Abstract
We consider the generalized minimax problem, that is, the problem of minimizing a function φ(x)=F(g 1(x),...,g m(x)), where F is a smooth function and each g i is the maximum of a finite number of smooth functions. We prove that, under suitable assumptions, it is possible to construct a continuously differentiable exact barrier function, whose minimizers yield the minimizers of the function φ. In this way, the nonsmooth original problem can be solved by usual minimization techniques for unconstrained differentiable functions.
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References
DEMYANOV, V. F., and RUBINOV, A. M., Quasidifferential Calculus, Optimization Software, New York, New York, 1986.
BERTSEKAS, D. P., Nondifferentiable Optimization via Approximation, Mathematical Programming Study, Vol. 3, pp. 1–25, 1975.
AUSLENDER, A., Minimisation de Fonctions Localement Lipschitziennes: Applications a la Programmation Miconvexe, Midifferentiable, Nonlinear Programming 3, Edited by O. L. Mangasarian, R. R. Meyer, and S. M. Robinson, Academic Press, New York, New York, pp. 429–460, 1978.
PAPAVASSILOPOULOS, G., Algorithms for a Class of Nondifferentiable Problems, Journal of Optimization Theory and Applications, Vol. 34, pp. 41–82, 1981.
BEN-TAL, A., and ZOWE, J., Necessary and Sufficient Optimality Conditions for a Class of Nonsmooth Minimization Problems, Mathematical Programming, Vol. 24, pp. 70–91, 1982.
KIWIEL, K. C., A Quadratic Approximation Method for Minimizing a Class of Quasidifferentiable Functions, Numerische Mathematik, Vol. 45, pp. 411–430, 1984.
LEMARECHAL, C., An Extension of the Davidon Method to Nondifferentiable Problems, Mathematical Programming Study, Vol. 3, pp. 95–109, 1975.
WOLFE, P., A Method of Conjugate Subgradients for Minimizing Nondifferentiable Convex Functions, Mathematical Programming Study, Vol. 3, pp. 145–173, 1975.
MIFFLIN, R., An Algorithm for Constrained Optimization with Semismooth Functions, Mathematics of Operations Research, Vol. 2, pp. 191–207, 1977.
GAUDIOSO, M., and MONACO, M. F., A Bundle-Type Approach to the Unconstrained Minimization of Convex Nonsmooth Functions, Mathematical Programming, Vol. 23, pp. 216–226, 1982.
LEMARECHAL, C., Numerical Experiments in Nonsmooth Optimization, Progress in Nondifferentiable Optimization, Edited by E. A. Nurminskii, IIASA Proceedings, Laxenburg, Austria, pp. 61–84, 1982.
SHOR, N. Z., Minimization Methods for Nondifferentiable Functions, Springer Verlag, Berlin, Germany, 1985.
KIWIEL, K. C., Methods of Descent for Nondifferentiable Optimization, Lecture Notes in Mathematics, Springer Verlag, Berlin, Germany, Vol. 1133, 1985.
POLAK, E., Basics of Minimax Algorithms, Nonsmooth Optimization and Related Topics, Edited by F. H. Clarke, V. F. Demyanov, and F. Giannessi, Plenum Press, New York, New York, pp. 343–369, 1989.
SCHRAMM, H., and ZOWE, J., A Version of the Bundle Idea for Minimizing a Nonsmooth Function: Conceptual Idea, Convergence Analysis, Numerical Results, Report 206, DFG-Schwerpunktprogramms, Anwendungsbezogene Optimierung und Steuerung, Universitat Bayreuth, Bayreuth, Germany, 1990.
POLAK, E., MAYNE, D. H., and HIGGINS, J. E., Superlinearly Convergent Algorithm for Min-Max Problems, Journal of Optimization Theory and Applications, Vol. 69, pp. 407–439, 1991.
MADSEN, K., Minimization of Nonlinear Approximation Functions, PhD Thesis, Technical University of Denmark, Lyngby, Denmark, 1985.
DI PILLO, G., Exact Penalty Methods, Algorithms for Continuous Optimization: The State of the Art, Edited by E. Spedicato, Kluwer, Dordrecht, Netherlands, pp. 209–253, 1994.
DI PILLO, G., GRIPPO, L., and LUCIDI, S., A Smooth Method for the Finite Minimax Problem, Mathematical Programming, Vol. 60, pp. 187–214, 1993.
GLAD, T., and POLAK, E., A Multiplier Method with Automatic Limitation of Penalty Growth, Mathematical Programming, Vol. 17, pp. 140–155, 1979.
LUCIDI, S., New Results on a Continuously Differentiable Exact Penalty Function, SIAM Journal on Optimization, Vol. 2, pp. 558–574, 1992.
POLAK, E., On the Global Stabilization of Locally Convergent Algorithms, Automatica, Vol. 12, pp. 337–342, 1976.
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Di Pillo, G., Grippo, L. & Lucidi, S. Smooth Transformation of the Generalized Minimax Problem. Journal of Optimization Theory and Applications 95, 1–24 (1997). https://doi.org/10.1023/A:1022627226891
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DOI: https://doi.org/10.1023/A:1022627226891