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

Amir Ali Ahmadi(a_a_a***at***princeton.edu)
Georgina Hall(gh4***at***princeton.edu)
Antonis Papachristodoulou(antonis***at***eng.ox.ac.uk)
James Saunderson(james.saunderson***at***monash.edu)
Yang Zheng(yang.zheng***at***eng.ox.ac.uk)

Abstract: It is well-known that any sum of squares (SOS) program can be cast as a semidefinite program (SDP) of a particular structure and that therein lies the computational bottleneck for SOS programs, as the SDPs generated by this procedure are large and costly to solve when the polynomials involved in the SOS programs have a large number of variables and degree. In this paper, we review SOS optimization techniques and present two new methods for improving their computational efficiency. The first method leverages the sparsity of the underlying SDP to obtain computational speed-ups. Further improvements can be obtained if the coefficients of the polynomials that describe the problem have a particular sparsity pattern, called \emph{chordal sparsity}. The second method bypasses semidefinite programming altogether and relies instead on solving a sequence of more tractable convex programs, namely linear and second order cone programs. This opens up the question as to how well one can approximate the cone of SOS polynomials by second order representable cones. In the last part of the paper, we present some recent negative results related to this question.

Keywords: Sum of squares, semidefinite programming

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Category 2: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Citation: Invited tutorial paper for CDC 2017.

Download: [PDF]

Entry Submitted: 10/09/2017
Entry Accepted: 10/09/2017
Entry Last Modified: 10/09/2017

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