DSOS and SDSOS optimization: LP and SOCP-based alternatives to sum of squares optimization

Sum of squares (SOS) optimization has been a powerful and influential addition to the theory of optimization in the past decade. Its reliance on relatively large-scale semidefinite programming, however, has seriously challenged its ability to scale in many practical applications. In this paper, we introduce DSOS and SDSOS optimization as more tractable alternatives to sum of squares optimization that rely instead on linear programming and second order cone programming. These are optimization problems over certain subsets of sum of squares polynomials and positive semidefinite matrices and can be of potential interest in general applications of semidefinite programming where scalability is a limitation.