Euler-Lagrange equation in the case of nonregular equality constraints

An optimization problem with inequality and equality constraints in Banach spaces is considered in the case when the operators which determine the equality constraints are nonregular. In this case, the classical Euler-Lagrange equation has the degenerate form, i.e., does not depend on the functional to be minimized. Applying some generalization of the Lusternik theorem to the Dubovitskii-Milyutin method, the family of Euler-Lagrange equations is obtained in the nondegenerate form under the assumption of twice Fréchet differentiability of the operators. The Pareto-optimal problem is also considered.The author would like to thank the reviewers for valuable suggestions and remarks. This research was partially supported by the SIUE Research Scholar Award.