In this talk we consider infinite-constrained optimization problems in Banach spaces with both equality and inequality constraints. Regularity conditions for these problems are usually expressed with the help of Robinson and Kurcyusz-Zowe constraint qualifications which are difficult to be checked and fail to hold in many practical applications. In general, Slater-type conditions and surjectivity of the derivative of active constraints imply Robinson, and Kurcyusz-Zowe regularity conditions. Our aim is to discuss regularity conditions when the derivative is not necessarily surjective.
We introduce sufficient conditions for the non-emptiness of the set of Lagrange multipliers.
Our basic tools is the rank theorem and a generalization of Lusternik's theorem. The talk is based on the joint work with Krzysztof Leśniewski and Krzysztof Rutkowski.