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Amir Ali Ahmadi (a_a_aprinceton.edu) Abstract: In recent years, optimization theory has been greatly impacted by the advent of sum of squares (SOS) optimization. The reliance of this technique on largescale semidefinite programs however, has limited the scale of problems to which it can be applied. In this paper, we introduce DSOS and SDSOS optimization as linear programming and secondorder cone programmingbased alternatives to sum of squares optimization that allow one to trade off computation time with solution quality. These are optimization problems over certain subsets of sum of squares polynomials (or equivalently subsets of positive semidefinite matrices), which can be of interest in general applications of semidefinite programming where scalability is a limitation. We show that some basic theorems from SOS optimization which rely on results from real algebraic geometry are still valid for DSOS and SDSOS optimization. Furthermore, we show with numerical experiments from diverse application areaspolynomial optimization, statistics and machine learning, derivative pricing, and control theorythat with reasonable tradeoffs in accuracy, we can handle problems at scales that are currently significantly beyond the reach of traditional sum of squares approaches. Finally, we provide a review of recent techniques that bridge the gap between our DSOS/SDSOS approach and the SOS approach at the expense of additional running time. The Supplementary Material of the paper introduces an accompanying MATLAB package for DSOS and SDSOS optimization. Keywords: Sum of squares optimization, polynomial optimization, nonnegative polynomials, semidefinite programming, linear programming, second order cone programming Category 1: Linear, Cone and Semidefinite Programming (Semidefinite Programming ) Category 2: Linear, Cone and Semidefinite Programming (SecondOrder Cone Programming ) Category 3: Linear, Cone and Semidefinite Programming (Linear Programming ) Citation: Download: [PDF] Entry Submitted: 06/08/2017 Modify/Update this entry  
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