Global infimum of strictly convex quadratic functions with bounded perturbations

The problem of minimizing $${\tilde f=f+p}$$ over some convex subset of a Euclidean space is investigated, where f(x) = x T Ax + b T x 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. Copyright Springer-Verlag 2010