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One approach for solving optimization problems with apriori estimates of approximation of admissible set

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Abstract

Properties and construction principles of a satisfactory approximation for the set of admissible solutions to a constrained optimization problem are studied. The replacement of the initial admissible set by its satisfactory approximation in the course of solution makes it possible to construct finite algorithms for the methods of interior and exterior points (the penalty function method, or the method of centers) with a stopping criterion ensuring a given accuracy of the obtained solution. Necessary and sufficient conditions for constructing outer and inner satisfactory approximations of the admissible set are obtained. A feasible procedure for specifying a set being a satisfactory approximation of the admissible set is described, which can be used for constructing algorithms ensuring the achievement of a given accuracy in a finite number of iterations.

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References

  1. C. Grossmann and A. A. Kaplan, Nonlinear Programming Based on Unconstrained Minimization (Teubner, Leipzig, 1979; Nauka, Novosibirsk, 1981).

    Google Scholar 

  2. Yu. G. Evtushenko and V.G. Zhadan, To the Problem of Systematizing Numerical Methods of Nonlinear Programming (Vychislitel’nyi Tsentr, Akad. Nauk SSSR, Moscow, 1988).

    Google Scholar 

  3. V. G. Zhadan, Numerical Methods of Linear and Nonlinear Programming: Auxiliary Functions in Constrained Minimization (Vychislitel’nyi Tsentr im. A.A. Dorodnitsyna, Ross. Akad. Nauk, Moscow, 2002).

    Google Scholar 

  4. Ya. I. Zabotin and I. A. Fukin, “On a modification of the penalty shifting method for nonlinear programming problems,” Russian Math. (Iz. VUZ) 44(12), 47–52 (2001).

    MathSciNet  Google Scholar 

  5. Ya. I. Zabotin and I. A. Fukin, “Algorithms in the penalty method with approximation of the admissible set,” Russian Math. (Iz. VUZ) 48(1), 33–44 (2004).

    MATH  MathSciNet  Google Scholar 

  6. Ya. I. Zabotin and A. A. Andrianova, “Algorithms in the method of centers with approximation of the set of admissible solutions,” Russian Math. (Iz. VUZ) 45(12), 38–42 (2002).

    MathSciNet  Google Scholar 

  7. A. A. Andrianova and Ya. I. Zabotin, “Control of the minimization process in the parametrized method of centers,” Russian Math. (Iz. VUZ) 46(12), 1–8 (2003).

    MathSciNet  Google Scholar 

  8. A. A. Andrianova, “Parameterization of the method of centers for minimizing explicitly quasi-convex functions,” in Research in Applied Mathematics and Computer Science (Izd. Kazan. Mat. O-va, Kazan, 2006), Vol. 26, pp. 3–12 [in Russian].

    Google Scholar 

  9. A. A. Andrianova, “Principles of constructing an approximation of admissible sets in solving constrained optimization problems with a given accuracy,” in Proceedings of the Fifteenth Baikal International Workshop “Optimization Methods and Their Applications, Vol. 2: Mathematical Programming (RIO IDSTU, Sibirsk. Otd., Ross. Akad. Nauk, Irkutsk, 2011), pp. 35–38.

    Google Scholar 

  10. A. G. Sukharev, A.V. Timokhov, and V.V. Fedorov, A Course in Optimization Methods (Fizmatlit, Moscow, 2005) [in Russian].

    Google Scholar 

  11. Ya. I. Zabotin and I. N. Dan’shin, “Algorithms with combination, parametrization and two-sided approximation in the center method,” Russian Math. (Iz. VUZ) 42(12), 38–46 (1999).

    MathSciNet  Google Scholar 

  12. Ya. I. Zabotin, “The minimax method for the solution of a mathematical programming problem,” Izv. Vyssh. Ucebn. Zaved. Mat., No. 6, 36–43 (1975).

    Google Scholar 

  13. I. A. Fukin, “ρ-Approximability of convex functions,” in All-Russia Conference “Algorithmic Analysis of Unstable problems,” Yekaterinburg, Russia, 2004 (Izd. Ural. Univ., Yekaterinburg, 2004), pp. 306–307.

    Google Scholar 

  14. A. I. Korablev, “Relaxation methods for minimizing pseudoconvex functions,” in Research on Applied Mathematics (Izd. Kazan. Univ., Kazan, 1980), pp. 3–8 [in Russian].

    Google Scholar 

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Authors and Affiliations

  1. Kazan Federal University, Kremlevskaya ul. 18, Kazan, 420008, Tatarstan, Russia

    A. A. Andrianova

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  1. A. A. Andrianova
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Correspondence to A. A. Andrianova.

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Original Russian Text © A.A. Andrianova, 2012, published in Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2012, Vol. 154, No. 3, pp. 190–201.

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Andrianova, A.A. One approach for solving optimization problems with apriori estimates of approximation of admissible set. Lobachevskii J Math 34, 368–376 (2013). https://doi.org/10.1134/S1995080213040021

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  • DOI: https://doi.org/10.1134/S1995080213040021

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