Some properties of boundedly perturbed strictly convex quadratic functions
نویسنده
چکیده
We investigate the problem (P̃ ) of minimizing f̃(x) := f(x) + p(x) subject to x ∈ D, where f(x) := x Ax + b x, A is a symmetric positive definite n-by-n matrix, b ∈ R, D ⊂ R is convex, and p : R → R satisfies supx∈D |p(x)| ≤ s for some given s < +∞. p is called perturbation, but it may describe some errors caused by modeling, measurement, approximation and calculation. We prove that the strict convexity of f is not completely destroyed by perturbation p, but the perturbed f̃ is still strictly outer Γconvex for some specified balanced set Γ ⊂ R. As 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∗ ∈ 12 Γ holds if x∗ is the optimal solution of the problem of minimizing f on D and x̃∗ is an arbitrary global optimal solution of (P̃ ), we show that the set Ss of global optimal solutions of (P̃ ) is stable with respect to the Hausdorff metric dH(., .). Moreover, the roughly generalized subdifferentiability of f̃ and a generalization of Kuhn-Tucker Theorem for (P̃ ) are presented.
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تاریخ انتشار 2009