TY - JOUR

T1 - Perturbed sums-of-squares theorem for polynomial optimization and its applications

AU - Muramatsu, Masakazu

AU - Waki, Hayato

AU - Tunçel, Levent

N1 - Funding Information:
We thank two anonymous referees for their valuable comments to improve the paper. The first author was supported in part by JSPS KAKENHI Grant Numbers 19560063 and 26330025. The second author was supported by JSPS KAKENHI Grant Numbers 22740056 and 26400203. The third author was supported in part by a Discovery Grant from NSERC, a research grant from University of Waterloo and by ONR research grant N00014-12-10049.

PY - 2016/1/2

Y1 - 2016/1/2

N2 - We consider a property of positive polynomials on a compact set with a small perturbation. When applied to a polynomial optimization problem (POP), the property implies that the optimal value of the corresponding SemiDefinite Programming (SDP) relaxation with sufficiently large relaxation order is bounded from below by (f∗-ε) and from above by f∗+ ε(n + 1), where f∗is the optimal value of the POP. We propose new SDP relaxations for POP based on modifications of existing sums-of-squares representation theorems. An advantage of our SDP relaxations is that in many cases they are of considerably smaller dimension than those originally proposed by Lasserre. We present some applications and the results of our computational experiments.

AB - We consider a property of positive polynomials on a compact set with a small perturbation. When applied to a polynomial optimization problem (POP), the property implies that the optimal value of the corresponding SemiDefinite Programming (SDP) relaxation with sufficiently large relaxation order is bounded from below by (f∗-ε) and from above by f∗+ ε(n + 1), where f∗is the optimal value of the POP. We propose new SDP relaxations for POP based on modifications of existing sums-of-squares representation theorems. An advantage of our SDP relaxations is that in many cases they are of considerably smaller dimension than those originally proposed by Lasserre. We present some applications and the results of our computational experiments.

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U2 - 10.1080/10556788.2015.1052969

DO - 10.1080/10556788.2015.1052969

M3 - Article

AN - SCOPUS:84953636058

VL - 31

SP - 134

EP - 156

JO - Optimization Methods and Software

JF - Optimization Methods and Software

SN - 1055-6788

IS - 1

ER -