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Relative Entropy Relaxations for Signomial Optimization

Venkat Chandrasekaran(venkatc***at***caltech.edu)
Parikshit Shah(pshah***at***discovery.wisc.edu)

Abstract: Signomial programs (SPs) are optimization problems specified in terms of signomials, which are weighted sums of exponentials composed with linear functionals of a decision variable. SPs are non convex optimization problems in general, and families of NP-hard problems can be reduced to SPs. In this paper we describe a hierarchy of convex relaxations to obtain successively tighter lower bounds of the optimal value of SPs. This sequence of lower bounds is computed by solving increasingly larger-sized relative entropy optimization problems, which are convex programs specified in terms of linear and relative entropy functions. Our approach relies crucially on the observation that the relative entropy function -- by virtue of its joint convexity with respect to both arguments -- provides a convex parametrization of certain sets of globally nonnegative signomials with efficiently computable nonnegativity certificates via the arithmetic geometric-mean inequality. By appealing to representation theorems from real algebraic geometry, we show that our sequences of lower bounds converge to the global optima for broad classes of SPs. Finally, we also demonstrate the effectiveness of our methods via numerical experiments.

Keywords: arithmetic-geometric-mean inequality; convex optimization; geometric programming; global optimization; real algebraic geometry

Category 1: Convex and Nonsmooth Optimization

Category 2: Global Optimization


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Entry Submitted: 09/26/2014
Entry Accepted: 09/26/2014
Entry Last Modified: 09/26/2014

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