Mathematical Programs with Geometric Constraints in Banach Spaces: Enhanced Optimality, Exact Penalty, and Sensitivity

In this paper we study the mathematical program with geometric constraints such that the image of a mapping from a Banach space is included in a nonempty and closed subset of a finite dimensional space. We obtain the nonsmooth enhanced Fritz John necessary optimality conditions in terms of the approximate subdifferential. In the case where the Banach space is a weakly compactly generated Asplund space, the optimality condition obtained can be expressed in terms of the limiting subdifferential, while in the general case it can be expressed in terms of the Clarke subdifferential. One of the technical difficulties in obtaining such a result in an infinite dimensional space is that no compactness result can be used to show the existence of local minimizers of a perturbed problem. In this paper we employ the celebrated Ekeland’s variational principle to obtain the results instead. The enhanced Fritz John condition allows us to obtain the enhanced Karush– Kuhn–Tucker condition under the pseudo-normality and the quasi-normality conditions which are weaker than the classical normality conditions. We then prove that the quasi-normality is a sufficient condition for the existence of local error bounds of the constraint system. Finally we obtain a tighter upper estimate for the subdifferentials of the value function of the perturbed problem in terms of the enhanced multipliers.