On Constrained Mixed-Integer DR-Submodular Minimization

TL;DR

This paper introduces a polynomial-time method for minimizing DR-submodular functions with mixed-integer variables under constraints, providing valid inequalities and a complete convex hull characterization.

Contribution

It develops the first polynomial-time exact separation algorithm for these problems and characterizes the convex hull of the epigraph of DR-submodular functions.

Findings

Polynomial-time solvability of the minimization problem.

Polyhedral description of the convex hull of the epigraph.

Valid inequalities for the problem's feasible region.

Abstract

DR-submodular functions encompass a broad class of functions which are generally non-convex and non-concave. We study the problem of minimizing any DR-submodular function, with continuous and general integer variables, under box constraints and possibly additional monotonicity constraints. We propose valid linear inequalities for the epigraph of any DR-submodular function under the constraints. We further provide the complete convex hull of such an epigraph, which, surprisingly, turns out to be polyhedral. We propose a polynomial-time exact separation algorithm for our proposed valid inequalities, with which we first establish the polynomial-time solvability of this class of mixed-integer nonlinear optimization problems.