{ \begin{array}{ll} \mu (x)-2 &{} (\text{ if } i, j \in N),\\ \mu (x)-1 &{} (\text{ if } i =0 \text{ or } j=0), \end{array}\right. } \\ M(x - \chi _i + \chi _j)&={M}'. \end{aligned}$$

However, neither (i) nor (ii) of Statement A is true for M$^{\natural }$-convex functions, as shown by the following example.

Example 2.13

This example shows that Statement A is not true for M$^{\natural }$-convex functions. Consider a function $f: \mathbb {Z}^2 \rightarrow \mathbb {R}\cup \{ +\infty \}$ given by

$$\begin{aligned}&{\textrm{dom}\,}f = \{x \in \mathbb {Z}^2 \mid 0 \le x(1) \le 2,\ 0 \le x(2) \le 1\},\\&f(x) = 2 - x(1) \qquad (x \in {\textrm{dom}\,}f) \end{aligned}$$

(see Fig. 2), for which $\arg \min f = \{ (2,0),(2,1) \}$. Function f satisfies the condition (M$^{\natural }$-EXC), and hence it is M$^{\natural }$-convex. For $x =(0,1)$, we have

Fig. 2

Values of function f in Example 2.13

$$\begin{aligned} \mu (x) = \Vert (2,1)- (0,1)\Vert _1 = 2, \qquad M(x) = \{(2,1)\}. \end{aligned}$$

We see that $(i,j)=(2,1)$ is a possible choice to minimize the value $f(x - \chi _i + \chi _j)$ among all $i,j \in N\cup \{0\}$. Then, we have

$$\begin{aligned} x'&= x - \chi _i + \chi _j = (1,0), \\ {M}'&= M(x) \cap \{x^* \in \mathbb {Z}^2 \mid x^*(i) \le x(i)-1,\ x^*(j) \ge x(j)+1 \}\\&= \{(2,1)\} \cap \{x^* \in \mathbb {Z}^2 \mid x^*(2) \le 0,\ x^*(1) \ge 1 \} = \emptyset , \\ \mu (x - \chi _i + \chi _j)&= \Vert (2,0) - (1,0)\Vert _1 = 1 \ne 0 = \mu (x) - 2. \end{aligned}$$

That is, Statement A does not hold for this M$^{\natural }$-convex function f. $\square$

While Statement A fails for M$^{\natural }$-convex functions, an alternative geodesic property holds for M$^{\natural }$-convex functions, which can be obtained from the geodesic property for M-convex functions with respect to the L$_1$-norm (see [14, Corollary 4.2], [3, Theorem 2.4]; see also Theorem A.3 in Appendix).

Let $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ be a function with $\arg \min f \ne \emptyset$. For $x \in \mathbb {Z}^n$, we define

$$\begin{aligned} \widetilde{\mu }(x)&= \min \{\Vert x^* - x \Vert _1 + |x^*(N)-x(N)| \mid x^* \in \arg \min f\},\\ \widetilde{M}(x)&= \{x^* \in \mathbb {Z}^n \mid x^* \in \arg \min f, \ \Vert x^* - x \Vert _1 + |x^*(N)-x(N)| = \widetilde{\mu }(x) \}. \end{aligned}$$

That is, $\widetilde{\mu }(x)$ is a kind of distance from x to the nearest minimizer of f, and $\widetilde{M}(x)$ is the set of the minimizers of f nearest to x with respect to this distance.

Theorem 2.14

(cf. [14, Corollary 4.2], [3, Theorem 2.4]) Let $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ be an M$^{\natural }$-convex function with $\arg \min f \ne \emptyset$, and $x \in {\textrm{dom}\,}f$ be a vector that is not a minimizer of f, i.e., $x \not \in \arg \min f$. Also, let (i, j) be a pair of distinct elements in $N\cup \{0\}$ minimizing the value $f(x - \chi _i + \chi _j)$, and define

$$\begin{aligned} \widetilde{M}'= {\left\{ \begin{array}{ll} \{x^* \in \widetilde{M}(x) \mid x^*(i) \le x(i)-1,\ x^*(j) \ge x(j)+1 \} &{} (\text{ if } i, j \in N),\\ \{x^* \in \widetilde{M}(x) \mid x^*(i) \le x(i)-1,\ x^*(N) \le x(N) -1 \} &{} (\text{ if } i \in N,\ j=0),\\ \{x^* \in \widetilde{M}(x) \mid x^*(j) \ge x(j)+1,\ x^*(N) \ge x(N) +1 \} &{} (\text{ if } i=0,\ j \in N). \end{array}\right. }\nonumber \\ \end{aligned}$$

(2.6)

• (i) There exists some $x^* \in \widetilde{M}(x)$ that is contained in $\widetilde{M}'$; we have $\widetilde{M}' \ne \emptyset$, in particular.

• (ii) It holds that $\widetilde{\mu }(x - \chi _i + \chi _j) = \widetilde{\mu }(x)-2$ and $\widetilde{M}(x - \chi _i + \chi _j) = \widetilde{M}'$.

Note that the statement (i) immediately implies the minimizer cut property (Theorem 2.6) for M$^{\natural }$-convex functions. The statement (ii) implies that in each iteration of Algorithm BasicSteepestDescent applied to an M$^{\natural }$-convex function, the distance $\widetilde{\mu }(x)$ to the nearest minimizer reduces by two. This fact yields an exact number of iterations required by BasicSteepestDescent.

Corollary 2.15

Let $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ be an M$^{\natural }$-convex function with $\arg \min f \ne \emptyset$. Suppose that Algorithm BasicSteepestDescent is applied to f with the initial vector $x_0 \in {\textrm{dom}\,}f$. Then, the number of iterations is equal to $\tilde{\mu }(x_0)/2$.

In contrast, s.s. quasi M$^{\natural }$-convex functions do not enjoy the geodesic property in the form of Theorem 2.14, as illustrated in the following example.

Example 2.16

This example shows that the statement of Theorem 2.14 is not true for s.s. quasi M$^{\natural }$-convex functions in the case of $i=0$ or $j=0$. Consider the s.s. quasi M$^{\natural }$-convex function $f: \mathbb {Z}^3 \rightarrow \mathbb {R}\cup \{ +\infty \}$ in Example 2.12 (see Fig. 1), for which $\arg \min f = \{ (2,1,0),(2,0,1) \}$. For $x =(0,1,2)$, we have

$$\begin{aligned} \widetilde{\mu }(x)= & {} \Vert (2,1,0)- (0,1,2)\Vert _1 = \Vert (2,0,1)- (0,1,2)\Vert _1 = 4, \\ \widetilde{M}(x)= & {} \{(2,1,0), (2,0,1)\}. \end{aligned}$$

For $(i,j) = (2,0)$, we have $x - \chi _i + \chi _j=(0,0,2)$ and

$$\begin{aligned} f(x - \chi _i + \chi _j) = 2 = \min _{i',j'\in N\cup \{0\}} f(x - \chi _{i'} + \chi _{j'}). \end{aligned}$$

However, we have

$$\begin{aligned} \widetilde{M}'&= \widetilde{M}(x) \cap \{x^* \in \mathbb {Z}^3 \mid x^*(i) \le x(i)-1,\ x^*(N) \le x(N) -1 \}\\&= \{(2,1,0), (2,0,1)\} \cap \{x^* \in \mathbb {Z}^3 \mid x^*(2) \le 0,\ x^*(N) \le 2 \} = \emptyset , \\ \widetilde{\mu }(x - \chi _i + \chi _j)&= \Vert (2,0,1)- (0,0,2)\Vert _1 = 3 \ne 2 = \widetilde{\mu }(x) -2. \end{aligned}$$

Thus, the statements (i) and (ii) in Theorem 2.14 fail for f. $\square$

2.4 Proximity property

Given a function $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$, $\hat{x} \in {\textrm{dom}\,}f$, and an integer $\alpha \ge 2$, we consider the following scaled minimization problem:

$$\begin{aligned} \hbox {(SP)}\quad \hbox {Minimize }\quad f(x) \qquad \text{ subject } \text{ to } \quad x= \hat{x} + \alpha y,\ y \in \mathbb {Z}^n. \end{aligned}$$

It is expected that an appropriately chosen neighborhood of a global (or local) optimal solution of the scaled minimization problem (SP) contains some minimizer of f; such a property is referred to as a proximity property in this paper. It is known that M-convex functions enjoy a proximity property [4, Theorem 3.4] (see also Theorem A.4 in Appendix), which can be rewritten in terms of M$^{\natural }$-convex functions as follows:

Theorem 2.17

(cf. [4, Theorem 3.4]) Let $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ be an M$^{\natural }$-convex function and $\alpha \ge 2$ be an integer. For every vector ${x} \in {\textrm{dom}\,}f$ satisfying

$$\begin{aligned} f({x}) \le \min \Big [ \min _{i\in N}f({x}\pm \alpha \chi _i),\ \min _{i,j \in N}f({x} - \alpha (\chi _i - \chi _j)) \Big ], \end{aligned}$$

there exists some minimizer $x^*$ of f satisfying

$$\begin{aligned} \Vert x^* - {x} \Vert _\infty \le n (\alpha -1), \qquad |x^*(N) - {x}(N)| \le n (\alpha -1). \end{aligned}$$

This theorem, in particular, implies that for every optimal solution x of (SP), there exists some minimizer $x^*$ of f such that $\Vert x^* - {x} \Vert _\infty \le n (\alpha -1)$.

In contrast, a proximity property of this form does not hold for s.s. quasi M$^{\natural }$-convex functions. Indeed, the following example shows that there exists a family of s.s. quasi M$^{\natural }$-convex functions such that the distance between an approximate global minimizer $\hat{x}$ and a unique exact global minimizer $x^*$ can be arbitrarily large.

Example 2.18

This example shows a function satisfying (SSQM$^{\natural }$) for which the statement of Theorem 2.17 does not hold. With an integer $k \ge 2$, define a function $f: \mathbb {Z}^3 \rightarrow \mathbb {R}\cup \{ +\infty \}$ as follows (see Fig. 3 for the case of $k=3$):

$$\begin{aligned}&{\textrm{dom}\,}f = \{x \in \mathbb {Z}^3 \mid 0 \le x_1 \le k,\ 0 \le x_2 \le 1,\ 0 \le x_3 \le 1 \},\\&f(\lambda , 0,0) = 0 \qquad (0 \le \lambda \le k),\\&f(\lambda , 1,0) = f(\lambda , 0,1) = \lambda -k-1 \qquad (0 \le \lambda \le k),\\&f(\lambda , 1,1) = 2(\lambda -k-1) \qquad (0 \le \lambda \le k). \end{aligned}$$

It can be verified that f satisfies the condition (SSQM$^{\natural }$). Note that the value $f(\lambda ,0,0)$ is constant for every $\lambda$ with $0 \le \lambda \le k$, while $f(\lambda ,1,0), f(\lambda ,0,1)$, and $f(\lambda ,1,1)$ are strictly increasing with respect to $\lambda$ in the interval $0 \le \lambda \le k$. We also see that f has a unique minimizer $y^*=(0,1,1)$. For the problem (SP) with $\hat{x} = (k,0,0)$ and $\alpha = 2$, $\hat{x}$ itself is an optimal solution. The distance between the unique minimizer $y^*$ of f and $\hat{x}$ is $\Vert \hat{x} - y^* \Vert _\infty = k$, which can be arbitrarily large by taking a sufficiently large k. $\square$

Fig. 3

Values of function f in Example 2.18 with $k=3$

3 Proofs

3.1 Minimizer cut property

We derive Theorem 2.9 from the following variants of the minimizer cut property for s.s. quasi M$^{\natural }$-convex functions.

Theorem 3.1

Let $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ be a function with (SSQM$^{\natural }$) satisfying $\arg \min f \ne \emptyset$, and $x \in {\textrm{dom}\,}f$.

• (i) Let $i \in N$ and suppose that the minimum of $f(x - \chi _{i} + \chi _{j'})$ over $j' \in N\cup \{0\}$ is attained by some $j \in N$ $(j \ne 0)$. Then, there exists some minimizer $x^*$ of f satisfying

  $$\begin{aligned} {\left\{ \begin{array}{ll} x^*(j) \ge x(j)+1 &{} (\text{ if } j \in N \setminus \{i\}),\\ x^*(i) \ge x(i) &{} (\text{ if } j =i). \end{array}\right. } \end{aligned}$$

• (ii) Symmetrically to (i), let $j \in N$ and suppose that the minimum of $f(x - \chi _{i'} + \chi _{j})$ over $i' \in N\cup \{0\}$ is attained by some $i \in N$ $(i \ne 0)$. Then, there exists some minimizer $x^*$ of f satisfying

  $$\begin{aligned} {\left\{ \begin{array}{ll} x^*(i) \le x(i)-1 &{} (\text{ if } i \in N \setminus \{j\}),\\ x^*(j) \le x(j) &{} (\text{ if } i =j). \end{array}\right. } \end{aligned}$$

• (iii) Suppose that the minimum of $f(x + \chi _{j'})$ over $j' \in N\cup \{0\}$ is attained by some $j \in N$ $(j \ne 0)$. Then, there exists some minimizer $x^*$ of f satisfying $x^*(j) \ge x(j)+1$.

• (iv) Symmetrically to (iii), suppose that the minimum of $f(x - \chi _{i'})$ over $i' \in N\cup \{0\}$ is attained by some $i \in N$ $(i \ne 0)$. Then, there exists some minimizer $x^*$ of f satisfying $x^*(i) \le x(i)-1$.

While postponing the proof of Theorem 3.1, we first give a proof of Theorem 2.9.

Proof of Theorem 2.9

We first consider the case of $j \in N$ (i.e., $j \ne 0$ and $i \in N\cup \{0\}$). By the choice of j, we have $f(x - \chi _i + \chi _j) = \min _{j' \in N\cup \{0\}} f(x - \chi _i + \chi _{j'})$. Hence, we can apply Theorem 3.1 (i) (if $i \ne 0$) or Theorem 3.1 (iii) (if $i =0$) to obtain some $x^* \in \arg \min f$ such that $x^*(j) \ge x(j)+1$. If $i = 0$ then we are done since $x^*$ satisfies the desired condition (2.3).

We consider the case $i\ne 0$. Let $\tilde{f}: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ be the restriction of f to $D = \{y \in \mathbb {Z}^n \mid y(j) \ge x(j) + 1\}$. Since $x^* \in D \cap \arg \min f$, it holds that

$$\begin{aligned} \min \tilde{f} = \min f, \qquad \arg \min \tilde{f} \subseteq \arg \min f. \end{aligned}$$

We can check easily that $\tilde{f}$ satisfies (SSQM$^{\natural }$) and $i \in N$ satisfies

$$\begin{aligned} \tilde{f}(x - \chi _i + \chi _j) = \min _{i' \in N\cup \{0\}} \tilde{f}(x - \chi _{i'} + \chi _{j}). \end{aligned}$$

Hence, we can apply Theorem 3.1 (ii) to obtain some minimizer $x^{**}$ of $\tilde{f}$ satisfying $x^{**}(i) \le x(i)-1$. This vector $x^{**}$ satisfies $x^{**}(i) \le x(i)-1$ and $x^{**}(j) \ge x(j)+1$ as desired in (2.3). Note that $x^{**}$ is a minimizer of f because $x^{**} \in \arg \min \tilde{f} \subseteq \arg \min f$.

The remaining case (i.e., $i \in N$ and $j = 0$) can be treated similarly by using Theorem 3.1 (iv). $\square$

We now give a proof of Theorem 3.1.

Proof of Theorem 3.1

In the following, we give proofs of (i) and (iii); proofs of (ii) and (iv) can be obtained by applying (i) and (iii) to $g(x) = f(-x)$, respectively.

[Proof of (i)]    Put $x' =x - \chi _i + \chi _j$. It suffices to show that $x^*(j) \ge x'(j)$ holds for some $x^* \in \arg \min f$. Let $x^*$ be a vector in $\arg \min f$ that maximizes the value $x^*(j)$. If $x^*$ satisfies $x^*(j) \ge x'(j)$, then we are done. Hence, we assume, to the contrary, that $x^*(j) < x'(j)$, and derive a contradiction.

The condition (SSQM$^{\natural }$) applied to $x'$, $x^*$, and $j \in \textrm{supp}^{+}(x' - x^*)$ implies that there exists some $r \in \textrm{supp}^{-}(x' - x^*)\cup \{0\}$ such that

$$\begin{aligned}&f(x^*) > f(x^* + \chi _j - \chi _r), \quad \text{ or } \end{aligned}$$

(3.1)

$$\begin{aligned}&f(x') > f(x' - \chi _j + \chi _r), \quad \text{ or } \end{aligned}$$

(3.2)

$$\begin{aligned}&f(x^*) = f(x^* + \chi _j - \chi _r) \text{ and } f(x') = f(x' - \chi _j + \chi _r). \end{aligned}$$

(3.3)

Note that $r \ne j$ holds. Since $x^* \in \arg \min f$, we have

$$\begin{aligned} f(x^*) \le f(x^* + \chi _j - \chi _r). \end{aligned}$$

(3.4)

By the choice of j and $x' =x - \chi _i + \chi _j$, we have

$$\begin{aligned} f(x') \le f(x - \chi _i + \chi _r) = f(x' - \chi _j + \chi _r), \end{aligned}$$

(3.5)

where $x - \chi _i + \chi _r = x' - \chi _j + \chi _r$. The inequalities (3.4) and (3.5) exclude the possibilities of (3.1) and (3.2), respectively. Therefore, we have (3.3). The former equation in (3.3) implies that $x^* + \chi _j - \chi _r$ is also a minimizer of f, a contradiction to the choice of $x^*$ since $(x^* + \chi _j - \chi _r)(j) = x^*(j)+1 > x^*(j)$.

[Proof of (iii)]    The proof given below is similar to that for (i). Put $x' =x + \chi _j$. It suffices to show that $x^*(j) \ge x'(j)$ holds for some $x^* \in \arg \min f$. Let $x^*$ be a vector in $\arg \min f$ that maximizes the value $x^*(j)$. If $x^*$ satisfies $x^*(j) \ge x'(j)$, then we are done. Hence, we assume, to the contrary, that $x^*(j) < x'(j)$, and derive a contradiction.

The condition (SSQM$^{\natural }$) applied to $x'$, $x^*$, and $j \in \textrm{supp}^{+}(x' - x^*)$ implies that there exists some $r \in \textrm{supp}^{-}(x' - x^*)\cup \{0\}$ such that

$$\begin{aligned}&f(x^*) > f(x^* + \chi _j - \chi _r), \quad \text{ or } \end{aligned}$$

(3.6)

$$\begin{aligned}&f(x') > f(x' - \chi _j + \chi _r), \quad \text{ or } \end{aligned}$$

(3.7)

$$\begin{aligned}&f(x^*) = f(x^* + \chi _j - \chi _r) \text{ and } f(x') = f(x' - \chi _j + \chi _r). \end{aligned}$$

(3.8)

Note that $r \ne j$ holds. Since $x^* \in \arg \min f$, we have

$$\begin{aligned} f(x^*) \le f(x^* + \chi _j - \chi _r). \end{aligned}$$

(3.9)

By the choice of j and $x' =x + \chi _j$, we have

$$\begin{aligned} f(x') \le f(x + \chi _r) = f(x' - \chi _j + \chi _r), \end{aligned}$$

(3.10)

where $x + \chi _r = x' - \chi _j + \chi _r$. The inequalities (3.9) and (3.10) exclude the possibilities of (3.6) and (3.7), respectively. Therefore, we have (3.8). The former equation in (3.8) implies that $x^* + \chi _j - \chi _r$ is also a minimizer of f, a contradiction to the choice of $x^*$ since $(x^* + \chi _j - \chi _r)(j) = x^*(j)+1 > x^*(j)$. $\square$

3.2 Domain reduction algorithm

To prove Corollary 2.11, we need to explain the general framework of the domain reduction algorithm for minimization of a function $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ with bounded ${\textrm{dom}\,}f$. For a nonempty bounded set $S \subseteq \mathbb {Z}^n$, we denote

$$\begin{aligned}&\ell (S; i) = \min \{x(i) \mid x \in S \}, \qquad u(S; i) = \max \{x(i) \mid x \in S \} \qquad (i \in N). \end{aligned}$$

We define the peeled set $\widehat{S}$ of S by

$$\begin{aligned} \widehat{S}&= \{x \in S \mid \ell '(i) \le x(i) \le u'(i) \ (i \in N)\},\\ \ell '(i)&= (1-1/n)\ell (S;i) + (1/n)u(S;i) \qquad (i \in N), \\ u'(i)&= (1/n)\ell (S;i) + (1-1/n)u(S;i) \qquad (i \in N). \end{aligned}$$

The outline of the algorithm is described as follows.

Algorithm DomainReduction

Step 0: Let $B:= {\textrm{dom}\,}f$.

Step 1: Find a vector x in the peeled set $\widehat{B}$.

Step 2: If x is a minimizer of f, then output x and stop.

Step 3: Find an axis-orthogonal hyperplane $y(i) = \alpha$ with some $i \in N$ and $\alpha \in \mathbb {Z}$ such that

(Case 1) $\arg \min f \cap \{y \mid y(i) \ge \alpha \} \ne \emptyset$ and $x \notin \{y \mid y(i) \ge \alpha \}$    or

(Case 2) $\arg \min f \cap \{y \mid y(i) \le \alpha \} \ne \emptyset$ and $x \notin \{y \mid y(i) \le \alpha \}$.

Step 4: Set

$$\begin{aligned} B:= {\left\{ \begin{array}{ll} B \cap \{y \mid y(i) \ge \alpha \} &{} (\text{ Case } \text{1 }), \\ B \cap \{y \mid y(i) \le \alpha \} &{} (\text{ Case } \text{2 }) \end{array}\right. } \end{aligned}$$

and go to Step 1.

The algorithm successfully finds a minimizer of f if the following are true in each iteration:

(DR1) the peeled set $\widehat{B}$ in Step 1 is nonempty, and

(DR2) there exists a hyperplane satisfying the desired condition in Step 3.

These conditions hold indeed if f is an s.s. quasi M$^{\natural }$-convex function, as we show later (after Lemma 3.2).

The number of iterations of the algorithm DomainReduction can be analyzed as follows.

Lemma 3.2

(cf. [12, Section 3]) If the conditions (DR1) and (DR2) are satisfied in each iteration of DomainReduction, then the algorithm terminates in $O(n^2 \log L)$ iterations with $L = L_\infty ({\textrm{dom}\,}f)$.

Proof

We provide a proof for completeness. We first note that the set B always contains a minimizer of f. For an iteration of the algorithm, we say that it is of type i if the hyperplane found in Step 3 is of the form $y(i)=\alpha$. In an iteration of type i, the value $u(B;i)- \ell (B;i)$ decreases by $(1/n)(u(B;i)- \ell (B;i))$ by the choices of the vector x in Step 1 and the hyperplane in Step 3. This implies that after $O(n \log L)$ iterations of type i, we have $u(B;i)- \ell (B;i) < 1$, i.e., $u(B;i)= \ell (B;i)$, and therefore an iteration of type i never occurs afterwards. Hence, after $O(n^2 \log L)$ iterations we have $u(B;i)= \ell (B;i)$ for all $i \in N$, i.e., B consists of a single vector, which must be a minimizer of f. $\square$

We now assume that f is an s.s. quasi M$^{\natural }$-convex function (i.e., satisfies (SSQM$^{\natural }$)) such that the effective domain of f is a bounded M$^{\natural }$-convex set, and show that the conditions (DR1) and (DR2) are satisfied in each iteration. The minimizer cut property (Theorem 2.9) guarantees the condition (DR2) for an s.s. quasi M$^{\natural }$-convex function. An axis-orthogonal hyperplane satisfying the condition in Step 3 can be found by evaluating function values $O(n^2)$ times.

The condition (DR1), i.e., the nonemptiness of the peeled set $\widehat{B}$, can be shown as follows. We say that a set $S \subseteq \mathbb {Z}^n$ is an M$^{\natural }$-convex set (see, e.g., [8, Section 4.7]) if it satisfies the following exchange axiom:

$\forall x, y \in S$, $\forall i \in \textrm{supp}^{+}(x - y)$, $\exists j \in \textrm{supp}^{-}(x - y)\cup \{0\}$ such that

$$\begin{aligned} x - \chi _{i} + \chi _{j} \in S, \qquad y + \chi _{i} - \chi _{j} \in S. \end{aligned}$$

(3.11)

It is known that for an M$^{\natural }$-convex set $S \subseteq \mathbb {Z}^n$ and an integer interval $[a, b]\ (\subseteq \mathbb {Z}^n)$, their intersection $S \cap [a,b]$ is again an M$^{\natural }$-convex set if it is nonempty. Hence, the set B in each iteration of the algorithm is always an M$^{\natural }$-convex set as far as it is nonempty. The following lemma shows that the set B is always nonempty.

Lemma 3.3

For a bounded M$^{\natural }$-convex set $B \subseteq \mathbb {Z}^n$, the peeled set $\widehat{B}$ is nonempty.

Proof

The proof can be reduced to a similar statement known for an M-convex set (see, e.g., [8] for the definition of M-convex set). First note that a set $B \subseteq \mathbb {Z}^n$ is an M$^{\natural }$-convex set if and only if the set $S = \{(y, -y(N)) \mid y \in B\}$ $(\subseteq \mathbb {Z}^n \times \mathbb {Z})$ is an M-convex set. For an M$^{\natural }$-convex set $B \subseteq \mathbb {Z}^n$, the peeled set $\widehat{S}$ of the associated M-convex set S is nonempty by [12, Theorem 2.4], whereas $\{y \mid (y, -y(N)) \in \widehat{S}\} \subseteq \widehat{B}$. Therefore, we have $\widehat{B} \ne \emptyset$. $\square$

We note that for a bounded M$^{\natural }$-convex set $B \subseteq \mathbb {Z}^n$, a vector in the peeled set $\widehat{B}$ can be found in $O(n^2 \log L)$ time with $L = L_\infty ({\textrm{dom}\,}f)$ as in [12]. Hence, the condition (DR1) is also satisfied for s.s. quasi M$^{\natural }$-convex functions. By Theorem 2.3, Step 2 can be done in $O(n^2)$ time. This concludes the proof of Corollary 2.11.

4 Concluding remarks

4.1 Remarks on minimizer cut property

We note that, in addition to Theorem 2.6, M$^{\natural }$-convex functions enjoy the following variants of the minimizer cut property, which are similar to, but stronger than, the statements of Theorem 3.1 for s.s. quasi M$^{\natural }$-convex functions. The statements in the theorem below can be obtained from the corresponding statements for M-convex functions [12, Theorem 2.2] (see Theorem A.2 (i), (ii) in Appendix).

Theorem 4.1

(cf.  [12, Theorem 2.2]) Let $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ be an M$^{\natural }$-convex function with $\arg \min f \ne \emptyset$, and $x \in {\textrm{dom}\,}f$.

• (i) Let $i \in N$ and let j be an element in $N\cup \{0\}$ minimizing the value $f(x - \chi _i + \chi _j)$. Then, there exists some minimizer $x^*$ of f satisfying

  $$\begin{aligned} {\left\{ \begin{array}{ll} x^*(j) \ge x(j)+1 &{} (\text{ if } j \in N \setminus \{i\}),\\ x^*(i) \ge x(i) &{} (\text{ if } j =i),\\ x^*(N) \le x(N) -1 &{} (\text{ if } j =0). \end{array}\right. } \end{aligned}$$

• (ii) Symmetrically to (i), let $j \in N$ and let i be an element in $N\cup \{0\}$ minimizing the value $f(x - \chi _i + \chi _j)$. Then, there exists some minimizer $x^*$ of f satisfying

  $$\begin{aligned} {\left\{ \begin{array}{ll} x^*(i) \le x(i)-1 &{} (\text{ if } i \in N \setminus \{j\}),\\ x^*(i) \le x(i) &{} (\text{ if } i =j),\\ x^*(N) \ge x(N) +1 &{} (\text{ if } i =0). \end{array}\right. } \end{aligned}$$

• (iii) Let j be an element in $N\cup \{0\}$ minimizing the value $f(x + \chi _j)$. Then, there exists some minimizer $x^*$ of f satisfying

  $$\begin{aligned} {\left\{ \begin{array}{ll} x^*(j) \ge x(j)+1 &{} (\text{ if } j \in N),\\ x^*(N) \le x(N) &{} (\text{ if } j =0). \end{array}\right. } \end{aligned}$$

• (iv) Symmetrically to (iii), let i be an element in $N\cup \{0\}$ minimizing the value $f(x - \chi _i)$. Then, there exists some minimizer $x^*$ of f satisfying

  $$\begin{aligned} {\left\{ \begin{array}{ll} x^*(i) \le x(i)-1 &{} (\text{ if } i \in N),\\ x^*(N) \ge x(N) &{} (\text{ if } i =0). \end{array}\right. } \end{aligned}$$

It is in order here to dwell on the difference between Theorem 4.1 and Theorem 3.1 by focusing on the statement (i). The statement (i) of Theorem 4.1 for M$^{\natural }$-convex functions covers all possible cases of $j \in N\cup \{0\}$. In contrast, the statement (i) of Theorem 3.1 for s.s. quasi M$^{\natural }$-convex functions puts an assumption that $j \ne 0$ attains the minimum, which means that we can obtain no conclusion if the minimum of $f(x - \chi _{i} + \chi _{j})$ over $j \in N\cup \{0\}$ is attained uniquely by $j = 0$. Thus, the statement (i) of Theorem 3.1 is strictly weaker than the statement (i) of Theorem 4.1.

We present an example to show that the statements (i) and (iii) of Theorem 4.1 in the case of $j=0$ do not hold for s.s. quasi M$^{\natural }$-convex functions.

Fig. 4

Values of function f in Example 4.2

Example 4.2

Here is an example to show that the statements (i) and (iii) of Theorem 4.1 in the case of $j=0$ are not true for s.s. quasi M$^{\natural }$-convex functions. Consider the function $f: \mathbb {Z}^2 \rightarrow \mathbb {R}\cup \{ +\infty \}$ given by¹

$$\begin{aligned}&{\textrm{dom}\,}f= \{(1,0), (2,0), (0,1), (1,1)\},\\&f(1,0) = 1,\ f(2,0)=0,\ f(0,1)=2,\ f(1,1)=3 \end{aligned}$$

(see Fig. 4). Function f satisfies the condition (SSQM$^{\natural }$).

For $x =(1,1)$ and $i = 1$, $j = 0$ minimizes the value $f(x - \chi _1 + \chi _j)$ among all $j \in N\cup \{0\}$. However, the unique minimizer $y^* = (2,0)$ does not satisfy $y^*(N) \le x(N)-1$, i.e., the statement (i) of Theorem 4.1 does not hold in the case of $j=0$.

For $y = (0,1)$, $j=0$ minimizes the value $f(y + \chi _j)$ among all $j \in N\cup \{0\}$. However, the unique minimizer $y^* = (2,0)$ does not satisfy $y^*(N) \le y(N)$, i.e., the statement (iii) of Theorem 4.1 does not hold in the case of $j=0$. $\square$

We can also show that the statements (ii) and (iv) of Theorem 3.1 in the case of $i=0$ do not hold for s.s. quasi M$^{\natural }$-convex functions; a counterexample to the statements is given by the function $g(x) = f(-x)$ with the function f in Example 4.2.

4.2 Connection with quasi M-convex functions

As mentioned in Introduction, the concept of s.s. quasi M-convex function is proposed in [8, 10] as a quasi-convex version of M-convex function. We explain the subtle difference between s.s. quasi M-convexity and s.s. quasi M$^{\natural }$-convexity in this section².

Recall that the condition (SSQM$^{\natural }$) defining s.s. quasi M$^{\natural }$-convexity is obtained by relaxing the condition (M$^{\natural }$-EXC) for M$^{\natural }$-convex functions. Similarly, the concept of s.s. quasi M-convex function is defined by using the relaxed version of the exchange axiom for M-convex functions as follows.

A function $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ is said to be M-convex if it satisfies the following exchange axiom:

(M-EXC) $\forall x, y \in {\textrm{dom}\,}f$, $\forall i \in \textrm{supp}^{+}(x - y)$, $\exists j \in \textrm{supp}^{-}(x - y)$ such that

$$\begin{aligned} f(x) + f(y) \ge f(x - \chi _{i} + \chi _{j}) + f(y + \chi _{i} - \chi _{j}). \end{aligned}$$

(4.1)

A semi-strictly quasi M-convex function is defined as a function satisfying the following relaxed version of (M-EXC):

(SSQM) $\forall x, y \in {\textrm{dom}\,}f$, $\forall i \in \textrm{supp}^{+}(x - y)$, $\exists j \in \textrm{supp}^{-}(x - y)$ satisfying at least one of the three conditions:

$$\begin{aligned}&f(x - \chi _i + \chi _j)< f(x), \end{aligned}$$

(4.2)

$$\begin{aligned}&f(y + \chi _i - \chi _j) < f(y), \end{aligned}$$

(4.3)

$$\begin{aligned}&f(x - \chi _i + \chi _j) = f(x) \text{ and } f(y + \chi _i - \chi _j) = f(y). \end{aligned}$$

(4.4)

Recall also that an M$^{\natural }$-convex function is originally defined as the projection of an M-convex function: an M$^{\natural }$-convex function is defined as a function $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ such that the function $\tilde{f}: \mathbb {Z}^n \times \mathbb {Z}\rightarrow \mathbb {R}\cup \{ +\infty \}$ given by

$$\begin{aligned} \tilde{f}(x, x_0) = {\left\{ \begin{array}{ll} f(x) &{} (\text{ if } x_0= - x(N)),\\ + \infty &{} (\text{ if } x_0 \ne - x(N)) \end{array}\right. } \end{aligned}$$

(4.5)

is M-convex (i.e., satisfies (M-EXC)). Hence, a function f is M$^{\natural }$-convex if and only if it satisfies the following exchange axiom obtained by the projection of (M-EXC):

(M$^{\natural }$-EXC-PRJ) $\forall x, y \in {\textrm{dom}\,}f$,

(i) if $x(N) > y(N)$, then $\forall i \in \textrm{supp}^{+}(x - y)$, $\exists j \in \textrm{supp}^{-}(x - y)\cup \{0\}$ satisfying (4.1),

(ii) if $x(N) \le y(N)$, then $\forall i \in \textrm{supp}^{+}(x - y)$, $\exists j \in \textrm{supp}^{-}(x - y)$ satisfying (4.1),

(iii) if $x(N) < y(N)$, then $\exists j \in \textrm{supp}^{-}(x - y)$ satisfying (4.1) with $i=0$.

That is, the following equivalence holds for M$^{\natural }$-convex functions.

Theorem 4.3

( [9, Theorem 4.2]) For $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$,

$$\begin{aligned} {}&{} \tilde{f}~\textrm{ in }~(4.5)~\textrm{ satisfies}~\mathrm {(M} {-}\mathrm {EXC)} \iff f~\textrm{ satisfies }~\mathrm {(M}^{\natural } {-}\textrm{EXC} {-}\mathrm {PRJ)} \nonumber \\{}&{} \quad \iff f~\textrm{ satisfies}~\mathrm {(M}^{\natural } {-}\mathrm {EXC)}. \end{aligned}$$

(4.6)

For s.s. quasi M$^{\natural }$-convex functions, we may similarly consider the function $\tilde{f}$ in (4.5) and the projected version of (SSQM):

(SSQM$^{\natural }$-EXC-PRJ) $\forall x, y \in {\textrm{dom}\,}f$,

(i) if $x(N) > y(N)$, then $\forall i \in \textrm{supp}^{+}(x - y)$, $\exists j \in \textrm{supp}^{-}(x - y)\cup \{0\}$ satisfying at least one of (4.2), (4.3), and (4.4),

(ii) if $x(N) \le y(N)$, then $\forall i \in \textrm{supp}^{+}(x - y)$, $\exists j \in \textrm{supp}^{-}(x - y)$ satisfying at least one of (4.2), (4.3), and (4.4),

(iii) if $x(N) < y(N)$, then $\exists j \in \textrm{supp}^{-}(x - y)$ satisfying at least one of (4.2), (4.3), and (4.4) with $i=0$.

While the conditions (M$^{\natural }$-EXC-PRJ) and (M$^{\natural }$-EXC) are equivalent to each other, as mentioned in Theorem 4.3, the condition (SSQM$^{\natural }$-EXC-PRJ) is not equivalent to, but strictly stronger than, (SSQM$^{\natural }$) for s.s. quasi M$^{\natural }$-convex functions. That is, we have

(4.7)

in contrast to (4.6). To be more specific, it is easy to see that

$\bullet$ (SSQM$^{\natural }$-EXC-PRJ) (i) and (ii) together imply (SSQM$^{\natural }$),

$\bullet$ (SSQM$^{\natural }$) implies (SSQM$^{\natural }$-EXC-PRJ) (i),

but (SSQM$^{\natural }$) does not imply (ii) and (iii) of (SSQM$^{\natural }$-EXC-PRJ), as shown in the examples below.

Example 4.4

Consider the function $f: \mathbb {Z}^2 \rightarrow \mathbb {R}\cup \{ +\infty \}$ given by

$$\begin{aligned}&{\textrm{dom}\,}f= \{x \in \mathbb {Z}^2 \mid x(1)\ge 0, \ x(2) \ge 0,\ x(1)+x(2) \le 2\},\\&f(0,0)=0,\ f(1,0)=f(0,1)=1,\ f(0,2)=2,\ f(2,0)=f(1,1)=3 \end{aligned}$$

(see Fig. 5). Function f satisfies the condition (SSQM$^{\natural }$). The condition (SSQM$^{\natural }$-EXC-PRJ) (ii) fails for this function. Indeed, for $x=(0,2)$, $y=(2,0)$, and $i=2 \in \textrm{supp}^{+}(x-y)$, we have a unique element $j=1 \in \textrm{supp}^{-}(x-y)$, for which

Fig. 5

Values of function f in Example 4.4

$$\begin{aligned}&x - \chi _i + \chi _j =y + \chi _i - \chi _j = (1,1),\\&f(x) = 2 < 3 = f(x - \chi _i + \chi _j),\qquad f(y) = 3 = f(y + \chi _i - \chi _j). \end{aligned}$$

$\square$

Example 4.5

Consider the function $f: \mathbb {Z}^2 \rightarrow \mathbb {R}\cup \{ +\infty \}$ in Example 4.2, which satisfies (SSQM$^{\natural }$). The condition (iii) of (SSQM$^{\natural }$-EXC-PRJ) fails for this function. Indeed, for $x= (0,1)$ and $y = (2,0)$, we have $x(N)=1 < 2 = y(N)$ and $\textrm{supp}^{-}(x-y)=\{1\}$, but

$$\begin{aligned}&x + \chi _1 = (1,1), \quad y- \chi _1 = (1,0), \\&f(x + \chi _1) = 3> 2 = f(x), \quad f(y - \chi _1) = 1 > 0 = f(y). \end{aligned}$$

We can also verify that $\tilde{f}$ in (4.5) does not satisfy (SSQM), which is consistent with (4.7). $\square$

It is known that an s.s. quasi M-convex function (i.e., function f satisfying (SSQM)) satisfies the minimizer cut property [10, Theorem 4.3] and the proximity property [10, Theorem 4.4]; it can be shown that an s.s. quasi M-convex function also enjoys the geodesic property (see Sect. A.2 in Appendix). It follows from this fact that if a function $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ satisfies the (stronger) condition (SSQM$^{\natural }$-EXC-PRJ), then it satisfies the same statements as in Theorem 2.6 (minimizer cut property), Theorem 2.14 (geodesic property), and Theorem 2.17 (proximity property) for M$^{\natural }$-convex functions. In this connection we emphasize that the s.s. quasi M$^{\natural }$-convex functions in Examples 2.12 and 2.18 do not satisfy (SSQM$^{\natural }$-EXC-PRJ) (iii).

Appendix

1.1 Properties on minimization of M-convex functions

For ease of comparison between M- and M$^{\natural }$-convexity, we recall four theorems for minimization of an M-convex function, which correspond, respectively, to Theorems 2.1, 2.6, 2.14, and 2.17 for an M$^{\natural }$-convex function.

Optimality Condition by Local Optimality

Theorem A.1

( [5, Theorem 2.4], [7, Theorem 2.2]) Let $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ be an M-convex function. A vector $x^* \in {\textrm{dom}\,}f$ is a minimizer of f if and only if

$$\begin{aligned} f(x^* - \chi _i + \chi _j) \ge f(x^*) \qquad (i, j \in N). \end{aligned}$$

The same statement holds also for s.s. quasi M-convex functions [10, Theorem 4.2].

Minimizer Cut Property

Theorem A.2

([12, Theorem 2.2]) Let $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ be an M-convex function with $\arg \min f \ne \emptyset$, and $x \in {\textrm{dom}\,}f$ be a vector with $x \not \in \arg \min f$.

• (i) For $i \in N$, let $j \in N$ be an element minimizing the value $f(x - \chi _i + \chi _j)$. Then, there exists some minimizer $x^*$ of f satisfying

  $$\begin{aligned} {\left\{ \begin{array}{ll} x^*(j) \ge x(j)+1 &{} (if~j \in N \setminus \{i\}),\\ x^*(i) \ge x(i) &{} (if~j=i). \end{array}\right. } \end{aligned}$$

• (ii) Symmetrically, for $j \in N$, let $i \in N$ be an element minimizing the value $f(x - \chi _i + \chi _j)$. Then, there exists some minimizer $x^*$ of f satisfying

  $$\begin{aligned} {\left\{ \begin{array}{ll} x^*(i) \le x(i)-1 &{} (if~i \in N \setminus \{j\}),\\ x^*(j) \le x(j) &{} (if~i =j). \end{array}\right. } \end{aligned}$$

• (iii) For a pair (i, j) of distinct elements in N minimizing the value $f(x - \chi _i + \chi _j)$, there exists some minimizer $x^*$ of f satisfying $x^*(i) \le x(i)-1$ and $x^*(j) \ge x(j)+1$.

The same statement holds also for s.s. quasi M-convex functions [10, Theorem 4.3].

Geodesic Property Recall the definitions of $\mu (x)$ and M(x) in (2.4) and (2.5), respectively.

Theorem A.3

([14, Corollary 4.2], [3, Theorem 2.4]) Let $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ be an M-convex function with $\arg \min f \ne \emptyset$, and $x \in {\textrm{dom}\,}f$ be a vector that is not a minimizer of f. Also, let (i, j) be a pair of distinct elements in N minimizing the value $f(x - \chi _i + \chi _j)$, and define

$$\begin{aligned} {M}'= \{x^* \in M(x) \mid x^*(i) \le x(i)-1,\ x^*(j) \ge x(j)+1 \}. \end{aligned}$$

• (i) There exists some $x^* \in M(x)$ that is contained in ${M}'$; we have ${M}' \ne \emptyset$, in particular.

• (ii) It holds that $\mu (x - \chi _i + \chi _j) = \mu (x)-2$ and $M(x - \chi _i + \chi _j) ={M}'$.

The same statement holds also for s.s. quasi M-convex functions; see Sect. A.2 for a proof.

Proximity Property

Theorem A.4

([4, Theorem 3.4], [10, Theorem 4.4]) Let $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ be an M-convex function and $\alpha \ge 2$ be an integer. For every vector ${x} \in {\textrm{dom}\,}f$ satisfying

$$\begin{aligned} f({x}) \le \min _{i,j \in N}f({x} - \alpha (\chi _i - \chi _j)), \end{aligned}$$

there exists some minimizer $x^*$ of f such that $\Vert x^* - {x} \Vert _\infty \le (n-1) (\alpha -1)$.

The same statement holds also for s.s. quasi M-convex functions [10, Theorem 4.4].

1.2 Proof of geodesic property for quasi M-convex functions

We give a proof for the following geodesic property for semi-strictly quasi M-convex functions.

Theorem A.5

Let $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ be a function with $\arg \min f \ne \emptyset$ satisfying (SSQM) (i.e., f is semi-strictly quasi M-convex), and $x \in {\textrm{dom}\,}f$ be a vector that is not a minimizer of f. Also, let (i, j) be a pair of distinct elements in N minimizing the value $f(x - \chi _i + \chi _j)$, and define

$$\begin{aligned} {M}'= \{x^* \in M(x) \mid x^*(i) \le x(i)-1,\ x^*(j) \ge x(j)+1 \}. \end{aligned}$$

• (i) There exists some $x^* \in M(x)$ that is contained in ${M}'$; we have ${M}' \ne \emptyset$, in particular.

• (ii) It holds that $\mu (x - \chi _i + \chi _j) = \mu (x)-2$ and $M(x - \chi _i + \chi _j) ={M}'$.

The proof given below is essentially the same as the one in [3] for Theorem A.3 on M-convex functions. In the proof of Theorem A.5 (i) we use the following lemma.

Lemma A.6

Let $f: \mathbb {Z}^n \rightarrow \mathbb {R}\cup \{ +\infty \}$ be a function with $\arg \min f \ne \emptyset$ satisfying (SSQM). Assume that $x \in {\textrm{dom}\,}f$ is not a minimizer of f, and let (i, j) be a pair of distinct elements in N minimizing the value $f(x - \chi _i + \chi _j)$. Define $y = x - \chi _i + \chi _j$.

• (i) For every $x^* \in \arg \min f$ with $i \in \textrm{supp}^{+}(x^* - y)$, there exists some $h \in \textrm{supp}^{-}(x^* - y)$ with $h \ne j$ such that $x^* - \chi _i + \chi _h \in \arg \min f$.

• (ii) For every $x^* \in \arg \min f$ with $j \in \textrm{supp}^{-}(x^* - y)$, there exists some $k \in \textrm{supp}^{+}(x^* - y)$ with $k \ne i$ such that $x^* + \chi _j - \chi _k \in \arg \min f$.

Proof

We prove (i) only since (ii) can be proven similarly. We first note that $f(y) < f(x)$ holds since x is not a minimizer [10, Theorem 4.2]. By definition, f satisfies the condition (SSQM), which, applied to $x^*, y$ and $i \in \textrm{supp}^{+}(x^* - y)$, implies that there exists some $h \in \textrm{supp}^{-}(x^* - y)$ such that at least one of the following conditions holds:

$$\begin{aligned}&f(x^* - \chi _i + \chi _h)< f(x^*), \end{aligned}$$

(A.1)

$$\begin{aligned}&f(y + \chi _i - \chi _h) < f(y), \end{aligned}$$

(A.2)

$$\begin{aligned}&f(x^* - \chi _i + \chi _h)= f(x^*) \text{ and } f(y + \chi _i - \chi _h) = f(y). \end{aligned}$$

(A.3)

Note that $h \ne i$ holds. We have

$$\begin{aligned} f(x^* - \chi _i + \chi _h) \ge f(x^*) \end{aligned}$$

(A.4)

since $x^* \in \arg \min f$. By the choice of $i, j \in N$, we have

$$\begin{aligned} f(y + \chi _i - \chi _h) \ge f(y), \end{aligned}$$

(A.5)

where $y + \chi _i - \chi _h = x - \chi _h + \chi _j$. The inequalities (A.4) and (A.5) exclude the possibility of (A.1) and (A.2), respectively. Therefore, we have (A.3). The former equation in (A.3) implies that $x^* - \chi _i + \chi _h$ is also a minimizer of f. We also have $h \ne j$, which follows from the latter equation in (A.3) and the inequality $f(y) < f(x)$ since $y + \chi _i - \chi _h=x - \chi _h + \chi _j$. $\square$

Proof of Theorem A.5 (i)

Putting $y = x - \chi _i + \chi _j$, we have $f(y)< f(x)$. We first show that there exists some $y^* \in M(x)$ such that $y^*(i) \le x(i) -1$. Assume, to the contrary, that $y^*(i) > x(i) - 1 =y(i)$ holds for every $y^* \in M(x)$. Let $y^* \in M(x)$ be a vector minimizing the value $y^*(i)$ among all such vectors. By Lemma A.6 (i), there exists some $h \in \textrm{supp}^{-}(y^* - y)$ with $h \ne j$ such that $y^* - \chi _i + \chi _h \in \arg \min f$. Since $y^*(h) < y(h) = x(h)$, we have $\Vert (y^* - \chi _i + \chi _h) - x\Vert _1 \le \Vert y^* - x\Vert _1$. Since $y^* \in M(x)$ and $y^* - \chi _i + \chi _h \in \arg \min f$, it follows that $y^* - \chi _i + \chi _h \in M(x)$. This, however, is a contradiction to the choice of $y^*$ since $(y^* - \chi _i + \chi _h)(i) = y^*(i) - 1 < y^*(i)$.

We then show that there exists some $x^* \in M(x)$ satisfying both of $x^*(i) \le x(i) -1$ and $x^*(j) \ge x(j) + 1$. Let $x^* \in M(x)$ be a vector with $x^*(i) \le x(i) - 1$. If there exists such $x^*$ with $x^*(j) \ge x(j) + 1$, then we are done. Hence, we assume, to the contrary, that $x^*(j) < x(j) + 1 = y(j)$ for every $x^* \in M(x)$ with $x^*(i) \le x(i) - 1$, and suppose that $x^*$ maximizes the value $x^*(j)$ among all such $x^*$. By Lemma A.6 (ii), there exists some $k \in \textrm{supp}^{+}(x^* - y)$ with $k \ne i$ such that $x^* + \chi _j - \chi _k \in \arg \min f$. We show that $x^* + \chi _j - \chi _k \in M(x)$ holds. Since $x^*(k) > y(k) = x(k)$, we have $\Vert (x^* + \chi _j - \chi _k) - x\Vert _1 \le \Vert x^* - x\Vert _1$. We also have $(x^* + \chi _j - \chi _k)(i) = x^*(i) \le x(i) - 1$ since $i \notin \{j, k\}$. This, however, is a contradiction to the choice of $x^*$ since $(x^* + \chi _j - \chi _k)(j) = x^*(j) + 1 > x^*(j)$. Hence, we have $x^*(j) \ge x(j) + 1$.

This concludes the proof of Theorem A.5 (i). $\square$

Proof of Theorem A.5 (ii)

We first prove the equation $\mu (x - \chi _i + \chi _j) = \mu (x) - 2$. It holds that

$$\begin{aligned} \forall y^* \in M(x - \chi _i + \chi _j): \quad \mu (x - \chi _i + \chi _j)&= \Vert y^* - (x - \chi _i + \chi _j)\Vert _1 \nonumber \\&\ge \Vert y^* - x\Vert _1 - \Vert \chi _i - \chi _j\Vert _1 \nonumber \\&= \Vert y^* - x\Vert _1 - 2 \ge \mu (x) - 2, \end{aligned}$$

(A.6)

where the first inequality is by the triangle inequality. We also have

$$\begin{aligned} \forall x^* \in M'&:\mu (x - \chi _i + \chi _j) \le \Vert x^* - (x - \chi _i + \chi _j)\Vert _1\nonumber \\ {}&= \Vert x^* - x\Vert _1 - 2 = \mu (x) - 2, \end{aligned}$$

(A.7)

where the first equality is by the inequalities $x^*(i) \le x(i) - 1,\ x^*(j) \ge x(j) + 1$ and the second equality is by $x^* \in M(x)$. The equation $\mu (x - \chi _i + \chi _j) = \mu (x) - 2$ follows from (A.6) and (A.7).

We then prove the equation $M(x - \chi _i + \chi _j) ={M}'$. It follows from $\mu (x - \chi _i + \chi _j) = \mu (x) - 2$, (A.6), and (A.7) that all the inequalities in (A.6) and (A.7) hold with equality; in particular, we have

$$\begin{aligned}&\forall y^* \in M(x - \chi _i + \chi _j): \quad \Vert y^* - (x - \chi _i + \chi _j)\Vert _1 = \Vert y^* - x\Vert _1 - 2 = \mu (x) - 2, \end{aligned}$$

(A.8)

$$\begin{aligned}&\forall x^* \in M': \quad \mu (x - \chi _i + \chi _j) = \Vert x^* - (x - \chi _i + \chi _j)\Vert _1. \end{aligned}$$

(A.9)

The equation (A.9) implies that $M(x - \chi _i + \chi _j) \supseteq M'$. We can also obtain the reverse inclusion $M(x - \chi _i + \chi _j) \subseteq M'$ from (A.8). Indeed, for every $y^* \in M(x - \chi _i + \chi _j)$, we have $y^* \in M(x)$ by $\Vert y^* - x\Vert _1 = \mu (x)$, and we also have $y^*(i) \le x(i) - 1$ and $y^*(j) \ge x(j) + 1$ since $\Vert y^* - (x - \chi _i + \chi _j)\Vert _1 = \Vert y^* - x\Vert _1 - 2$. Hence, $M(x - \chi _i + \chi _j) \subseteq M'$ holds, implying the equation $M(x - \chi _i + \chi _j) ={M}'$. $\square$