Perturbed sums-of-squares theorem for polynomial optimization and its applications. (2nd January 2016)

• Record Type: Journal Article
• Authors: Muramatsu, Masakazu; Waki, Hayato; Tunçel, Levent
• Is Part Of: Optimization methods and software. Volume 31:Number 1(2016)
• Page Start: 134
• Page End: 156
• Publication Date: 2016-01-02
• Subjects: polynomial optimization -- semidefinite relaxation -- sums-of-squares relaxation; Mathematical optimization -- Periodicals; Algorithms -- Periodicals; 519.7
• DOI: 10.1080/10556788.2015.1052969
• Languages: English
• ISSNs: 1055-6788

Abstract

Abstract : 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 and from above by, where 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.