Journal
EURO JOURNAL ON COMPUTATIONAL OPTIMIZATION
Volume 5, Issue 1-2, Pages 87-117Publisher
ELSEVIER
DOI: 10.1007/s13675-015-0050-y
Keywords
Global optimization; Polynomial optimization; convex relaxations; LP and semidefinite hierarchies
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We consider a new hierarchy of semidefinite relaxations for the general polynomial optimization problem (P) : f* = min{f(x) : x is an element of K} on a compact basic semi-algebraic set K subset of 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.
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