Abstract
This paper deals with new variable-metric algorithms for nonsmooth optimization problems, the so-called adaptive algorithms. The essence of these algorithms is that there are two simultaneously working gradient algorithms: the first is in the main space and the second is in the space of the matrices that modify the main variables. The convergence of these algorithms is proved for different cases. The results of numerical experiments are also given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Eaves, B. C., andZangwill, W.,Generalized Cutting Plane Algorithms, SIAM Journal on Control, Vol. 9, pp. 529–542, 1971.
Bertsekas, D. P., andMitter, S. K.,A Descent Numerical Method for Optimization Problems with Nondifferentiable Cost Functionals, SIAM Journal on Control, Vol. 11, pp. 637–652, 1973.
Wolfe, P.,A Method of Conjugate Subgradients for Minimizing Nondifferentiable Functions, Mathematical Programming Study, Vol. 3, pp. 145–173, 1975.
Lemarechal, C., Strodiat, J. J., andBihain, A.,On a Bundle Algorithm for Nonsmooth Optimization, Nonlinear Programming, Vol. 4, Edited by O. L. Mangasarian, R. R. Meyer, and S. M. Robinson, Academic Press, New York, New York, pp. 245–282, 1981.
Mifflin, R.,A Modification and an Extension of Lemarechal's Algorithm for Nonsmooth Minimization, Mathematical Programming Study, Vol. 17, pp. 77–90, 1982.
Kiwiel, K. C.,Methods of Descent for Nondifferentiable Optimization, Springer-Verlag, Berlin, Germany, 1985.
Dem'janov, V. F., andVasil'ev, L. V.,Nondifferentiable Optimization, Springer, New York, New York, 1985.
Robinson, S. M.,Newton's Method for a Class of Nonsmooth Functions, Preprint, Department of Industrial Engineering, University of Wisconsin-Madison, Madison, Wisconsin, 1988.
Shor, N. Z.,On the Structure of Algorithms for the Numerical Solution of Optimal Planning and Designing, PhD Thesis, V. Glushkov Institute of Cybernetics, Kiev, USSR, 1964 (in Russian).
Dennis, J. M., andMoré, J. J.,Quasi-Newton Methods: Motivation and Theory, SIAM Review, Vol. 19, pp. 46–89, 1977.
Shor, N. Z.,Minimization Methods for Nondifferentiable Functions, Springer-Verlag, Berlin, Germany, 1985.
Poljak, B. T.,Subgradient Methods: A Survey of Soviet Research, Proceedings of the IIASA Workshop on Nonsmooth Optimization, Edited by C. Lemarechal and R. Mifflin, Sopron, Hungary, pp. 5–29, 1977.
Uryas'ev, S. P.,Stochastic Quasigradient Algorithms with Adaptively Controlled Parameters, Working Paper WP-86-32, IIASA, Laxenburg, Austria, 1986.
Ermoliev, Yu., andWets, R. J. B., Editors,Numerical Techniques for Stochastic Optimization, Springer-Verlag, Berlin, Germany, 1988.
Uryas'ev, S. P.,Adaptive Variable-Metric Algorithms for Nonsmooth Optimization Problems, Working Paper WP-88-60, IIASA, Laxenburg, Austria, 1988.
Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.
Pshenychnyi, B. N.,Necessary Conditions for an Extremum, Marcel Dekker, New York, New York, 1971.
Nesterov, Yu, E.,Minimization Methods for Nonsmooth Convex and Quasiconvex Functions, Ekonomika i Matematicheskie Metody, USSR, Vol. 20, pp. 519–531, 1984 (in Russian).
Nurminski, E.,Numerical Methods for Solving Deterministic and Stochastic Minimax Problems, Naukova Dumka, Kiev, USSR, 1979 (in Russian).
Hoffman, A.,Weak Convex Functions, Multifunctions, and Optimization, 27th Internationales Wissenschaftliches Kolloquium der Technische Hochschule Ilmenau, Vol. 5, pp. 33–36, 1982.
Ritch, P. S.,Discrete Optimal Control with Multiple Constraints, I: Constraint Separation and Transformation Technique, Automatica, Vol. 9, p. 415–429, 1973.
Outrata, J. V., andSchindler, Z.,On Some Nondifferentiable Problems in Optimal Control, Proceedings of the IIASA Workshop on Nondifferentiable Optimization, Edited by V. F. Demyanov and D. Pallaschke, Sopron, Hungary, pp. 118–128, 1984.
Ermoliev, Yu. M., Lyashko, I. I., Mikhalevich, V. S., andTyuptya, V. I.,Mathematical Methods of Operation Research, Visha Shkola, Kiev, USSR, 1979 (in Russian).
Author information
Authors and Affiliations
Additional information
Communicated by O. L. Mangasarian
Rights and permissions
About this article
Cite this article
Uryas'ev, S.P. New variable-metric algorithms for nondifferentiable optimization problems. J Optim Theory Appl 71, 359–388 (1991). https://doi.org/10.1007/BF00939925
Issue Date:
DOI: https://doi.org/10.1007/BF00939925