A bounded degree SOS hierarchy for polynomial optimization

We consider a new hierarchy of semidefinite relaxations for the general polynomial optimization problem (P):f∗=min{f(x):x∈K} on a compact basic semi-algebraic set K⊂Rn. This hierarchy combines some advantages of the standard LP-relaxations associated with Krivine’s positivity certificate and some ad...

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Bibliographic Details
Published in:EURO journal on computational optimization Vol. 5; no. 1-2; pp. 87 - 117
Main Authors: Lasserre, JeanB, Toh, Kim-Chuan, Yang, Shouguang
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Elsevier Ltd 01.03.2017
Springer Berlin Heidelberg
Springer Nature B.V
Springer
Elsevier
Subjects:
90C22
90C26
Analysis
Business and Management
Computer programming
Convergence
convex relaxations
Global optimization
Hierarchies
LP and semidefinite hierarchies
Management Science
Mathematical analysis
Mathematics
Matrices (mathematics)
Matrix methods
Operations Management
Operations Research
Operations Research/Decision Theory
Optimization
Optimization and Control
Optimization techniques
Original Paper
Polynomial optimization
Polynomials
Software
Studies
Variables
Global optimization
90C26
Polynomial optimization
LP and semidefinite hierarchies
90C22
convex relaxations
Polynomial Optimization
LP-hierarchies
Semidefinite hierarchies
ISSN:2192-4406, 2192-4414
Online Access:Get full text
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Summary:We consider a new hierarchy of semidefinite relaxations for the general polynomial optimization problem (P):f∗=min{f(x):x∈K} on a compact basic semi-algebraic set K⊂Rn. This hierarchy combines some advantages of the standard LP-relaxations associated with Krivine’s positivity certificate and some advantages of the standard SOS-hierarchy. In particular it has the following attractive features: (a) in contrast to the standard SOS-hierarchy, for each relaxation in the hierarchy, the size of the matrix associated with the semidefinite constraint is the same and fixed in advance by the user; (b) in contrast to the LP-hierarchy, finite convergence occurs at the first step of the hierarchy for an important class of convex problems; and (c) some important techniques related to the use of point evaluations for declaring a polynomial to be zero and to the use of rank-one matrices make an efficient implementation possible. Preliminary results on a sample of non convex problems are encouraging.
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ISSN:2192-4406
2192-4414
DOI:10.1007/s13675-015-0050-y