Abstract
We propose a method of outer approximations, with each approximate problem smoothed using entropic regularization, to solve continuous min-max problems. By using a well-known uniform error estimate for entropic regularization, convergence of the overall method is shown while allowing each smoothed problem to be solved inexactly. In the case of convex objective function and linear constraints, an interior-point algorithm is proposed to solve the smoothed problem inexactly. Numerical examples are presented to illustrate the behavior of the proposed method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bertsekas, D. P., Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, NY, 1982.
Polyak, R. A., Smooth Optimization Methods for Minmax Problems, SIAM Journal on Control and Optimization, Vol. 26, pp. 1274–1286, 1988.
Sheu, R. L., and Wu, S. Y., Combined Entropic Regularization and Path-Following Method for Solving Finite Convex Min-Max Problems Subject to Infinitely Many Linear Constraints, Journal of Optimization Theory and Applications, Vol. 101, pp. 167–190,1999.
Chang, P. L., A Minimax Approach to Nonlinear Programming, Doctoral Dissertation, Department of Mathematics, University of Washington, Seattle, Washington, 1980.
Ben-tal, A., and Teboulle, M., A Smoothing Technique for Nondifferentiable Optimization Problems, Lecture Notes in Mathematics, Springer Verlag, Berlin, Germany, Vol. 1405, pp. 1–11, 1989.
Wu, S. Y., and Fang, S. C., Solving Min-Max Problems and Linear Semi-Infinite Programs, Computers and Mathematics with Applications, Vol. 32, pp. 87–93, 1996.
Polak, E., Higgins, J. E., and Mayne, D. Q., A Barrier Function Method for Minimax Problems, Mathematical Programming, Vol. 54, pp. 155–176, 1992.
Polak, E., Optimization: Algorithms and Consistent Approximations, Springer Verlag, New York, NY, 1994.
Hettich, R., and Kortanek, K. O., Semi-Infinite Programming: Theory, Methods, and Applications, SIAM Review, Vol. 35, pp. 380–429, 1993.
Reemtsen, R., and GÖrner, S., Numerical Methods for Semi-Infinite Programming: A Survey, Semi-Infinite Programming, Edited by R. Reemtsen and J.J. Rückmann, Kluwer Academic Publishers, Dordrecht, Netherlands, pp. 195–275, 1998.
Dieudonne, J., Foundations of Modern Analysis 10-1, Academic Press, New York, NY, 1969.
Denhertog, D., Interior-Point Approach to Linear, Quadratic, and Convex Programming, Kluwer Academic Publishers, Dordrecht, Netherlands, 1994.
Sheu, R. L., A Generalized Interior-Point Barrier Function Approach for Smooth Convex Programming with Linear Constraints, Journal of Information and Optimization Sciences, Vol. 20, pp. 187–202, 1999.
Gonzaga, C. C., and Polak, E., On Constraint Dropping Schemes and Optimality Functions for a Class of Outer Approximations Algorithms, SIAM Journal on Control and Optimization, Vol. 17, pp. 477–493, 1979.
Powell, M. J. D., A Tolerant Algorithm for Linearly Constrained Optimization Calculations, Mathematical Programming, Vol. 45, pp. 547–566, 1989.
Abbe, L., A Logarithmic Barrier Approach and Its Regularization Applied to Convex Semi-Infinite Programming Problems, PhD Dissertation, Universität Trier, Trier, Germany, 2001.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Sheu, R.L., Lin, J.Y. Solving Continuous Min-Max Problems by an Iterative Entropic Regularization Method. Journal of Optimization Theory and Applications 121, 597–612 (2004). https://doi.org/10.1023/B:JOTA.0000037605.19435.63
Issue Date:
DOI: https://doi.org/10.1023/B:JOTA.0000037605.19435.63