A Generalization of the Karush–Kuhn–Tucker Theorem for Approximate Solutions of Mathematical Programming Problems Based on Quadratic Approximation
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In computations related to mathematical programming problems, one often has to consider approximate, rather than exact, solutions satisfying the constraints of the problem and the optimality criterion with a certain error. For determining stopping rules for iterative procedures, in the stability analysis of solutions with respect to errors in the initial data, etc., a justified characteristic of such solutions that is independent of the numerical method used to obtain them is needed. A necessary δ-optimality condition in the smooth mathematical programming problem that generalizes the Karush–Kuhn–Tucker theorem for the case of approximate solutions is obtained. The Lagrange multipliers corresponding to the approximate solution are determined by solving an approximating quadratic programming problem.
Keywordsapproximate solutions mathematical programming Karush–Kuhn–Tucker theorem quadratic programming
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- 2.E. G. Gol’shtein, Duality Theory in Mathematical Programming and Its Applications (Nauka, Moscow, 1971) [in Russian].Google Scholar
- 5.Yu. G. Evtushenko, Optimization and Fast Automatic Differentiation (Vychisl. Tsentr, Ross. Akad. Nauk, Moscow, 2013) [in Russian].Google Scholar
- 10.A. V. Dmitruk, A. A. Milyutin, and N. P. Osmolovskii, “The Lyusternik theorem and the extremum value theory,” Usp. Mat. Nauk 35 (6), 11–46 (1980).Google Scholar
- 12.E. Belousov, “On the calculation of the exact Lipschita and Hoffman constants for a system of linear inequalities,” Vestn. Tambov Univ., Ser. Estestv. Tekhn. Nauki, 5, 416–417 (2000).Google Scholar
- 14.E. S. Levitin, A. A. Milyutin, and N. P. Osmolovskii, “Theory of high-order conditions in smooth constrained extremum value problems,” in Theoretical and Applied Issues of Optimal Control (Nauka, Novosibirsk, 1985), pp. 4–40 [in Russian].Google Scholar
- 22.F. H. Clarke, “Nonsmooth analysis and optimization,” in Proc. of the Int. Congress of Mathematicians, Helsinki, 1978, pp. 847–853.Google Scholar
- 25.V. V. Voloshinov and E. S. Levitin, “Characterization of δ-optimal solutions in mathematical programming problems,” in Proc. of the 11th Int. Baikal Seminar, Irkutsk, 1998, Vol. 1, pp. 69–70.Google Scholar