On the Stable Sequential Kuhn-Tucker Theorem and Its Applications

Abstract

The Kuhn-Tucker theorem in nondifferential form is a well-known classical optimality criterion for a convex programming problems which is true for a convex problem in the case when a Kuhn-Tucker vector exists. It is natural to extract two features connected with the classical theorem. The first of them consists in its possible "impracticability" (the Kuhn-Tucker vector does not exist). The second feature is connected with possible "instability" of the classical theorem with respect to the errors in the initial data. The article deals with the so-called regularized Kuhn-Tucker theorem in nondifferential sequential form which contains its classical analogue. A proof of the regularized theorem is based on the dual regularization method. This theorem is an assertion without regularity assumptions in terms of minimizing sequences about possibility of approximation of the solution of the convex programming problem by minimizers of its regular Lagrangian, that are constructively generated by means of the dual regularization method. The major distinctive property of the regularized Kuhn-Tucker theorem consists that it is free from two lacks of its classical analogue specified above. The last circumstance opens possibilities of its application for solving various ill-posed problems of optimization, optimal control, inverse problems.

Inverse scattering problems in subsurface diagnostics of inhomogeneous media

“…Of course, supercomputers should be used to realize this algorithm. As a next step in the development of inverse scattering methods of subsurface diagnostics, we intend to work out yet more effective and stable dual-regularization algorithms based on sequential approach to Kuhn-Tucker theorem [11].…”

Stable sequential Kuhn-Tucker theorem in iterative form or a regularized Uzawa algorithm in a regular nonlinear programming problem

“…To overcome difficulties associated with the instability of classical optimality conditions, an approach relying heavily on regularization theory [2,9] was proposed in [3,4,7,8] for deriving conditions on min imizing sequences that are stable against errors in initial data in the case of convex programming problems in a Hilbert space. It was found that, despite the substantial instability of optimization problems, stable conditions on elements of minimizing sequences can be obtained by combining optimization methods underlying classical optimality conditions and regularization methods based on duality theory [3,4,10,11].…”

“…It was found that, despite the substantial instability of optimization problems, stable conditions on elements of minimizing sequences can be obtained by combining optimization methods underlying classical optimality conditions and regularization methods based on duality theory [3,4,10,11]. Following this approach, conditions expressed in terms of minimizing sequences were obtained in [3,4,7,8]. An important feature of these conditions is that they are nondifferential in nature and entirely similar in structure to classical nondifferential optimality conditions.…”

“…Moreover, due to the appli cation of dual regularization [3,4,10,11], their formulations explicitly (constructively) indicate how a corresponding sequence of dual variables has to be chosen. Specifically, each element of this sequence is the solution of a Tikhonov regularized dual problem [3,4,7,8] in the case of the dual regularization method, while, in the case of the iterative dual regularization method, its elements are generated by an iterative formula representing a regularized gradient ascent related to maximization in the dual problem [7] or, equivalently, regularization of the classical Uzawa algorithm (see [12,13]). In both cases, the con dition imposed on every element of the minimizing sequence of feasible elements represents the usual optimality condition for the regular Lagrangian taken at the corresponding dual variable value.…”

“…In both cases, the con dition imposed on every element of the minimizing sequence of feasible elements represents the usual optimality condition for the regular Lagrangian taken at the corresponding dual variable value. In other words, obtained in [3,4,7,8], the conditions on minimizing sequences ensure a parallel stable construc tion of a minimizing primal variable sequence and a corresponding dual variable sequence, which is max imizing in the dual of the original problem. The dual variable sequence is bounded if the original problem has a Kuhn-Tucker vector solving the dual problem and converges to the minimum norm Kuhn-Tucker vector.…”