Constraint Qualifications for Infinite-Dimensional General Optimization Problems via Pseudo-Jacobians and Application to Infinite Programming

This paper concerns applications of advanced techniques of pseudo-Jacobians for studying a nonsmooth and nonconvex general optimization problem in Banach spaces. To this end, calculus rules for pseudo-Jacobians in infinite-dimensional spaces are presented; in particular, thwee chain rule for pseudo-Jacobians is developed. Then, using the idea of pseudo-Jacobian, the Robinson, Mangasarian- Fromovitz and Abadie constraint qualifications are generalized. The relations between these constraint qualifications and the local error bound property are investigated. Furthermore, a necessary optimality condition is derived under the Abadie constraint qualification. As an application, new constraint qualifications and necessary optimality conditions are obtained for nonsmooth and nonconvex infinite programming problems. In this way, a pseudo-Jacobian for the pointwise supremum of an arbitrary family of real-valued functions defined on a Banach space is provided. Some examples are given to clarify the results.