Global infimum of strictly convex quadratic functions with bounded perturbations
The problem of minimizing <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${\tilde f=f+p}$$</EquationSource> </InlineEquation> over some convex subset of a Euclidean space is investigated, where f(x) = x <Superscript> T </Superscript> Ax + b <Superscript> T </Superscript> x is strictly convex and |p| is only assumed to be bounded by some positive number s. It is shown that the function <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$${\tilde f}$$</EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$${\tilde f}$$</EquationSource> </InlineEquation> is its global minimal solution and the diameter of the set of global minimal solutions of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$${\tilde f}$$</EquationSource> </InlineEquation> is less than or equal to γ*. Especially, the distance between the global minimal solution of f and any global minimal solution of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$${\tilde f}$$</EquationSource> </InlineEquation> is less than or equal to γ*/2. This property is used to prove a roughly generalized support property of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$${\tilde f}$$</EquationSource> </InlineEquation> and some generalized optimality conditions. Copyright Springer-Verlag 2010
