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It is well-known that any sum of squares (SOS) program can be cast as asemidefinite program (SDP) of a particular structure and that therein lies thecomputational bottleneck for SOS programs, as the SDPs generated by thisprocedure are large and costly to solve when the polynomials involved in theSOS programs have a large number of variables and degree. In this paper, wereview SOS optimization techniques and present two new methods for improvingtheir computational efficiency. The first method leverages the sparsity of theunderlying SDP to obtain computational speed-ups. Further improvements can beobtained if the coefficients of the polynomials that describe the problem havea particular sparsity pattern, called chordal sparsity. The second methodbypasses semidefinite programming altogether and relies instead on solving asequence of more tractable convex programs, namely linear and second order coneprograms. This opens up the question as to how well one can approximate thecone of SOS polynomials by second order representable cones. In the last partof the paper, we present some recent negative results related to this question.

Improving Efficiency and Scalability of Sum of Squares Optimization: Recent Advances and Limitations