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

Dmytro O. Plutenko

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.

Quadratic Programming over Linearly Ordered Fields: Decidability and Attainment of Optimal 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.
Paper Structure (14 sections, 10 theorems, 15 equations, 1 algorithm)

This paper contains 14 sections, 10 theorems, 15 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linearly Ordered Fields
  4. Polyhedra and Implicit Equalities
  5. Full-Dimensionality and Interior Points
  6. Facets and Formal Facets
  7. Decoupled Diagonal Decomposition of Quadratic Functions
  8. Existence of Minima
  9. Algorithmic Decidability of Boundedness
  10. Algorithmic Resolution and BSS Model
  11. Supporting Lemmas
  12. Proof of Theorem \ref{['algo_correctness']}
  13. Results and Discussion
  14. Conclusions

Key Result

proposition thmcounterproposition

Any non-empty polyhedron $P$ of dimension $k<n$ is affinely isomorphic to a full-dimensional polyhedron $P'\subseteq\mathbb{F}^{k}$.

Theorems & Definitions (25)

  • definition thmcounterdefinition: Full-Dimensionality
  • proposition thmcounterproposition: Dimension Reduction
  • proof
  • definition thmcounterdefinition: Formal Facet
  • lemma thmcounterlemma: Decoupled Diagonal Decomposition
  • proof
  • corollary thmcountercorollary: Single-Direction Linear Term
  • proof
  • remark thmcounterremark: Algorithmic Aspects
  • theorem 1: Generalized Eaves' Theorem
  • ...and 15 more