Global infimum of strictly convex quadratic functions with bounded perturbations

The problem of minimizing f̃ = f +p over some convex subset of a Euclidean space is investigated, where f(x) = x Ax + b x is a strictly convex quadratic function and |p| is only assumed to be bounded by some positive number s. It is shown that the function f̃ 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 f̃ is its global minimal solution and the diameter of the set of global minimal solutions of f̃ is less than or equal to γ∗. Especially, the distance between the global minimal solution of f and any global minimal solution of f̃ is less than or equal to γ∗/2. This property is used to prove the rough support property of f̃ and some generalized optimality conditions.