Improving efficiency and scalability of sum of squares optimization: Recent advances and limitations

Amir Ali Ahmadi, Georgina Hall, Antonis Papachristodoulou, James Saunderson, Yang Zheng

Research output: Chapter in Book/Report/Conference proceedingConference PaperResearchpeer-review

13 Citations (Scopus)


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 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.
Original languageEnglish
Title of host publication2017 IEEE 56th Annual Conference on Decision and Control (CDC 2017)
EditorsMario Sznaier
Place of PublicationPiscataway NJ USA
PublisherIEEE, Institute of Electrical and Electronics Engineers
Number of pages10
ISBN (Electronic)9781509028733 , 9781509028726
ISBN (Print)9781509028740
Publication statusPublished - 2017
EventIEEE Conference on Decision and Control 2017 - Melbourne Convention Center, Melbourne, Australia
Duration: 12 Dec 201715 Dec 2017
Conference number: 56th (Proceedings)


ConferenceIEEE Conference on Decision and Control 2017
Abbreviated titleCDC 2017
Internet address

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