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In recent years, optimization theory has been greatly impacted by the adventof sum of squares (SOS) optimization. The reliance of this technique onlarge-scale semidefinite programs however, has limited the scale of problems towhich it can be applied. In this paper, we introduce DSOS and SDSOSoptimization as linear programming and second-order cone programming-basedalternatives to sum of squares optimization that allow one to trade offcomputation time with solution quality. These are optimization problems overcertain subsets of sum of squares polynomials (or equivalently subsets ofpositive semidefinite matrices), which can be of interest in generalapplications of semidefinite programming where scalability is a limitation. Weshow that some basic theorems from SOS optimization which rely on results fromreal algebraic geometry are still valid for DSOS and SDSOS optimization.Furthermore, we show with numerical experiments from diverse applicationareas---polynomial optimization, statistics and machine learning, derivativepricing, and control theory---that with reasonable tradeoffs in accuracy, wecan handle problems at scales that are currently significantly beyond the reachof traditional sum of squares approaches. Finally, we provide a review ofrecent techniques that bridge the gap between our DSOS/SDSOS approach and theSOS approach at the expense of additional running time. The SupplementaryMaterial of the paper introduces an accompanying MATLAB package for DSOS andSDSOS optimization.

DSOS and SDSOS Optimization: More Tractable Alternatives to Sum of Squares and Semidefinite Optimization