On regularization of the nondifferential Kuhn–Tucker theorem in a nonlinear problem for constrained extremum

We consider a regular parametric nonlinear (nonconvex) problem for constrained extremum with an operator equality constraint and a finite number of functional inequality constraints. The constraints of the problem contain additive parameters, which makes it possible to use the apparatus of the “nonlinear” perturbation method for its study. The set of admissible elements of the problem is a complete metric space, and the problem itself may not have a solution. The regularity of the problem is understood in the sense that it has a generalized Kuhn–Tucker vector. Within the framework of the ideology of the Lagrange multiplier method, a regularized nondifferential Kuhn–Tucker theorem is formulated and proved, the main purpose of which is the stable generation of generalized minimizing sequences in the problem under consideration. These minimizing sequences are constructed from subminimals (minimals) of the modified Lagrange function taken at the values of the dual variable generated by the corresponding regularization procedure for the dual problem. The construction of the modified Lagrange function is a direct consequence of the subdifferential properties of a lower semicontinuous and, generally speaking, nonconvex value function as a function of the problem parameters. The regularized Kuhn–Tucker theorem “overcomes” the instability properties of its classical counterpart, is a regularizing algorithm, and serves as a theoretical basis for creating algorithms of practical solving problems for constrained extremum.