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We consider a new hierarchy of semidefinite relaxations for the generalpolynomial optimization problem $(P):\:f^{\ast}=\min \{\,f(x):x\in K\,\}$ on acompact basic semi-algebraic set $K\subset\R^n$. This hierarchy combines someadvantages of the standard LP-relaxations associated with Krivine's positivitycertificate and some advantages of the standard SOS-hierarchy. In particular ithas the following attractive features: (a) In contrast to the standardSOS-hierarchy, for each relaxation in the hierarchy, the size of the matrixassociated with the semidefinite constraint is the same and fixed in advance bythe user. (b) In contrast to the LP-hierarchy, finite convergence occurs at thefirst step of the hierarchy for an important class of convex problems. Finally(c) some important techniques related to the use of point evaluations fordeclaring a polynomial to be zero and to the use of rank-one matrices make anefficient implementation possible. Preliminary results on a sample of nonconvex problems are encouraging.

A bounded degree SOS hierarchy for polynomial optimization