Maximizing a class of submodular utility functions with constraints

Motivated by stochastic 0-1 integer programming problems with an expected utility objective, we study the mixed-integer nonlinear set: P = { (w, x) ∈ R× {0, 1}N : w ≤ f (a′x + d), b′x ≤ B } where N is a positive integer, f : R 7→ R is a concave function, a, b ∈ RN 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 ≤ B. Computational effectiveness of the proposed inequalities within a branch-and-cut framework is illustrated using instances of an expected utility capital budgeting problem.