For a function defined on the integer lattice, we consider discrete versions of midpoint convexity, which offer a unifying framework for discrete convexity of functions, including integral convexity, L -convexity, and submodularity. By considering discrete midpoint convexity for all pairs at ∞ -distance equal to two or not smaller than two, we identify new classes of discrete convex functions, called locally and globally discrete midpoint convex functions. These functions enjoy nice structural properties. They are stable under scaling and addition, and satisfy a family of inequalities named parallelogram inequalities. Furthermore, they admit a proximity theorem with the same small proximity bound as that for L -convex functions. These structural properties allow us to develop an algorithm for the minimization of locally and globally discrete midpoint convex functions based on the proximity-scaling approach and on a novel 2-neighborhood steepest descent algorithm.

The concept of M-convexity for functions in integer variables, introduced by Murota (1995), plays a primary role in the theory of discrete convex analysis. In this paper, we consider the problem of minimizing an M-convex function, which is a natural generalization of the separable convex resource allocation problem under a submodular constraint and contains some classes of nonseparable convex function minimization on integer lattice points. We propose a new approach for M-convex function minimization based on continuous relaxation. We show proximity theorems for M-convex function minimization and its continuous relaxation, and develop a new algorithm based on continuous relaxation by using the proximity theorems. The practical performance of the proposed algorithm is evaluated by computational experiments. 1 Below we give some important special cases of the problem (MC). Example 1.1 (Resource Allocation Problem under a Submodular Constraint). Let f i : R → R (i ∈ N) be a family of univariate convex functions. Also, let ρ : 2 N → Z ∪ {+∞} be a submodular function, i.e., ρ satisfies ρ(X) + ρ(Y) ≥ ρ(X ∩ Y) + ρ(X ∪ Y) for every X, Y ∈ 2 N. We assume ρ(∅) = 0, ρ(Y) ≥ 0 (∀Y ⊆ N), and ρ(N) < +∞. The (separable convex) resource allocation problem under a submodular constraint [1, 6, 7] is formulated as follows: (SC) Minimize n i=1 f i (x(i)) subject to x(N) = ρ(N), x(Y) ≤ ρ(Y) (Y ∈ 2 N), x ≥ 0, x ∈ Z n ,

In discrete convex analysis, the scaling and proximity properties for the class of Lconvex functions were established more than a decade ago and have been used to design efficient minimization algorithms. For the larger class of integrally convex functions of n variables, we show here that the scaling property only holds when n ≤ 2, while a proximity theorem can be established for any n, but only with a superexponential bound. This is, however, sufficient to extend the classical logarithmic complexity result for minimizing a discrete convex function of one variable to the case of integrally convex functions of any fixed number of variables.∞ -distance for f α . Proximity-scaling algorithmS0: Find an initial vector x with f (x) < +∞, and set α := 2 log 2 K ∞ .

It is pointed out that the polynomial-time scaling algorithm by Hochbaum does not work correctly for the general resource allocation problem. Hochbaum's algorithm increases a variable by one unit if the variable cannot feasibly be increased by the scaling unit. We modify the algorithm to increase such a variable by the largest possible amount and show that with this modification the algorithm works correctly. The effect is to modify the factor F in the running time of Hochbaum's algorithm for finding whether a certain solution is feasible by the factor F̃ of finding the maximum feasible increment (also called the saturation capacity). Therefore, the corrected algorithm runs in O(n(log n + F̃) log(B/n)) time.

This paper considers projection and convolution operations for integrally convex functions, which constitute a fundamental function class in discrete convex analysis. It is shown that the class of integrally convex functions is stable under projection, and this is also the case with the subclasses of integrally convex functions satisfying local or global discrete midpoint convexity. As is known in the literature, the convolution of two integrally convex functions may possibly fail to be integrally convex. We show that the convolution of an integrally convex function with a separable convex function remains integrally convex. We also point out in terms of examples that the similar statement is false for integrally convex functions with local or global discrete midpoint convexity.