Global infimum of strictly convex quadratic functions with bounded perturbations

Hoang Xuan Phu; Vo Minh Pho
October 2010
Mathematical Methods of Operations Research;Oct2010, Vol. 72 Issue 2, p327
Academic Journal
The problem of minimizing $${\tilde f=f+p}$$ over some convex subset of a Euclidean space is investigated, where f( x) = x Ax + b 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 ?-convex for any ? > ?*, where ?* is determined by s and the smallest eigenvalue of A. As consequence, a ?*-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 ?*. Especially, the distance between the global minimal solution of f and any global minimal solution of $${\tilde f}$$ is less than or equal to ?*/2. This property is used to prove a roughly generalized support property of $${\tilde f}$$ and some generalized optimality conditions.


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