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Methods of descent over groups of variables for a class of constrained minimization problems. (English. Russian original) Zbl 0845.90112

J. Math. Sci., New York 73, No. 5, 550-556 (1995); translation from Issled. Prikl. Mat. 18, 48-59 (1992).
Summary: Optimization problems arise frequently in practice in which the restrictions can be presented as the direct product of sets of spaces of generally different dimensions. This property of defining the sets makes it possible to construct special methods of mathematical programming to solve such problems. In the present article, we propose two methods of minimizing a convex differentiable function on a set of this form. The proposed methods belong to the class of decomposition methods and are characterized by the fact that at each step the transition to a new recursion point is carried out not over all the variables of the problem but only over groups of variables.
Methods of descent over groups of variables for both unconstrained and constrained minimization have been studied by many authors. Among the methods of this class, for example, are different versions of the method of coordinatewise descent, gradient methods of descent over the fast variables for unconstrained minimization and minimization of a function on a parallelepiped, and the method of cyclic componentwise descent. Here, we propose a principle different from the known method for choosing the variables over which the descent is carried out at each step. This principle is based on the known conditions of extremality for an admissible point, guarantees the convergence of the proposed methods, and guarantees the construction of various implementations of them. We note that these methods allow an alternation of steps in the subspaces of fast and slow variables; they also provide the possibility of choosing the dimension of these subspaces and a parallelizing the computational process of minimization on multiprocessor computers.

MSC:

90C30 Nonlinear programming
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[1] I. Ya. Zabotin and A. V. Kobchikov, ”Choice of optimal running times for subsystems in information-control systems,”Priem i Obr. Inf. v. Slozh. Inf. Sys., No. 16, 10–19 (1987).
[2] A. V. Kohchikov and I. Ya. Zabotin, ”On the choice of the optimal relations between the masses of the blocks of a radioelectronic system,”Priem i Obr. Inf. v Slozh. Inf. Sys. No. 18, 64–70 (1990).
[3] V. G. Karmano,Mathematical Programming [in Russian], Nauka, Moscow (1986).
[4] F. P. Vasillev,Numerical Methods of Solving Extremal Problems [in Russian], Nauka, Moscow (1988).
[5] N. N. Moiseev,Numerical Methods in the Theory of Optimal Systems [in Russian], Nauka, Moscow (1971). · Zbl 0224.35071
[6] L. T. Ashchepkov, B. I. Belov, V. P. Bulatov, et al.,Methods of Solving Problems of Mathematical Programming and Optimal Control [in Russian], Nauka, Novosibirsk (1984).
[7] E. G. Gol’shtein and D. B. Yudin, ”Methods of Computing and Synthesizing Pulsed Automated Systems,”Avtomat. i Telemekh.,24, No. 12, 1643–1659 (1963).
[8] Yu. M. Ermol’ev, I. I. Lyashko, V. S. Mikhalevich, and V. I. Tyuptya,Mathematical Methods of Operations Research [in Russian], Vishcha Shkola, Kiev (1979). · Zbl 0446.90046
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