One approach for solving optimization problems with apriori estimates of approximation of admissible set
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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.
Keywords and phrasesmethods of sequential unconstrained minimization penalty function method method of centers solution of an optimization problem with a given accuracy satisfactory approximation of the admissible set feasible stopping criterion
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- 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
- 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
- 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