A bounded degree SOS hierarchy for polynomial optimization

Jean B. Lasserre (), Kim-Chuan Toh () and Shouguang Yang ()
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Jean B. Lasserre: University of Toulouse
Kim-Chuan Toh: National University of Singapore
Shouguang Yang: National University of Singapore

EURO Journal on Computational Optimization, 2017, vol. 5, issue 1, No 4, 87-117

Abstract: Abstract We consider a new hierarchy of semidefinite relaxations for the general polynomial optimization problem $$(P):\,f^{*}=\min \{f(x):x\in K\}$$ ( P ) : f ∗ = min { f ( x ) : x ∈ K } on a compact basic semi-algebraic set $$K\subset \mathbb {R}^n$$ K ⊂ R n . 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.

Keywords: Global optimization; Polynomial optimization; convex relaxations; LP and semidefinite hierarchies; 90C26; 90C22 (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (11)

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DOI: 10.1007/s13675-015-0050-y

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