Some properties of boundedly perturbed strictly convex quadratic functions

We investigate the problem (P̃) of minimizing f̃(x):f(x)+P(x) subject to x ∈ D, where f(x):= x TAx + b Tx, A is a symmetric positive definite n-by-n matrix, b ∈ ℝ n, D ⊂ ℝ n is convex and p: ℝ n → ℝ satisfies sup x∈D{pipe}p(x){pipe} ≤ s for some given s < +∞. Function p is called a perturbation, but it may also describe some correcting term, which arises when investigating a real inconvenient objective function f̃ by means of an idealized convex quadratic function f. We prove that f̃ is strictly outer Γ-convex for some specified balanced set Γ ⊂ ℝ n. As a consequence, a Γ-local optimal solution of (P̃) is global optimal and the difference of two arbitrary global optimal solutions of (P̃)is contained in Γ. By the property that x* - x̃* ∈ 1/2 holds if x* is the optimal solution of the problem of minimizing f on D and is an arbitrary global optimal solution of (P̃), we show that the set S s of global optimal solutions of (P̃) is stable with respect to the Hausdorff metric d H(.,.). Moreover, the roughly generalized subdifferentiability of f̃ and a generalization of Kuhn-Tucker theorem for (P̃) are presented. © 2012 Copyright Taylor and Francis Group, LLC.