Directed discrete midpoint convexity

For continuous functions, midpoint convexity characterizes convex functions. By considering discrete versions of midpoint convexity, several types of discrete convexities of functions, including integral convexity, L♮\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^\natural$$\end{document}-convexity and global/local discrete midpoint convexity, have been studied. We propose a new type of discrete midpoint convexity that lies between L♮\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^\natural$$\end{document}-convexity and integral convexity and is independent of global/local discrete midpoint convexity. The new convexity, named DDM-convexity, has nice properties satisfied by L♮\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^\natural$$\end{document}-convexity and global/local discrete midpoint convexity. DDM-convex functions are stable under scaling, satisfy the so-called parallelogram inequality and a proximity theorem with the same small proximity bound as that for L♮\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{\natural }$$\end{document}-convex functions. Several characterizations of DDM-convexity are given and algorithms for DDM-convex function minimization are developed. We also propose DDM-convexity in continuous variables and give proximity theorems on these functions.


Introduction
For a continuous function f defined on a convex set S ⊆ ℝ n , it was proved by Jensen [12] that midpoint convexity defined by is equivalent to the inequality defining convex functions By capturing the concept of midpoint convexity, several types of 'discrete' midpoint convexities for functions defined on the integer lattice ℤ n have been proposed. A weak version of 'discrete' midpoint convexity is obtained by replacing f ((x + y)∕2) by the smallest value of a linear extension of f among the integer points neighboring (x + y)∕2 . More precisely, for any point x ∈ ℝ n , we consider its integer neighborhood and the set Λ(x) of all coefficients ( z | z ∈ N(x)) for convex combinations indexed by N(x). For a function f ∶ ℤ n → ℝ ∪ {+∞} , we define the local convex envelope f of f by We say that f satisfies weak discrete midpoint convexity if the following inequality holds for all x, y ∈ ℤ n . On the other hand, f is said to be integrally convex [3] if f is convex on ℝ n . Characterizations of integral convexity by using weak discrete midpoint convexity have been discussed in [3,17,18]. The simplest characterization, Theorem A.1 in [18], says that f is integrally convex if and only if f satisfies (1.1) for all x, y ∈ dom f with 1 ‖x − y‖ ∞ ≥ 2 , where the effective domain dom f of f is defined by The class of integrally convex functions establishes a general framework of discrete convex functions, including separable convex, L ♮ -convex, M ♮ -convex, L ♮ 2 -convex, M ♮ 2 -convex functions [21], BS-convex and UJ-convex functions [4], and globally/ locally discrete midpoint convex functions [18]. The concept of integral convexity is used in formulating discrete fixed point theorems [8,9,29], designing algorithms for discrete systems of nonlinear equations [14,28], and guaranteeing the existence of a pure strategy equilibrium in finite symmetric games [10].
A strong version of 'discrete' midpoint convexity is obtained by replacing f ((x + y)∕2) by the average of the values of f at two integer points obtained by rounding-up and rounding-down of all components of (x + y)∕2 . More precisely, for a function f ∶ ℤ n → ℝ ∪ {+∞} , we say that f satisfies discrete midpoint convexity if it has f (x) + (1 − )f (y) ≥ f ( x + (1 − )y) (∀x, y ∈ S; ∀ ∈ [0, 1]).

3
Directed discrete midpoint convexity for all x, y ∈ ℤ n , where ⌈⋅⌉ and ⌊⋅⌋ denote the integer vectors obtained by rounding up and rounding down all components of a given real vector, respectively. It is known that discrete midpoint convexity characterizes the class of L ♮ -convex functions [5,20] which play important roles in both theoretical and practical aspects. L ♮ -convex functions are applied to several fields, including auction theory [15,25], image processing [13], inventory theory [2,27,30] and scheduling [1]. Since discrete midpoint convexity (1.2) obviously implies weak discrete midpoint convexity (1.1), L ♮ -convex functions forms a subclass of integrally convex functions.
Moriguchi et al. [18] classified discrete convex functions between L ♮ -convex and integrally convex functions in terms of discrete midpoint convexity with ∞ -distance requirements, and proposed two new classes of discrete convex functions, namely, globally/locally discrete midpoint convex functions. A function f ∶ ℤ n → ℝ ∪ {+∞} is said to be globally discrete midpoint convex if (1.2) holds for any pair (x, y) ∈ ℤ n × ℤ n with ‖x − y‖ ∞ ≥ 2 . A set S ⊆ ℤ n is called a discrete midpoint convex set if its indicator function S defined by is globally discrete midpoint convex, that is, if A function f ∶ ℤ n → ℝ ∪ {+∞} is said to be locally discrete midpoint convex if dom f is a discrete midpoint convex set and (1.2) holds for any pair (x, y) ∈ ℤ n × ℤ n with ‖x − y‖ ∞ = 2 . It is shown in [18] that the following inclusion relations among function classes hold: and globally/locally discrete midpoint convex functions inherit nice features from L ♮ -convex functions, that is, for a globally/locally discrete midpoint convex f and a positive integer , • the scaled function f defined by f (x) = f ( x) (x ∈ ℤ n ) belongs to the same class, that is, global/local discrete midpoint convexity is closed with respect to scaling operations, • a proximity theorem with the same proximity distance with L ♮ -convexity holds, that is, given an x with f (x ) ≤ f (x + d) for all d ∈ {−1, 0, +1} n , there exists a minimizer x * of f with ‖x − x * ‖ ∞ ≤ n( − 1), • when f has a minimizer, a steepest descent algorithm for the minimization of f is developed such that the number of local minimizations in the neighborhood {L ♮ -convex } ⫋ { globally discrete midpoint convex } ⫋ { locally discrete midpoint convex } ⫋ { integrally convex }, 1 3 of ∞ -distance 2 (the 2-neighborhood minimizations) is bounded by the shortest ∞ -distance from a given initial feasible point to a minimizer of f, and • when dom f is bounded and K ∞ denotes the ∞ -size of dom f , a scaling algorithm minimizing f with O(n log 2 K ∞ ) calls of the 2-neighborhood minimization is developed.
This paper, strongly motivated by [18], proposes a new type of discrete midpoint convexity between L ♮ -convexity and integral convexity, but it is independent of global/local discrete midpoint convexity with respect to inclusion relation. We name the new convexity directed discrete midpoint convexity (DDM-convexity) which forms the following classification The same features as mentioned above are satisfied by DDM-convexity. The merits of DDM-convexity relative to global/local discrete midpoint convexity are the following properties: • DDM-convexity is closed with respect to individual sign inversion of variables, that is, for a DDM-convex function f and (3)). Neither L ♮ -convexity nor global nor local discrete midpoint convexity has this property, while integral convexity is closed with respect to individual sign inversion of variables. • For a quadratic function f (x) = x ⊤ Qx with a symmetric matrix Q = [q ij ] , DDMconvexity is characterized by the diagonal dominance with nonnegative diagonals of Q: (see Theorem 9). While L ♮ -convexity is characterized by the combination of diagonal dominance with nonnegative diagonals and nonpositivity of all offdiagonal components of Q, global/local discrete midpoint convexity is independent of the diagonal dominance with nonnegative diagonals.
, a 2-separable convex function [7] defined as a function represented as The class of DDM-convex functions includes all 2-separable convex functions (see Theorem 4). It is known that if all ij are identically zero, then f is L ♮ -convex, whereas there exists a 2-separable convex function not contained in the class of globally/locally discrete midpoint convex functions.

3
Directed discrete midpoint convexity • A steepest descent algorithm for the minimization of DDM-convex functions requires only the 1-neighborhood minimization in contrast to the 2-neighborhood minimization (see Sect. 8.1).
In the next section, we give the definition of DDM-convexity and basic properties of DDM-convex functions. In Sect. 3, we discuss a relationship between DDMconvexity and known discrete convexities. For globally/locally discrete midpoint convex functions, Moriguchi et al. [18] revealed a useful property, which is expressed by the so-called parallelogram inequality. We show that a similar parallelogram inequality holds for DDM-convex functions in Sect. 4. Sections 5 and 6 are devoted to characterizations and operations for DDM-convexity. We prove a proximity theorem for DDM-convex functions in Sect. 7, while in Sect. 8 we propose a steepest descent algorithm and a scaling algorithm for DDM-convex function minimization. In Sect. 9, we define DDM-convex functions in continuous variables and give proximity theorems for such functions.

Directed discrete midpoint convexity
We give the definition of directed discrete midpoint convexity and show its basic properties.
For an ordered pair (x, y) of x, y ∈ ℤ n , we define (x, y) ∈ ℤ n by That is, each component (x, y) i of (x, y) is defined by rounding up or rounding down x i +y i 2 to the integer in the direction of x i − y i . It is easy to show the next characterization of (x, y) and (y, x). Proposition 1 For x, y, p, q ∈ ℤ n , p = (x, y) and q = (y, x) hold if and only if the following conditions (a)∼(c) hold: For every a, b ∈ ℝ n , let us denote the n-dimensional vector (a 1 b 1 , … , a n b n ) by a ⊙ b . The next proposition gives fundamental properties of (⋅, ⋅).

Proposition 2
For every x, y, d ∈ ℤ n , the following properties hold.

holds. ◻
By using the introduced (⋅, ⋅) , we propose new classes of functions and sets. We say that a function f ∶ ℤ n → ℝ ∪ {+∞} satisfies directed discrete midpoint convexity (DDM-convexity) or is a directed discrete midpoint convex function (DDM-convex function) if for all x, y ∈ ℤ n . We call S ⊆ ℤ n a directed discrete midpoint convex set (DDM-convex set) if its indicator function S is DDM-convex, that is, if holds.
The next propositions are direct consequences of Proposition 2 and the definition (2.1).

Proposition 3 The following statements hold:
(1) Any function defined on {0, 1} n is a DDM-convex function.

Relationships with known discrete convexities
We discuss relationships between DDM-convexity and known discrete convexities, including integral convexity, L ♮ -convexity, global/local discrete midpoint convexity and 2-separable convexity.
As mentioned in Sect. 1, the class of integrally convex functions is characterized by weak discrete midpoint convexity (1.1). Since DDM-convexity (2.1) trivially implies (1.1), any DDM-convex function is integrally convex. Therefore, DDM-convex functions inherit many properties of integrally convex functions. We introduce a good property of integrally convex functions as well as DDM-convex functions, box-barrier property.
By setting p =x − 1 and q =x + 1 where 1 denotes the vector of all ones, boxbarrier property implies the minimality criterion of integrally convex functions.
As a special case of Theorem 2, we have the minimality criterion of DDM-convex functions.
We next discuss the relationship between L ♮ -convexity and DDM-convexity. L ♮ -convex functions are originally defined by translation-submodularity: for all x, y ∈ ℤ n and nonnegative integer , where p ∨ q and p ∧ q denote the componentwise maximum and minimum of the vectors p and q, respectively. Translation-submodularity is a generalization of submodularity: L ♮ -convexity has several equivalent characterizations as below. (1) f is L ♮ -convex, that is, (3.1) holds for all x, y ∈ ℤ n and nonnegative integer .
(3) f is integrally convex and submodular. (4) For every x, y ∈ ℤ n with x ≱ y and A = argmax

3
Directed discrete midpoint convexity Theorem 3 yields the next property.

Proposition 6 Any L ♮ -convex function is DDM-convex.
Proof Let f ∶ ℤ → ℝ ∪ {+∞} be an L ♮ -convex function. We arbitrarily fix x, y ∈ dom f and show that (2.1) holds for x and y. By Proposition 2 (3), We next discuss the independence between global/local discrete midpoint convexity and DDM-convexity by showing the independence between discrete midpoint convex sets and DDM-convex sets.

Example 3
It is easy to show that the set S defined by is discrete midpoint convex, but S is not DDM-convex because for x = (0, 0, 0) and y = (2, 1, −1) , we have (x, y) = (1, 0, 0) ∉ S and (y, On the other hand, is DDM-convex. However, T is not discrete midpoint convex, and moreover, for any is not discrete midpoint convex while it is DDM-convex by Proposition 5 (3). The reason is as follows. Since T is symmetric on the third component, we can assume [7] is defined as a function represented as It is known that the function g defined by is L ♮ -convex [21, Proposition 7.9]. By Propositions 4 (4) and 6, it is enough to show that each ij is DDM-convex in order to prove DDM-convexity of 2-separable convex function

Parallelogram inequality
Parallelogram inequality was originally proposed in [18] for globally/locally discrete midpoint convex functions. By borrowing arguments from [18], we show that DDMconvex sets/functions have similar properties. For every pair (x, y) ∈ ℤ n × ℤ n with ‖y − x‖ ∞ = m , we consider sets defined by We first show the following property of DDM-convex sets.
To show this theorem, it is enough to verify For every x ∈ ℤ n , let us consider multiset D(x) of vectors by the following recursive formula: Directed discrete midpoint convexity where 0 denotes the n-dimensional zero vector. We give several propositions.
Proof We prove the assertion by induction on m. The assertion obviously holds if m ≤ 1.
Furthermore, the claim below guarantees that By combining (4.3), (4.6) and (4.7), we have Proof We show (i) (and can show (ii) in the same way). Let us fix i ∈ {1, … , n} .
In the case where � ∑ Thus, (i) holds. (End of the proof of Claim). ◻ , condition (4.4) of Proposition 7 holds. The assertion is an immediate consequence of (4.5). ◻ By Proposition 8, (4.2) can be rewritten as Therefore, Theorem 5 can be shown by the following proposition.
Theorem 6 Let f ∶ ℤ n → ℝ ∪ {+∞} be a function in DDMC (2) such that dom f is DDM-convex. For x ∈ dom f and y ∈ ℤ n with ‖y − x‖ ∞ = m , and for any partition (I, J) of {1, … , m} , we consider where A k , B k (k = 1, … , m) are the sets defined by (4.1). Then we have Proof We note that y = x + d 1 + d 2 . If y ∉ dom f , by f (y) = +∞ , (4.11) trivially holds. In the sequel, we assume that y ∈ dom f . Let I be denoted by For every k = 0, 1, … , |I| and for every l = 0, 1, … , |J| , define (4.9) By Theorem 5, for every k, l, we have x(k, l) ∈ dom f . We note that (4.11) is equivalent to Fix k ∈ {1, … , |I|} and l ∈ {1, … , |J|} . According to whether i k > j l or i k < j l , either or Thus, in the case where i k > j l , we have (2), we obtain On the other hand, the facts in the both cases where i k > j l and i k < j l , yield the right-hand side of (4.13) is equal to f (x(k, l − 1)) + f (x (k − 1, l)) . Therefore, we obtain By adding the above inequalities for (k, l) with 1 ≤ k ≤ |I| and 1 ≤ l ≤ |J| , we obtain (4.12). We emphasize that all the terms that are canceled in this addition of inequalities are finite valued because x(k, l) ∈ dom f for all (k, l) with 0 ≤ k ≤ |I| and 0 ≤ l ≤ |J| . ◻ The next theorem is an immediate consequence of Theorem 6, because a DDM-convex function f ∶ ℤ n → ℝ ∪ {+∞} belongs to DDMC(2) and dom f is DDM-convex. − 1, l)).

Then we have
We call the inequality (4.14) parallelogram inequality of DDM-convex functions.

Characterizations
In this section, we give several equivalent conditions of DDM-convexity and a simple characterization of quadratic DDM-convex functions.
For every pair (x, y) ∈ ℤ n × ℤ n , we recall that the families {A k | k = 1, … , m} and

Theorem 8
For a function f ∶ ℤ n → ℝ ∪ {+∞} , the following properties are equivalent to each other.
and q = y − 1 A m + 1 B m in (5.1). Vectors p and q satisfy conditions (a) and (c) of Proposition 1, ‖p − q‖ ∞ < m and p, q ∈ dom f . By repeating (5.1) for (p, q) until ‖p − q‖ ∞ ≤ 1 , the final p and q satisfy all conditions of Proposition 1, and hence, p = (x, y) and q = (y, x) are satisfied. This shows f ∈ DDMC(m) holds.
On the other hand, in (3.1), two vectors (x − 1) ∨ y and x ∧ (y + 1) can be rewritten as

Directed discrete midpoint convexity
In the same way as the relation between (4) of Theorem 8 and (4) of Theorem 3, two separate operations for L ♮ -convex functions must be executed simultaneously for DDM-convex functions. ◻ For a quadratic function f (x) = x ⊤ Qx (x ∈ ℤ n ) with a symmetric matrix Q = [q ij ] , we show that f is DDM-convex if and only if Q is diagonally dominant with nonnegative diagonals: For each p ∈ ℝ , let p + = max{p, 0} and p − = max{−p, 0} . Note that |p| = p + + p − . Quadratic function f (x) = x ⊤ Qx can be written as 2 In (5.3), the condition (5.2) of Q implies the nonnegativity of coefficients of x 2 i . Thus, if Q is diagonally dominant with nonnegative diagonals, then f is 2-separable convex, and hence, DDM-convex by Theorem 4. By proving the opposite implication, we obtain the following property.

is DDM-convex if and only if Q is diagonally dominant with nonnegative diagonals.
Proof It is enough to show that if f is DDM-convex, then Q is diagonally dominant with nonnegative diagonals. For each i ∈ {1, … , n} , define z i ∈ ℤ n by By DDM-convexity of f, the inequality must hold. Since (z i , 0) = z i − 1 i and (0, z i ) = 1 i , we have

3 which implies
By Q ⊤ = Q , we obtain the diagonal dominance with nonnegative diagonals of Q. ◻ The minimizers of DDM-convex functions are DDM-convex sets, while the minimizers of L ♮ -convex functions are L ♮ -convex sets. The class of L ♮ -convex functions has a characterization in terms of minimizers. For a function f ∶ ℤ n → ℝ ∪ {+∞} and p ∈ ℝ n , we denote by f − p the function given by

Theorem 10 ([21, 23]) Under some regularity condition, a function
Unfortunately, the class of DDM-convex functions does not have a similar characterization.

Example 4 Let us consider the function f ∶ ℤ 3 → ℝ ∪ {+∞} given by
The function f is not DDM-convex because while dom f is a DDM-convex set (in fact, an L ♮ -convex set). Furthermore, argmin(f − p) is a DDM-convex set for every p ∈ ℝ 3 as follows. There exists no p ∈ ℝ 3 such that {(0, 0, 0), (2, 1, 1)} ⊆ argmin(f − p) , because we have (1, 0, 1) . For any p ∈ ℝ 3 and for any x, y ∈ argmin(f − p) , this fact implies that ‖x − y‖ ∞ ≤ 1 must hold, and hence, argmin(f − p) is a DDMconvex set. We note that this example also shows that a similar characterization does not hold for the classes of globally/locally discrete midpoint convex functions. ◻

Scaling operations
Scaling operations are useful techniques for designing efficient algorithms in discrete optimization. It is shown in [18] that global/local discrete midpoint convexity, including L ♮ -convexity, is closed under scaling operations. We show that DDM-convexity is also closed under scaling operations.
Given a function f ∶ ℤ n → ℝ ∪ {+∞} and a positive integer , the -scaling of f is the function f defined by We also define the -scaling S of a set S ⊆ ℤ n by Theorem 11 Given a DDM-convex function f ∶ ℤ n → ℝ ∪ {+∞} and a positive integer , the scaled function f ∶ ℤ n → ℝ ∪ {+∞} is also DDM-convex.
Proof By the equivalence between (1) and (4) of Theorem 8, it is sufficient to show that for every x, y ∈ ℤ n with ‖x − y‖ ∞ = m and for families {A k | k = 1, … , m} and {B k | k = 1, … , m} defined by (4.1). The above inequality is written as By (5) of Theorem 8 for we have that is, (6.1). ◻

Corollary 3
For a DDM-convex set S ⊆ ℤ n and a positive integer , the -scaled set S is also DDM-convex.

Restrictions
For a function f ∶ ℤ n+m → ℝ ∪ {+∞} , the restriction of f on ℤ n is the function g defined by For a set S ⊆ ℤ n+m , the restriction of S on ℤ n is also defined by Obviously, the following properties hold.

Proposition 10
For a DDM-convex function, its restrictions are also DDM-convex.

Proposition 11
For a DDM-convex set, its restrictions are also DDM-convex.

Projections
For a function f ∶ ℤ n+m → ℝ ∪ {+∞} , the projection of f to ℤ n is the function defined by where we assume that g(x) > −∞ for all x ∈ ℤ n . For a set S ⊆ ℤ n+m , the projection of S to ℤ n is also defined by In the same way as the proof for globally discrete midpoint convex functions in [16,Theorem 3.5] we can show the following property.

Proposition 12
For a DDM-convex function, its projections are DDM-convex.

3
Directed discrete midpoint convexity Proof Let g be the projection defined by (6.2) of a DDM-convex function f. For every x (1) , x (2) ∈ dom g and every > 0 , by the definition of the projection, there exist y (1) , y (2) ∈ ℤ m with g(x (i) ) ≥ f (x (i) , y (i) ) − for i = 1, 2 . Thus, we have By DDM-convexity of f and the definition of the projection, we have By (6.3) and (6.4), we obtain for any > 0 , which guarantees DDM-convexity of g. ◻

Corollary 4
For a DDM-convex set, its projections are also DDM-convex.

Convolutions
For two functions f 1 , f 2 ∶ ℤ n → ℝ ∪ {+∞} , the convolution f 1 ◻f 2 is the function defined by where we assume (f 1 ◻f 2 )(x) > −∞ for every x ∈ ℤ n . For two sets S 1 , S 2 ⊆ ℤ n , the Minkowski sum S 1 + S 2 defined by corresponds to the convolution of indicator functions S 1 and S 2 . The next example shows that the Minkowski sum of two DDM-convex sets may not be DDM-convex, and hence, DDM-convexity is not closed under the convolutions.

Proposition 14 The convolution of a DDM-convex function and a separable convex function is also DDM-convex.
Proof Let f ∶ ℤ n → ℝ ∪ {+∞} be a DDM-convex function, ∶ ℤ n → ℝ ∪ {+∞} a separable convex function represented as ∑ n i=1 i and let g = f ◻ . For every x (1) , x (2) ∈ dom g and > 0 , by the definition of convolutions, there exist y (i) , z (i) (i = 1, 2) such that It follows from DDM-convexity of f that

Corollary 6
Minkowski sum of a DDM-convex set and an integral box is also DDMconvex, where an integral box is the set defined by {x ∈ ℤ n | a ≤ x ≤ b} for some a ∈ (ℤ ∪ {−∞}) n and b ∈ (ℤ ∪ {+∞}) n with a ≤ b.

Proximity theorems
For a function f ∶ ℤ n → ℝ ∪ {+∞} and a positive integer , a proximity theorem estimates the distance between a given local minimizer x • of the -scaled function f and a minimizer x * of f. For instance, the following proximity theorems for L ♮ -convex functions and globally/locally discrete midpoint convex functions are known.
Theorem 12 ([11, 21]) Let f ∶ ℤ n → ℝ ∪ {+∞} be an L ♮ -convex function, be a positive integer and In the same way as the arguments in [18], we can show the following proximity theorem for DDM-convex functions.
Theorem 14 Let f ∶ ℤ n → ℝ ∪ {+∞} be a DDM-convex function, be a positive integer and We note that f is also DDM-convex by Theorem 11 and x • corresponds to a minimizer 0 of f (y) = f (x • + y) by Corollary 1. We emphasize that the bound n( − 1) for DDM-convex functions is the same as that for L ♮ -convex functions and globally/locally discrete midpoint convex functions.
To prove Theorem 14, we assume x • = 0 without loss of generality. Let Then Theorem 1 (box-barrier property) implies that f (z) ≥ for all z ∈ ℤ n . Fix y = (y 1 , … , y n ) ∈ W , and let ‖y‖ ∞ = m(= n( − 1) + 1) . By using we can write y as s) . Because m = n( − 1) + 1 and the length of a strictly decreasing chain connecting (s, n) to (1, s) in ℤ 2 is bounded by n, there exists a constant subsequence of length ≥ in the sequence {(a k , b k )} k=1,…,m by the pigeonhole principle. Hence the assertion holds. ◻ By using k 0 in Lemma 2, we define a subset J of {1, … , m} by J = {k 0 , … , k 0 + − 1} . By the parallelogram inequality (4.14) in Theorem 7, where By the assumption, we have f By the definition of , f (y − d 0 ) ≥ must hold. Therefore, which implies (7.1), completing the proof of Theorem 14.

Minimization algorithms
In this section, we propose two algorithms for DDM-convex function minimization.

The 1-neighborhood steepest descent algorithm
We first propose a variant of steepest descent algorithm for DDM-convex function minimization problem. Let f ∶ ℤ n → ℝ ∪ {+∞} be a DDM-convex function with argmin f ≠ ∅ . We suppose that an initial point is given. Let L denote the minimum l ∞ -distance between x (0) and a minimizer of f, that is, L is defined by For all k = 0, 1, … , L we define sets S k by The idea of our algorithm is to generate a sequence of minimizers in S k for k = 1, … , L . The next proposition guarantees that consecutive minimizers can be chosen to be close to each other.

Proposition 15
For each k = 1, … , L and for any Proof If k = 1 , the assertion is obvious. Suppose that k ≥ 2 and y is any point in S k . By (2.1) for x (k−1) and y, we have −1) , y)) + f ( (y, x (k−1) )).
Since x (k−1) , y ∈ S k and S k is a DDM-convex set, we also have Next, we show To show this we arbitrarily fix i ∈ {1, … , n} , and consider the two cases: Case 1:  (8.2), this implies f (y * ) < f ( (y * , x (k−1) )) . Moreover, by (8.3), we have f (x (k−1) ) ≤ f ( (x (k−1) , y * )) . These two inequalities contradict (8.1) for x (k−1) and y * . Hence ‖y * − x (k−1) ‖ ∞ ≤ 1 must hold. ◻ By Proposition 15, it seems be natural to assume that we can find a minimizer of f within the 1-neighborhood N 1 (x) of x defined by With the use of a 1-neighborhood minimization oracle, which finds a point minimizing f in N 1 (x) for any x ∈ dom f , our algorithm can be described as below. .
where a symmetric matrix Q ∈ ℝ n×n is nonsingular and diagonally dominant with nonnegative diagonals, and c ∈ ℝ n . Since Q is nonsingular, the (convex) continuous relaxation problem has a unique minimizer −Q −1 c . Furthermore, because the objective function is 2-separable convex, it follows from Theorem 18 in the next section that there exists an optimal solution in the box: Therefore, the 1-neighborhood steepest descent algorithm with an initial point then F(n, k) = O(n3 k+1 ) as in [6]. ◻

Scaling algorithm
In the same way as the scaling algorithm for minimization of globally/locally discrete midpoint convex functions in [18], the scaling property (Theorem 11) and the proximity theorem (Theorem 14) enable us to design a scaling algorithm for the minimization of DDM-convex functions with bounded effective domains. Let f ∶ ℤ n → ℝ ∪ {+∞} be a DDM-convex function with bounded effective domain. We suppose that K ∞ = max{‖x − y‖ ∞ | x, y ∈ dom f } (K ∞ < +∞) and an initial point x ∈ dom f are given. Our algorithm can be described as follows.

Scaling algorithm for DDM-convex functions
S0: Let x ∈ dom f and ∶= 2 ⌈log 2 (K ∞ +1)⌉ . S1: Find a vector y that minimizes f (y) = f (x+ y) subject to ‖y‖ ∞ ≤ n (by the 1-neighborhood steepest descent algorithm), and set x ∶= x + y. S2: If = 1 , then stop (x is a minimizer of f). S3: Set ∶= ∕2 , and go to S1. Proof The correctness of the algorithm can be shown by induction on . If = 2 ⌈log 2 (K ∞ +1)⌉ , then x is a unique point of dom f because = 2 ⌈log 2 (K ∞ +1)⌉ > K ∞ , that is, a minimizer of f . Let x 2 denote the point x at the beginning of S1 for and assume that x 2 is a minimizer of f 2 . The function f (y) = f (x 2 + y) is DDM-convex by Theorem 11. Let y = argmin{f (y) | ‖y‖ ∞ ≤ n} and x = x 2 + y . Theorem 14 guarantees that x is a minimizer of f because of x 2 ∈ argmin f 2 . At the termination of the algorithm, we have = 1 and f = f . The output of the algorithm, which is computed by the 1-neighborhood steepest descent algorithm, satisfies the condition of Corollary 1, and hence, the output is indeed a minimizer of f. The time complexity of the algorithm can be analyzed as follows: by Theorem 15, S1 terminates in O(n) calls of the 1-neighborhood minimization oracles in each iteration. The number of iterations is O(log 2 K ∞ ) . Hence, the assertion holds. ◻

DDM-convex functions in continuous variables
In [19], proximity theorems between L ♮ -convex functions and their continuous relaxations are proposed. We extend these results to DDM-convexity. It is known that the continuous version of L ♮ -convexity can naturally be defined by using translation-submodularity (3.1). In this section, we define DDM-convexity in continuous variables in a different way. We call a continuous convex function F ∶ ℝ n → ℝ ∪ {+∞} a directed discrete midpoint convex function in continuous variables ( ℝ-DDM-convex function) if for any positive integer , the function f 1∕ ∶ ℤ n → ℝ ∪ {+∞} defined by is DDM-convex. We denote by f the DDM-convex function f 1∕1 which is nothing but the restriction of F to ℤ n .
An example of an ℝ-DDM-convex function is a continuous 2-separable convex function F which is defined as for univariate continuous convex functions i , ij , ij ∶ ℝ → ℝ ∪ {+∞} (i = 1, … , n; j ∈ {1, … , n}⧵{i} ) as below. The restriction f of F to ℤ n is trivially a 2-separable convex function on ℤ n defined by (3.4). Furthermore, the function F 1∕ ∶ ℝ n → ℝ ∪ {+∞} defined by is also a continuous 2-separable convex function, and hence, the restriction f 1∕ of F 1∕ to ℤ n is also a 2-separable convex function on ℤ n .
We have the following proximity theorems between an ℝ-DDM-convex function F and its restriction f to ℤ n .
for all i, guarantees that We finally show F(x) = min F , that is, x ∈ argmin F . Suppose to the contrary that there exists x ′ with F(x � ) < F(x) . Let = F(x) − F(x � ) > 0 . By the continuity of F, there exists such that Because there exist N ∈ {k i | i = 1, 2, … } and y ∈ ℝ n such that 2 N y ∈ ℤ n and ‖x � − y‖ ∞ < , by (9.4), we have which contradicts (9.3). Therefore, x must be a minimizer of F. ◻ If F has a unique minimizer, the converse of Theorem 17 also holds.

3
Directed discrete midpoint convexity Proof Let x be a unique minimizer of F. If f has a minimizer x * , then ‖x * − x‖ ∞ ≤ n must hold by Theorem 17. Thus, it is enough to show that f has a minimizer. Suppose to the contrary that f has no minimizer and let B = {x ∈ ℝ n | x − n1 ≤ x ≤ x + n1} . Then, there exists y ∈ dom f ⧵B such that f (y) < f (x) for all x ∈ dom f ∩ B . Let = ‖y − x‖ ∞ and B � = {x ∈ ℝ n | x − 1 ≤ x ≤ x + 1} . Note that > n and B ′ ⊃ B . Let us consider the restriction G of F to B ′ defined by Obviously, G is ℝ-DDM-convex and x is a unique minimizer of G. In particular, the restriction g of G to ℤ n is DDM-convex and has a minimizer z since B ′ is bounded. This point z does not belong to B since y ∉ B and f (y) < f (x) for all x ∈ dom f ∩ B . However, this contradicts Theorem 17 for G and g. ◻ If F has a bounded effective domain, a similar statement holds. Let K ∞ = sup{‖x − y‖ ∞ | x, y ∈ dom F}.
Proof If dom f = argmin f , the assertion holds. In the sequel, we assume that dom f ≠ argmin f . We fix a minimizer x of F, arbitrarily. For a sufficiently small > 0 , let us consider functions F ∶ ℝ n → ℝ ∪ {+∞} and f ∶ ℤ n → ℝ ∪ {+∞} defined by Function F has the unique minimizer x and satisfies the conditions of Theorem 18, by Proposition 4 (4), because f 1∕ defined by (9.1) for F is the sum of f 1∕ and a separable convex function which are DDM-convex. Thus, by Theorem 18 for F and f , there exists x ∈ argmin f with ‖x − x‖ ∞ ≤ n. Let = min{f (x) | x ∈ dom f ⧵ argmin f } > 0 . Note that is well-defined by boundedness of dom f . We show that if < ( − min f )∕(nK 2 ∞ ) , then x ∈ argmin f . For any x ∈ argmin f , by f (x ) ≤ f (x) , we have which says x ∈ argmin f . ◻

Remark 5
There is a convex function which is not ℝ-DDM-convex. For example, for a positive definite matrix Q = 5 2 2 1 , the function is convex, but the restriction f of F to ℤ 2 is not DDM-convex by Theorem 9, and hence, F is not ℝ-DDM-convex.