Nondifferential Kuhn–Tucker theorems in constrained extremum problems via subdifferentials of nonsmooth analysis

The paper is devoted to obtaining Kuhn-Tucker theorems in nondifferential form in constrained extremum problems in a Hilbert space. The constraints of the problems are specified by operators whose images are also embedded in a Hilbert space. These constraints contain parameters that are additively included in them. The basis for obtaining nondifferential Kuhn-Tucker theorems is the so-called perturbation method. The article consists of two main sections. The first of them is devoted to obtaining the nondifferential Lagrange principle in the case when the constrained extremum problem is convex. In this case, the Kuhn-Tucker theorem is its “regular part”. Various statements are also presented here that relate the Lagrange multipliers to the subdifferentiability properties of the convex value function of the problem. The main purpose of the first section is to trace how the classical construction of the Lagrange function in its regular and nonregular forms is “generated” by subdifferentials and asymptotic subdifferentials of the value function. This circumstance and the results of the first section make it possible to transfer the natural bridge from the convex parametric constrained extremum problems to similar nonlinear parametric problems of the second section in which the value function, generally speaking, is not convex. The central role here is played not by subdifferentials in the sense of convex analysis, but by subdifferentials of nonsmooth (nonlinear) analysis. As a result, in this case, the so-called modified (not classical) Lagrange function acts as the main construction. Its construction depends entirely on how subdifferentiability is understood in the sense of nonsmooth (nonlinear) analysis.