On semidefinite-representable sets over valued fields

TL;DR

This paper extends the theory of semidefinite-representable sets to valued fields, providing algorithms and demonstrating fundamental properties and examples in this generalized setting.

Contribution

It introduces algorithms for $K$-polyhedra, proves properties of semidefinite-representable sets over valued fields, and presents novel examples distinguishing these sets.

Findings

Algorithms for $K$-polyhedra based on Smith normal forms.

Fundamental properties of semidefinite-representable sets extend to valued fields.

Existence of non-polyhedral $K$-spectrahedra and sets that are semidefinite-representable but not $K$-spectrahedra.

Abstract

Polyhedra and spectrahedra over the real numbers, or more generally their images under linear maps, are respectively the feasible sets of linear and semidefinite programming, and form the family of semidefinite-representable sets. This paper studies analogues of these sets, as well as the associated optimization problems, when the data are taken over a valued field $K$. For $K$-polyhedra and linear programming over $K$ we present an algorithm based on the computation of Smith normal forms. We prove that fundamental properties of semidefinite-representable sets extend to the valued setting. In particular, we exhibit examples of non-polyhedral $K$-spectrahedra, as well as sets that are semidefinite-representable over $K$ but are not $K$-spectrahedra.