Abstract
Given a finite ground set N and a value vector \({a \in \mathbb{R}^N}\), we consider optimization problems involving maximization of a submodular set utility function of the form \({h(S)= f \left(\sum_{i \in S} a_i \right ), S \subseteq N}\), 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.
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The authors are thankful to the associate editor and a referee for their constructive comments.
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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.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Ahmed, S., Atamtürk, A. Maximizing a class of submodular utility functions. Math. Program. 128, 149–169 (2011). https://doi.org/10.1007/s10107-009-0298-1
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DOI: https://doi.org/10.1007/s10107-009-0298-1