Quadratic Programming over Linearly Ordered Fields: Decidability and Attainment of Optimal Solutions

TL;DR

This paper develops an algebraic framework for solving quadratic programming problems over general linearly ordered fields, proving that bounded below functions attain their minimum and providing an exact algorithm for finding solutions within these fields.

Contribution

It introduces a unified algebraic approach to quadratic programming over ordered fields, extending classical results beyond real numbers and offering a deterministic decision algorithm.

Findings

Proves bounded below quadratic functions attain minima in the field.

Develops an exact, deterministic algorithm for quadratic programming over ordered fields.

Establishes linearly constrained quadratic programming as the maximal class with guaranteed exact solutions.

Abstract

Classical existence theorems and solution methods for quadratic programming traditionally rely on the analytical properties of real numbers, specifically compactness and completeness. These tools are unavailable in general linearly ordered fields, such as the field of rational numbers or non-Archimedean structures, rendering standard analytical proofs insufficient in these general algebraic settings. In this paper, we establish a unified algebraic framework for the decidability of indefinite quadratic programming subject to linear constraints over general linearly ordered fields. We prove a generalized Eaves' theorem, demonstrating that if a quadratic function -- encompassing convex, non-convex, or degenerate (linear) cases -- is bounded from below on a polyhedron, the minimum is attained within the field itself, regardless of topological completeness. Our approach replaces classical analytical arguments with algebraic induction on dimension and polyhedral decomposition. Based on this foundation, we propose an exact, deterministic algorithm within the Blum--Shub--Smale model of computation that decides boundedness and computes a global minimizer using only field operations. We show that the problem is solvable in finite time via a recursive search over orthant-restricted facets. Finally, we note that linearly constrained quadratic programming represents the maximal class of polynomial optimization problems where exact solutions are structurally guaranteed within the original field, thereby demarcating the algebraic boundary of exact optimization over ordered structures.