A framework of discrete DC programming by discrete convex analysis

Abstract

A theoretical framework of difference of discrete convex functions (discrete DC functions) and optimization problems for discrete DC functions is established. Standard results in continuous DC theory are exported to the discrete DC theory with the use of discrete convex analysis. A discrete DC algorithm, which is a discrete analogue of the continuous DC algorithm (Concave-Convex procedure in machine learning) is proposed. The algorithm contains the submodular-supermodular procedure as a special case. Exploiting the polyhedral structure of discrete convex functions, the algorithms tailored to specific types of discrete DC functions are proposed. Mathematics Subject Classification 90C27 · 90C46

The complexity of minimizing the difference of two M♮-convex set functions

“…Thus, Lemma 3 shows that Problem (1) is a special case of minimizing the difference of two M ♮ -convex set functions. Hence, by Theorem 1, we obtain the following theorem, which answers the open question posed by Maehara and Murota [3].…”

“…Maehara and Murota [3] established a theoretical framework of difference of discrete convex functions and studied the problem of minimizing the difference of two discrete convex functions (discrete DC programs). The computational complexities of several types of discrete DC programs were revealed in their paper, but determining the complexity of minimizing the difference of two M ♮ -convex set functions was posed as an open question (see Section 4.3 and Table 1 in [3]). Although the term "M ♮ -convex functions on {0, 1} n " is used in [3], they are equivalent to "M ♮ -convex set functions" by identifying a subset of the ground set and its characteristic vector.…”

“…The computational complexities of several types of discrete DC programs were revealed in their paper, but determining the complexity of minimizing the difference of two M ♮ -convex set functions was posed as an open question (see Section 4.3 and Table 1 in [3]). Although the term "M ♮ -convex functions on {0, 1} n " is used in [3], they are equivalent to "M ♮ -convex set functions" by identifying a subset of the ground set and its characteristic vector. In this paper, we show the NP-hardness of this minimization problem.…”

A new approach for solving mixed integer DC programs using a continuous relaxation with no integrality gap and smoothing techniques

“…2 is strongly convex by Assumption (A2) and {f u k (z l )} can be shown to be bounded from below by using Proposition 5.3 and Lemma 5.4. Hence, φ k is coercive and convex in F , from which we can conclude that (22) has an optimum. It remains to show that the optimal value is finite.…”

“…, n}, is still a relatively unexplored area. Recently, a promising approach was proposed by Maehara and Murota [22], who showed how the framework of discrete convex analysis can be applied, to export results in continuous DC theory to a discrete setting. This was further pursued in Maehara, Marumo and Murota [23], who proved a powerful result in constructing continuous relaxations of discrete DC programs.…”