On the Stable Sequential Kuhn-Tucker Theorem and Its Applications

Mikhail I.  Sumin

doi:10.4236/am.2012.330190

Mikhail I. Sumin^*

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.

Keywords: Sequential Optimization; Minimizing Sequence; Stable Kuhn-Tucker Theorem in Nondifferential Form; Convex Programming; Duality; Regularization; Optimal Control; Inverse Problems

Cite this paper: M. Sumin, "On the Stable Sequential Kuhn-Tucker Theorem and Its Applications," Applied Mathematics, Vol. 3 No. 10, 2012, pp. 1334-1350. doi: 10.4236/am.2012.330190.

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