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.

翻译：实数域上的多面体与谱面体，或其在线性映射下的像，分别是线性规划与半定规划的可行集，并构成了半定可表示集族。本文研究当数据取自赋值域 $K$ 时，此类集合及其相关优化问题的类比。对于 $K$-多面体及 $K$ 上的线性规划，我们提出了一种基于计算史密斯标准形的算法。我们证明了半定可表示集的基本性质在赋值设定下依然成立。特别地，我们展示了非多面体的 $K$-谱面体实例，以及那些在 $K$ 上半定可表示但并非 $K$-谱面体的集合。