M-Convex Function Minimization by Continuous Relaxation Approach: Proximity Theorem and Algorithm

Abstract

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 ,

Separable Convex Optimization with Nested Lower and Upper Constraints

“…• For the specific case of quadratic functions, with continuous or integer variables, the method runs in O(n log m) time, hence extending the short list of quadratic problems known to be solvable in strongly polynomial time. This also resolves an open question from Moriguchi et al (2011): "It is an open question whether there exist O(n log n) algorithms for (Nest) with quadratic objective functions".…”

Algorithms for separable convex optimization with linear ascending constraints

“…of H + e and H − e , we also have H + e = H − e at all points of continuity of H + e , and H − e is the left continuous version of the right continuous H + e . Given (A.2) and these observations, it follows that we can find an η that satisfies (14). Proposition 4.…”

“…For indices l that satisfy (14) in the j th iteration with l < s(j) and η j l = Γ j (i.e., l is a tied index), there is some l ′ satisfying s(j − 1) > l ′ ≥ s(j) and Hence η j+1 l > Γ j for all such tied l. For all nontied indices l < s(j), i.e., indices that satisfy (14) in j th iteration but with η j l > Γ j or η j l does not exist, we must have…”

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“…The Mconvexity of this problem has been proved in [13], [16]. In [17]- [19], various algorithms based on M -convexity have been developed for solving problem (17).…”