Some properties of boundedly perturbed strictly convex quadratic functions
Hoàng Xuân Phú, Vo Minh Pho · Optimization · 2010
We investigate the problem of minimizing subject to x ∈ D, where f(x) := x T Ax + b T x, 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 |p(x)| ≤ 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 by means of an idealized convex quadratic function f. We prove that is strictly outer Γ-convex for some specified balanced set Γ ⊂ ℝ n . As a consequence, a Γ-local optimal solution of is global optimal and the difference of two arbitrary global optimal solutions of is contained in Γ. By the property that holds if x* is the optimal solution of the problem of minimizing f on D and is an arbitrary global optimal solution of , we show that the set S s of global optimal solutions of is stable with respect to the Hausdorff metric d H (.,.). Moreover, the roughly generalized subdifferentiability of and a generalization of Kuhn–Tucker theorem for are presented.