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