Global infimum of strictly convex quadratic functions with bounded perturbations

Abstract

The problem of minimizing \({\tilde f=f+p}\) over some convex subset of a Euclidean space is investigated, where f(x) = xTAx + bTx is strictly convex and |p| is only assumed to be bounded by some positive number s. It is shown that the function \({\tilde f}\) is strictly outer γ-convex for any γ > γ*, where γ* is determined by s and the smallest eigenvalue of A. As consequence, a γ*-local minimal solution of \({\tilde f}\) is its global minimal solution and the diameter of the set of global minimal solutions of \({\tilde f}\) is less than or equal to γ*. Especially, the distance between the global minimal solution of f and any global minimal solution of \({\tilde f}\) is less than or equal to γ*/2. This property is used to prove a roughly generalized support property of \({\tilde f}\) and some generalized optimality conditions.