Crease Structure of the Karush-Kuhn-Tucker Set in Parametric Optimization

摘要

Nonlinear optimization problems P(y) in R (the set of real numbers) to the npower depending on assumed parameter Y is a member of the set R to the P power are addressed. The input data are assumed to be twice continuously differentiable in both variables x and y. Concern is on the set closed Sigma of pairs (x, y) with the following property: x is a Karush-Kuhn-Tucker point for P(y). Special interest is met by the subset Sigma of closed Sigma with the additional property, that the Mangasarian-Fromovitz constraint qualification for the problem P(y) holds at point x. For input data in general position, the following results are obtained. Firstly, the set Sigma is (open and) dense in closed Sigma. Secondly, a (nontrivial) weak crease structure for Sigma is obtained. The latter constitutes a partition of Sigma being refined by any stratification of Sigma which satisfies the frontier condition with connected elements. This main result gives a lower bound for the complexity of the real crease structure and also works as a tool in the search for stratifications.