Title: 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.