Chandrasekaran, Venkat and Shah, Parikshit (2016) Relative Entropy Relaxations for Signomial Optimization. SIAM Journal on Optimization, 26 (2). pp. 1147-1173. ISSN 1052-6234. http://resolver.caltech.edu/CaltechAUTHORS:20161017-134459797
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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 nonconvex 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.
|Additional Information:||© 2016 SIAM. Received by the editors September 26, 2014; accepted for publication (in revised form) September 15, 2015; published electronically May 12, 2016. This work was supported in part by the following grants: National Science Foundation Career award CCF-1350590 and Air Force Office of Scientific Research grant FA9550-14-1-0098.|
|Subject Keywords:||arithmetic-geometric-mean inequality, convex optimization, geometric programming, global optimization, real algebraic geometry|
|Classification Code:||AMS subject classifications. 90C25, 90C26|
|Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Kristin Buxton|
|Deposited On:||17 Oct 2016 20:52|
|Last Modified:||17 Oct 2016 20:52|
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