Abstract
A special variable metric method is given for finding minima of convex functions that are not necessarily differentiable. Time-consuming quadratic programming subproblems do not need to be solved. Global convergence of the method is established. Some encouraging numerical experience is reported.
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References
Clarke, F. H., Optimization and Nonsmooth Analysis, Wiley (Interscience), New York, New York, 1983.
Kiwiel, K. C., Methods of Descent for Nondifferentiable Optimization, Lecture Notes in Mathematics, Springer Verlag, Berlin, Germany, Vol. 1133, 1985.
MÄkelÄ, M. M., and NeittaanmÄki, P., Nonsmooth Optimization, World Scientific Publishing Company, London, England, 1992.
Schramm, H., and Zowe, J., A Version of the Bundle Idea for Minimizing a Nonsmooth Function: Conceptual Idea, Convergence Analysis, Numerical Results, SIAM Journal on Optimization, Vol. 2, pp. 121–152, 1992.
LemarÉchal, C., and SagastizÁbal, C., Variable Metric Bundle Methods: From Conceptual to Implementable Forms, Mathematical Programming, Vol. 76, pp. 393–410, 1997.
LukŠan, L., and VlČek, A Bundle-Newton Method for Nonsmooth Unconstrained Minimization, Mathematical Programming, Vol. 83, pp. 373–391, 1998.
LukŠan, L., Dual Method for Solving a Special Problem of Quadratic Programming as a Subproblem of Linearly Constrained Nonlinear Minimax Approximation, Kybernetika, Vol. 20, pp. 445–457, 1984.
LemarÉchal, C., Numerical Experiments in Nonsmooth Optimization, Progress in Nonsmooth Optimization, Edited by E. A. Nurminski, IIASA, Laxenburg, Austraia, pp. 61–84, 1982.
Fletcher, R., Practical Methods of Optimization, John Wiley and Sons, Chichester, England, 1987.
LukŠan, L., Computational Experience with Known Variable Metric Updates, Journal of Optimization Theory and Applications, Vol. 83, pp. 27–47, 1994.
Zowe, J., Nondifferentiable Optimization, Computational Mathematical Programming, Edited by K. Schittkowski, Springer Verlag, Berlin, Germany, pp. 323–356, 1985.
Kiwiel, K. C., An Ellipsoid Trust Region Bundle Method for Nonsmooth Convex Minimization, SIAM Journal on Control and Optimization, Vol. 27, pp. 737–757, 1989.
Bihain, A., Optimization of Upper Semidifferentiable Functions, Journal of Optimization Theory and Applications, Vol. 4, pp. 545–568, 1984.
Facchinei, F., and Lucidi, S., Nonmonotone Bundle-Type Scheme for Convex Nonsmooth Minimization, Journal of Optimization Theory and Applications, Vol. 76, pp. 241–257, 1993.
LukŠan, L., A Compact Variable Metric Algorithm for Linear Minimax Approximation, Computing, Vol. 36, pp. 355–373, 1986.
Bandler, J. W., Srinivasan, T. V., and Charalambous, C., Minimax Optimization of Networks by Grazor Search, IEEE Transactions on Microwave Theory and Techniques, Vol. 20, pp. 596–604, 1972.
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Lukšan, L., Vlček, J. Globally Convergent Variable Metric Method for Convex Nonsmooth Unconstrained Minimization1. Journal of Optimization Theory and Applications 102, 593–613 (1999). https://doi.org/10.1023/A:1022650107080
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DOI: https://doi.org/10.1023/A:1022650107080