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A novel approach is introduced to a very widely occurring problem, providinga complete, explicit resolution of it: minimisation of a convex quadratic undera general quadratic, equality or inequality, constraint. Completeness comes viaidentification 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. Inparticular, centrally to this new approach, affine equivalence is exploited tore-express the same problem in simpler coordinate systems. Overall, theanalysis presented provides insight into the diverse forms taken both by theproblem itself and its solution set, showing how each may be intrinsicallyunstable. Comparisons of this global, analytic approach with the, intrinsicallycomplementary, local, computational approach of (generalised) trust regionmethods point to potential synergies between them. Points of contact withsimultaneous diagonalisation results are noted.

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