Supermodular covering knapsack polytope

The supermodular covering knapsack set is the discrete upper level set of a non-decreasing supermodular function. Submodular and supermodular knapsack sets arise naturally when modeling utilities, risk and probabilistic constraints on discrete variables. In a recent paper Atamtürk and Narayanan (2009) study the lower level set of a non-decreasing submodular function.In this complementary paper we describe pack inequalities for the supermodular covering knapsack set and investigate their separation, extensions and lifting. We give sequence-independent upper bounds and lower bounds on the lifting coefficients. Furthermore, we present a computational study on using the polyhedral results derived for solving 0–1 optimization problems over conic quadratic constraints with a branch-and-cut algorithm.