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Maximizing a class of submodular utility functions with constraints

Jiajin Yu(jiajinyu***at***gatech.edu)
Shabbir Ahmed(sahmed***at***isye.gatech.edu)

Abstract: Motivated by stochastic 0-1 integer programming problems with an expected utility objective, we study the mixed-integer nonlinear set: $P = \cset{(w,x)\in \reals \times \set{0,1}^N}{w \leq f(a'x + d), b'x \leq B}$ where $N$ is a positive integer, $f:\reals \mapsto \reals$ is a concave function, $a, b \in \reals^N$ are nonnegative vectors, $d$ is a real number and $B$ is a positive real number. We propose a family of inequalities for the convex hull of $P$ by exploiting submodularity of the function $f(a'x + d)$ over $\{0,1\}^N$ and the knapsack constraint $b'x \leq B$. Computational effectiveness of the proposed inequalities within a branch-and-cut framework is illustrated using instances of an expected utility capital budgeting problem.

Keywords: Submodularity, cutting planes, lifting, mixed integer nonlinear programming, branch-and-cut

Category 1: Integer Programming (0-1 Programming )

Category 2: Integer Programming (Cutting Plane Approaches )

Category 3: Stochastic Programming

Citation: Submitted for publication, December 2014.

Download: [PDF]

Entry Submitted: 12/30/2014
Entry Accepted: 12/30/2014
Entry Last Modified: 12/30/2014

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