Regularized parametric Kuhn–Tucker theorem in a Hilbert space

For a parametric convex programming problem in a Hilbert space with a strongly convex objective functional, a regularized Kuhn–Tucker theorem in nondifferential form is proved by the dual regularization method. The theorem states (in terms of minimizing sequences) that the solution to the convex programming problem can be approximated by minimizers of its regular Lagrangian (which means that the Lagrange multiplier for the objective functional is unity) with no assumptions made about the regularity of the optimization problem. Points approximating the solution are constructively specified. They are stable with respect to the errors in the initial data, which makes it possible to effectively use the regularized Kuhn–Tucker theorem for solving a broad class of inverse, optimization, and optimal control problems. The relation between this assertion and the differential properties of the value function (S-function) is established. The classical Kuhn–Tucker theorem in nondifferential form is contained in the above theorem as a particular case. A version of the regularized Kuhn–Tucker theorem for convex objective functionals is also considered.