On the Stable Sequential Kuhn-Tucker Theorem and Its Applications

Affiliation(s)

Mathematical Department, Nizhnii Novgorod State University, Nizhnii Novgorod, Russia.

Mathematical Department, Nizhnii Novgorod State University, Nizhnii Novgorod, Russia.

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.

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.

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.

M. Sumin, "On the Stable Sequential Kuhn-Tucker Theorem and Its Applications,"

References

[1] V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, “Optimal Control,” Nauka, Moscow, 1979.

[2] F. P. Vasil’ev, “Optimization Methods,” Faktorial Press, Moscow, 2002.

[3] M. Minoux, “Mathematical Programming. Theory and Algorithms,” Wiley, New York, 1986.

[4] M. I. Sumin, “Duality-Based Regularization in a Linear Convex Mathematical Programming Problem,” Journal of Computational Mathematics and Mathematical Physics, Vol. 47, No. 4, 2007, pp. 579-600. doi:10.1134/S0965542507040045

[5] M. I. Sumin, “Ill-Posed Problems and Solution Methods. Materials to Lectures for Students of Older Years. TextBook,” Lobachevskii State University of Nizhnii Novgorod, Nizhnii Novgorod, 2009.

[6] M. I. Sumin, “Parametric Dual Regularization in a Linear-Convex Mathematical Programming,” In: R. F. Linton and T. B. Carrol, Jr., Eds., Computational Optimization: New Research Developments, Nova Science Publishers Inc., New-York, 2010, pp. 265-311.

[7] M. I. Sumin, “Regularized Parametric Kuhn-Tucker Theorem in a Hilbert Space,” Journal of Computational Mathematics and Mathematical Physics, Vol. 51, No. 9, 2011, pp. 1489-1509. doi:10.1134/S0965542511090156

[8] J. Warga, “Optimal Control of Differential and Functional Equations,” Academic Press, New York, 1972.

[9] A. N. Tikhonov and V. Ya Arsenin, “Methods of Solution of Ill-Posed Problems,” Halsted, New York, 1977.

[10] M. I. Sumin, “A Regularized Gradient Dual Method for Solving Inverse Problem of Final Observation for a Parabolic Equation,” Journal of Computational Mathematics and Mathematical Physics, Vol. 44, No. 12, 2004, pp. 1903-1921.

[11] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural’tseva, “Linear and Quasilinear Equations of Parabolic Type,” Nauka, Moscow, 1967.

[12] V. I. Plotnikov, “Uniqueness and Existence Theorems and a Priori Properties of Generalized Solutions,” Doklady Akademii Nauk SSSR, Vol. 165, No. 1, 1965, pp. 33-35.

[1] V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, “Optimal Control,” Nauka, Moscow, 1979.

[2] F. P. Vasil’ev, “Optimization Methods,” Faktorial Press, Moscow, 2002.

[3] M. Minoux, “Mathematical Programming. Theory and Algorithms,” Wiley, New York, 1986.

[4] M. I. Sumin, “Duality-Based Regularization in a Linear Convex Mathematical Programming Problem,” Journal of Computational Mathematics and Mathematical Physics, Vol. 47, No. 4, 2007, pp. 579-600. doi:10.1134/S0965542507040045

[5] M. I. Sumin, “Ill-Posed Problems and Solution Methods. Materials to Lectures for Students of Older Years. TextBook,” Lobachevskii State University of Nizhnii Novgorod, Nizhnii Novgorod, 2009.

[6] M. I. Sumin, “Parametric Dual Regularization in a Linear-Convex Mathematical Programming,” In: R. F. Linton and T. B. Carrol, Jr., Eds., Computational Optimization: New Research Developments, Nova Science Publishers Inc., New-York, 2010, pp. 265-311.

[7] M. I. Sumin, “Regularized Parametric Kuhn-Tucker Theorem in a Hilbert Space,” Journal of Computational Mathematics and Mathematical Physics, Vol. 51, No. 9, 2011, pp. 1489-1509. doi:10.1134/S0965542511090156

[8] J. Warga, “Optimal Control of Differential and Functional Equations,” Academic Press, New York, 1972.

[9] A. N. Tikhonov and V. Ya Arsenin, “Methods of Solution of Ill-Posed Problems,” Halsted, New York, 1977.

[10] M. I. Sumin, “A Regularized Gradient Dual Method for Solving Inverse Problem of Final Observation for a Parabolic Equation,” Journal of Computational Mathematics and Mathematical Physics, Vol. 44, No. 12, 2004, pp. 1903-1921.

[11] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural’tseva, “Linear and Quasilinear Equations of Parabolic Type,” Nauka, Moscow, 1967.

[12] V. I. Plotnikov, “Uniqueness and Existence Theorems and a Priori Properties of Generalized Solutions,” Doklady Akademii Nauk SSSR, Vol. 165, No. 1, 1965, pp. 33-35.