Maximizing a class of submodular utility functions

• 成果类型: Article
• 署名作者: Ahmed, Shabbir; Atamtuerk, Alper
• 署名单位: University of California System; University of California Berkeley; University System of Georgia; Georgia Institute of Technology
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-009-0298-1
• 发表日期: 2011
• 页码: 149-169
• 关键词: model

摘要

Given a finite ground set N and a value vector a is an element of R-N, we consider optimization problems involving maximization of a submodular set utility function of the form h(S) = f(Sigma(i is an element of S)a(i)), S subset of 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, 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.