Some properties of boundedly perturbed strictly convex quadratic functions

We investigate the problem ((P) over tilde) of minimizing (f) over tilde (x) := f(x) + p(x) subject to x is an element of D, where f(x) := x(T)Ax + b(T)x, A is a symmetric positive definite n-by-n matrix, b is an element of R-n, D subset of R-n is convex and p : R-n -> R satisfies sup(x is an element of D)vertical bar p(x)vertical bar <= s for some given s < +infinity. Function p is called a perturbation, but it may also describe some correcting term, which arises when investigating a real inconvenient objective function (f) over tilde by means of an idealized convex quadratic function f. We prove that (f) over tilde is strictly outer Gamma-convex for some specified balanced set Gamma subset of R-n. As a consequence, a Gamma-local optimal solution of ((P) over tilde) is global optimal and the difference of two arbitrary global optimal solutions of ((P) over tilde) is contained in Gamma. By the property that x* - (x) over tilde* is an element of 1/2 Gamma holds if x* is the optimal solution of the problem of minimizing f on D and (x) over tilde* is an arbitrary global optimal solution of ((P) over tilde),we show that the set S-s of global optimal solutions of ((P) over tilde) is stable with respect to the Hausdorff metric d(H)(.,.). Moreover, the roughly generalized subdifferentiability of (f) over tilde and a generalization of Kuhn-Tucker theorem for ((P) over tilde) are presented.