A note on the representation of positive polynomials with structured sparsity

Abstract

Abstract. We consider real polynomials in finitely many variables. Let the variables consist of finitely many blocks that are allowed to overlap in a certain way. Let the solution set of a finite system of polynomial inequalities be given where each inequality involves only variables of one block. We investigate polynomials that are positive on such a set and sparse in the sense that each monomial involves only variables of one block. In particular, we derive a short and direct proof for Lasserre's theorem of the existence of sums of squares certificates respecting the block structure. The motivation for the results can be found in the literature and stems from numerical methods using semidefinite programming to simulate or control discrete-time behaviour of systems.

Correlative sparsity structures and semidefinite relaxations for concave cost transportation problems with change of variables

“…Lasserre later showed the convergence of their hierarchy in [19]. Similar convergence results were obtained by Grimm et al [8] in 2007 and Kojima and Muramatsu [14] in 2009 under weaker assumptions.…”

“…The above theorem follows from Theorem 5 of [8] by Grimm et al In fact, the proof is in Corollary 8.10 of the survey paper [20]. The assumption of the above theorem is slightly weaker than one of Lasserre in [19].…”

“…Thus, it is possible to evaluate the size of the matrix variables, and the maximum size is O ((min{p, q}) ω ) in the relaxation order ω. The hierarchies are guaranteed to converge to the global minimum of the CCTPs by the theorem of Grimm et al in [8]. Furthermore, we show that the SDP relaxations can be reduced in size by using Reznick's theorem in [25].…”

Sums of Squares, Moment Matrices and Optimization Over Polynomials

“…, k. Kojima and Muramatsu [62] proved the result for compact K with possibly empty interior. Grimm, Netzer and Schweighofer [43] give a simpler proof, which does not need the presence of ball constraints in the description of K but instead assumes that each set of polynomials g j (j ∈ J h ) generates an Archimedean module.…”

“…We give the proof of [43] which is elementary except it uses the following special case of Schmüdgen's theorem (Theorem 3.16):…”