Maximizing a class of submodular utility functions

Given a finite ground set N and a value vector a ∈ RN , we consider optimization problems involving maximization of a submodular set utility function of the form h(S) = f i∈S ai ) , S ⊆ N , where f is a strictly concave, increasing, differentiable function. This utility function appears frequently in combinatorial optimization problems whenmodeling risk aversion and decreasing marginal preferences, for instance, in risk-averse capital budgeting under uncertainty, competitive facility location, and combinatorial auctions. These problems can be formulated as linear mixed 0-1 programs. However, the standard formulation of these problems using submodular inequalities is ineffective for their solution, except for very small instances. In this paper, we perform a polyhedral analysis of a relevant mixed-integer set and, by exploiting the structure of the utility function h, strengthen the standard submodular formulation significantly. We show the lifting problem of the submodular inequalities to be a submodularmaximization problemwith a special structure solvable by a greedy algorithm, which leads to an easily-computable strengthening by subadditive lifting The research of the first author has been supported, in part, by Grant # FA9550-08-1-0117 from the Air Force Office of Scientific Research. The research of the second author has been supported, in part, by Grant # DMI0700203 from the National Science Foundation. The second author is grateful for the hospitality of the Georgia Institute of Technology, where part of this research was conducted. S. Ahmed School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA e-mail: [email protected] A. Atamtürk (B) Department of Industrial Engineering and Operations Research, University of California, Berkeley, CA 94720-1777, USA e-mail: [email protected]