On Polynomial Optimization over Non-compact Semi-algebraic Sets

Vaithilingam Jeyakumar (1) , Jean-Bernard Lasserre (2, 3) , G. Li (1)

1 UNSW - University of New South Wales [Sydney]
2 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
3 IMT - Institut de Mathématiques de Toulouse UMR5219

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.

Mots clés: Polynomial optimization; non-compact semi-algebraic sets; semidefinite programming relaxations; Positivstellensatze

Domaines: Optimisation et contrôle [math.OC]

Vaithilingam Jeyakumar, Jean-Bernard Lasserre, G. Li. On Polynomial Optimization over Non-compact Semi-algebraic Sets. Journal of Optimization Theory and Applications, 2014, 163, pp.707--718. ⟨10.1007/s10957-014-0545-3⟩. ⟨hal-00813962v2⟩