Spectrahedral Shadows and Completely Positive Maps on Real Closed Fields

Abstract

In this article we develop new methods for exhibiting convex semialgebraic sets that are not spectrahedral shadows. We characterize when the set of nonnegative polynomials with a given support is a spectrahedral shadow in terms of sums of squares. As an application of this result we prove that the cone of copositive matrices of size n ≥ 5 is not a spectrahedral shadow, answering a question of Scheiderer. Our arguments are based on the model theoretic observation that any formula defining a spectrahedral shadow must be preserved by every unital R-linear completely positive map R → R on a real closed field extension R of R.

Copositive matrices, sums of squares and the stability number of a graph

“…Hence, the cone K ( ) is semidefinite representable, which means that membership in it can be modeled using semidefinite programming. In [4] it is shown that COP 5 is not semidefinite representable, which is thus a stronger result that implies Theorem 12. On the other hand, it was shown recently in [52] that every 5 × 5 copositive matrix belongs to the cone K ( ) 5 for some ∈ N.…”

“…Based on the result in Putinar's theorem, Lasserre [28] proposed a hierarchy of approximations ( ( ) ) ∈N for problem (4). Given an integer ∈ N, the quadratic module truncated at degree (generated by the set = { 1 , .…”

“…Clearly, ( ) ≤ ( +1) ≤ * for all ∈ N. The hierarchy of parameters ( ) is also known as Lasserre sum-of-squares hierarchy for problem (4).…”

“…Theorem 5 (Nie [40]) Assume that the Archimedean condition (15) holds for the polynomial sets and ℎ in problem (4). If the constraint qualification condition (CQC), the strict complementarity condition (SCC), and the second order sufficiency condition (SOSC) hold at every global minimizer of ( 4), then the Lasserre hierarchy (17) has finite convergence, i.e., ( ) = * for some ∈ N.…”