Maximizing a class of submodular utility functions

Shabbir Ahmed, Alper Atamtürk
2009 Mathematical programming  
Given a finite ground set N and a value vector a ∈ R N , we consider optimization problems involving maximization of a submodular set utility function of the form h(S) = f i∈S a i , S ⊆ N , where f is a strictly concave, increasing, differentiable function. This utility function appears frequently in combinatorial optimization problems when modeling risk aversion and decreasing marginal preferences, for instance, in risk-averse capital budgeting under uncertainty, competitive facility location,
more » ... 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 submodular maximization problem with a special structure solvable by a greedy algorithm, which leads to an easily-computable strengthening by subadditive lifting of the inequalities. Computational experiments on expected utility maximization in capital budgeting show the effectiveness of the new formulation. Mathematics Subject Classification (2000) 90C57 · 91B16 · 91B28
doi:10.1007/s10107-009-0298-1 fatcat:kisqitzwm5h65l5v5keyc3m55y