Abstract
This paper presents a descent method for minimizing a sum of possibly nonsmooth convex functions. Search directions are found by solving subproblems obtained by replacing all but one of the component functions with their polyhedral approximations and adding a quadratic term. The algorithm is globally convergent and terminates when the objective function happens to be polyhedral. It yields a new decomposition method for solving large-scale linear programs with dual block-angular structure.
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Kiwiel, K. C.,An Aggregate Subgradient Method for Nonsmooth Convex Minimization, Mathematical Programming, Vol. 27, pp. 320–341, 1983.
Nurmiński, E.,Convergence and Numerical Experiments with a Decomposition Algorithm, International Institute for Applied Systems Analysis, Laxenburg, Austria, Report No. WP-82-08, 1982.
Nurmiński, E.,Decomposition Algorithm Based on the Primal-Dual Approximation, International Institute for Applied Systems Analysis, Laxenburg, Austria, Report No. WP-82-46, 1982.
Lasdon, L. S.,Optimization Theory for Large Systems, Macmillan Company, London, England, 1970.
Shor, N. Z.,Minimization Methods for Nondifferentiable Functions and Their Applications, Springer-Verlag, Heidelberg, Germany, 1984.
Tsurkov, V. I.,Decomposition in Large-Scale Problems, Nauka, Moscow, USSR, 1981 (in Russian).
Murtagh, B. A., andSaunders, M. A.,Large-Scale Linearly Constrained Optimization, Mathematical Programming, Vol. 14, pp. 41–72, 1978.
Nurmiński, E., andBalabanov, T.,Decomposition of Large-Scale Energy Model, Large Scale Systems, Vol. 4, pp. 295–308, 1983.
Cohen, G., andBalducchi, J. F.,Computation of Saddle Points of Nonstrictly Convex Lagrangians, Current Problems in Computational and Applied Mathematics, Edited by A. S. Alekseyev, Nauka, Novosibirsk, USSR, 1983 (in Russian).
Cohen, G., andZhu, D. L.,Decomposition of Nonsmooth Optimization Problems, Large Scale Systems: Theory and Applications, Edited by A. Straszak, Pergamon Press, Oxford, England, 1984.
Pierra, G.,Decompostion through Formalization in a Product Space, Mathematical Programming, Vol. 28, pp. 96–115, 1984.
Spingarn, J. E.,Partial Inverse of a Monotone Operator, Applied Mathematics and Optimization, Vol. 10, pp. 247–265, 1983.
Kiwiel, K. C.,Linearization Algorithm for Constrained Nonsmooth Minimization, System Modelling and Optimization, Edited by P. Thoft-Christensen, Springer-Verlag, Berlin, Germany, 1984.
Lemarechal, C., andMifflin, R.,Global and Superlinear Convergence of an Algorithm for One-Dimensional Minimization of Convex Functions, Mathematical Programming, Vol. 24, pp. 241–256, 1982.
Mifflin, R.,A Modification and an Extension of Lemarechal's Algorithm for Nonsmooth Minimization. Nondifferential and Variational Techniques in Optimization, Edited by D. C. Sorensen and R. J.-B. Wets, Mathematical Programming Study, Vol. 17, pp. 77–90, 1982.
Rockfellar, R. T.,Convex Analysis, Princeton University Press, Princeton, Nes Jersey, 1970.
Wierzbicki, A. P.,Lagrangian Functions in Nondifferentiable Optimization. Progress in Nondifferentiable Optimization, Edited by E. A. Nurminski, International Institute for Applied Systems Analysis, Laxenburg, Austria, Report No. CP-82-S8, 1982.
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Communicated by D. Q. Mayne
Supported by Program CPBP 02.15.
The author thanks the two referees for their helpful suggestions.
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Kiwiel, K.C. Decomposition method of descent for minimizing the sum of convex nonsmooth functions. J Optim Theory Appl 52, 255–271 (1987). https://doi.org/10.1007/BF00941285
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DOI: https://doi.org/10.1007/BF00941285