On Polynomial Optimization over Non-compact Semi-algebraic Sets
Résumé
We first prove that a simplified form of Krivine-Stengle's representation for positive polynomials over a non-compact basic semi-algebraic set $K$ holds generically. This representation is much simpler as it only involves the quadratic module associated with the polynomials that define $K$ and an SOS multiplier for the positive polynomial. Then inspired by this representation we consider the class of polynomial optimization problems $\inf \{f(x):x\in K\}$ for which the quadratic module generated by the polynomials that define $K$ and the polynomial $c-f$ (for some scalar $c$) is Archimedean. For such problems, the optimal value can be approximated as closely as desired by solving a hierarchy of semidefinite programs and the convergence is finite generically. Moreover, the Archimedean condition (as well as a sufficient coercivity condition) can also be checked numerically by solving a similar hierarchy of semidefinite programs. In other words, under reasonable assumptions the now standard hierarchy of SDP-relaxations extends to the non-compact case via a suitable modification.
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