Global infimum of strictly convex quadratic functions with bounded perturbations

Hoang Phu () and Vo Pho ()

Mathematical Methods of Operations Research, 2010, vol. 72, issue 2, 327-345

Abstract: 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

Keywords: Quadratic function; Convexity modulus; Generalized convexity; Outer γ-convexity; Bounded perturbation; Global minimizer; Support property; Optimality condition; 52A01; 52A41; 47H14; 90C20; 90C26 (search for similar items in EconPapers)
Date: 2010
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DOI: 10.1007/s00186-010-0324-3

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