Maximizing a class of submodular utility functions with constraints

Abstract

Motivated by stochastic 0–1 integer programming problems with an expected utility objective, we study the mixed-integer nonlinear set: $P = \big \{(w,x)\in \mathbb {R}\times \left\{ 0,1\right\} ^N: w \le f(a'x + d), b'x \le B\big \}$ where N is a positive integer, $f:\mathbb {R}\mapsto \mathbb {R}$ is a concave function, $a, b \in \mathbb {R}^N$ are nonnegative vectors, d is a real number and B is a positive real number. We propose a family of inequalities for the convex hull of P by exploiting submodularity of the function $f(a'x + d)$ over $\{0,1\}^N$ and the knapsack constraint $b'x \le B$. Computational effectiveness of the proposed inequalities within a branch-and-cut framework is illustrated using instances of an expected utility capital budgeting problem.