Sum of squares basis pursuit with linear and second order cone programming

Amir Ali Ahmadi, Georgina Hall

Research output: Chapter in Book/Report/Conference proceedingChapter

8 Scopus citations

Abstract

We devise a scheme for solving an iterative sequence of linear programs (LPs) or second order cone programs (SOCPs) to approximate the optimal value of any semidefinite program (SDP) or sum of squares (SOS) program. The first LP and SOCP-based bounds in the sequence come from the recent work of Ahmadi and Majumdar on diagonally dominant sum of squares (DSOS) and scaled diagonally dominant sum of squares (SDSOS) polynomials. We then iteratively improve on these bounds by pursuing better bases in which more relevant SOS polynomials admit a DSOS or SDSOS representation. Different interpretations of the procedure from primal and dual perspectives are given. While the approach is applicable to SDP relaxations of general polynomial programs, we apply it to two problems of discrete optimization: the maximum independent set problem and the partition problem. We further show that some completely trivial instances of the partition problem lead to strictly positive polynomials on the boundary of the sum of squares cone and hence make the SOS relaxation fail.

Original languageEnglish (US)
Title of host publicationContemporary Mathematics
PublisherAmerican Mathematical Society
Pages27-53
Number of pages27
DOIs
StatePublished - 2017

Publication series

NameContemporary Mathematics
Volume685
ISSN (Print)0271-4132
ISSN (Electronic)1098-3627

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Fingerprint Dive into the research topics of 'Sum of squares basis pursuit with linear and second order cone programming'. Together they form a unique fingerprint.

  • Cite this

    Ahmadi, A. A., & Hall, G. (2017). Sum of squares basis pursuit with linear and second order cone programming. In Contemporary Mathematics (pp. 27-53). (Contemporary Mathematics; Vol. 685). American Mathematical Society. https://doi.org/10.1090/conm/685/13712