Convergent SDP-relaxations for polynomial optimization with sparsity

Abstract

We consider a polynomial programming problem ℙ on a compact basic semi-algebraic set double-script K sign ⊂ ℝn, described by m polynomial inequalities gj(X) ≥ 0, and with criterion f ∈ ℝ[X]. We propose a hierarchy of semidefinite relaxations that take sparsity of the original data into account, in the spirit of those of Waki et al. [7]. The novelty with respect to [7] is that we prove convergence to the global optimum of P when the sparsity pattern satisfies a condition often encountered in large size problems of practical applications, and known as the running intersection property in graph theory. © Springer-Verlag Berlin Heidelberg 2006.