Conclusion
Transformation (6) smoothing thef(x) level lines explains the effectiveness ofr(α)-algorithms from visual geometrical considerations. It may be regarded as a satisfactory interpretation of space dilation in the direction of the difference of two successive subgradients. On the other hand, it preserves the gradient flavor of the method, in contrast to the classical ellipsoid method [11, 12], which is a successful interpretation of the subgradient method with space dilation in the direction of the subgradient. A sensible combination of ellipsoids of a special kind [5] with the ellipsoids ell(x0,a, b, c) is quite capable of producing, on the basis of a one-dimensional space dilation operator, effective algorithms that solve a broader class of problems than convex programming problems, e.g., the problem to find saddle points of convex-concave functions, particular cases of the problem of solving variational inequalities, and also special classes of linear and nonlinear complementarity problems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N. Z. Shor and N. G. Zhurbenko, “Minimization method using space dilation in the direction of the difference of two successive gradients”, Kibernetika, No. 3, 51–59 (1971).
N. Z. Shor, Minimization Methods for Nondifferentiable Functions and Their Applications [in Russian], Naukova Dumka, Kiev (1979).
P. I. Stetsyuk, “On convergence ofr-algorithms,” Kibern. Sist. Anal., No. 6, 173–177 (1995).
N. Z. Shor and S. I. Stetsenko, Quadratic Extremal Problems and Nondifferentiable Optimization [in Russian], Naukova Dumka, Kiev (1989).
P. I. Stetsyuk, “Convergence proof for space-dilation algorithms,” in: Optimal Decision Theory [in Russian], Inst. Kibern. im. V. M. Glushkova NAN Ukr., Kiev (1995), pp. 4–8.
B. T. Polyak, An Introduction to Optimization [in Russian], Nauka, Moscow (1983).
K. C. Kiewel, “An aggregate subgradient method for nonsmooth convex programming,” Math. Progr., No. 27/3, 320–341 (1983).
C. Lemarechal and R. Mifflin, eds. Nonsmooth Optimization, Pergamon Press, Oxford (1978).
M. B. Shchepakin, “Orthogonal descent method,” Kibernetika, No. 1, 58–62 (1987).
C. Lemarechal, “An extension of Davidon methods to nondifferentiable problems,” Math. Progr. Study 3, North-Holland, Amsterdam (1975), pp. 95–109.
D. B. Yudin and A. S. Nemirovskii, “Information complexity and effective methods for convex extremal problems,” Ekon. Mat. Met., No. 2, 357–359 (1976).
N. Z. Shor, “Cutting plane method with space dilation for the convex programming problem,” Kibernetika, No. 1, 94–95 (1977).
Additional information
Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 113–134, January–February, 1996.
Rights and permissions
About this article
Cite this article
Stetsyuk, P.I. r-Algorithms and ellipsoids. Cybern Syst Anal 32, 93–110 (1996). https://doi.org/10.1007/BF02366587
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02366587