Relative Entropy Relaxations for Signomial Optimization

TL;DR

This paper introduces a hierarchy of convex relaxations based on relative entropy to find tighter lower bounds for signomial programs, enabling global optimization of these generally NP-hard problems.

Contribution

It presents a novel hierarchy of convex relaxations using relative entropy functions that converge to the global optimum for broad classes of signomial programs.

Findings

Hierarchy of relaxations provides successively tighter bounds.

Method converges to global optima for broad classes of SPs.

Numerical experiments demonstrate effectiveness.

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.