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

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.