Explicit minimisation of a convex quadratic under a general quadratic constraint: a global, analytic approach

TL;DR

This paper presents a comprehensive, explicit algebraic solution to the problem of minimizing a convex quadratic function subject to a general quadratic constraint, using geometric insights and affine transformations.

Contribution

It introduces a complete, explicit, and algebraic resolution for quadratic minimization under quadratic constraints, identifying special cases and leveraging geometric and affine equivalence techniques.

Findings

Provides explicit algebraic solutions for all cases

Highlights geometric and affine methods in quadratic optimization

Suggests potential integration with trust region methods

Abstract

A novel approach is introduced to a very widely occurring problem, providing a complete, explicit resolution of it: minimisation of a convex quadratic under a general quadratic, equality or inequality, constraint. Completeness comes via identification of a set of mutually exclusive and exhaustive special cases. Explicitness, via algebraic expressions for each solution set. Throughout, underlying geometry illuminates and informs algebraic development. In particular, centrally to this new approach, affine equivalence is exploited to re-express the same problem in simpler coordinate systems. Overall, the analysis presented provides insight into the diverse forms taken both by the problem itself and its solution set, showing how each may be intrinsically unstable. Comparisons of this global, analytic approach with the, intrinsically complementary, local, computational approach of (generalised) trust region methods point to potential synergies between them. Points of contact with simultaneous diagonalisation results are noted.