On semidefinite-representable sets over valued fields

2026-02-10

AI Summary

This study investigates the structure and optimization of semidefinite representable sets over valued fields, extending the classical notions of polyhedra and spectrahedra from the real numbers to this more general setting. Semidefinite representable sets over a valued field \( K \) are defined as linear images of feasible sets of linear matrix inequalities, and Smith normal form algorithms are employed to analyze \( K \)-polyhedra and associated linear programs. The main contributions include the first explicit construction of a \( K \)-spectrahedron that is not a \( K \)-polyhedron, the demonstration that key structural properties of real semidefinite representable sets carry over to the valued-field context, and the discovery of sets that admit semidefinite representations yet fail to be \( K \)-spectrahedra—thereby transcending the classical framework and significantly broadening the scope of optimization theory over valued fields.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Non-convex Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

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.

Problem

• semidefinite-representable sets
• valued fields
• spectrahedra
• linear programming
• optimization

Innovation

• semidefinite-representable sets
• valued fields
• Smith normal form
• K-spectrahedra
• linear programming over K