Continuity of the optimal value function in indefinite quadratic programming
Topic of the paper is the continuity and semicontinuity of the optimal value function \[ (D,A,c,b)\to\varphi(D,A,c,b) \] of the quadratic programming problem \[ \text{minimize } c^{T}x+\tfrac{1}{2}x^{T}Dx \;\text{s.t.} Ax\geq b, \] where, in contrast to previous results of other authors, \(D\) is an arbitrary symmetric and hence not necessarily positive semidefinite matrix. The continuity of \(\varphi\) at \((D,A,c,b)\) is completely characterized by the Slater condition for the system \(Ax\geq b\) (C1) and the condition that \(x=0\) is the only solution of the problem for \(c=0\) and \(b=0\) (C2). Moreover, the conditions (C1) and (C2) are shown to be sufficient but not necessary for the upper and lower semicontinuity of \(\varphi\) at \((D,A,c,b)\) respectively. It is finally verified that the sufficient assumptions of \textit{M. J. Best} and \textit{B. Ding} [J. Optim. Theory Appl. 86, 245-250 (1995; Zbl 0838.90118)] for the lower semicontinuity and the continuity of \(\varphi\) are somewhat weaker than the corresponding assumptions of the author, but they only apply to the case of positive semidefinite \(D\) and, as is shown by examples, do not suffice to guarantee the respective property for general indefinite matrices \(D\).
scientific article; zbMATH DE number 1746310