Abstract
We present a new active-set strategy which can be used in conjunction with exponential (entropic) smoothing for solving large-scale minimax problems arising from the discretization of semi-infinite minimax problems. The main effect of the active-set strategy is to dramatically reduce the number of gradient calculations needed in the optimization. Discretization of multidimensional domains gives rise to minimax problems with thousands of component functions. We present an application to minimizing the sum of squares of the Lagrange polynomials to find good points for polynomial interpolation on the unit sphere in ℝ3. Our numerical results show that the active-set strategy results in a modified Armijo gradient or Gauss-Newton like methods requiring less than a quarter of the gradients, as compared to the use of these methods without our active-set strategy. Finally, we show how this strategy can be incorporated in an algorithm for solving semi-infinite minimax problems.
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References
Outendijk, G.Z.: Methods of Feasible Directions. Elsevier, Amsterdam (1960)
Polak, E.: Computational Methods in Optimization: A Unified Approach. Academic, New York (1971)
Fletcher, R.: Practical Methods of Optimization. Wiley, New York (2000)
Bertsekas, D.P.: Nonlinear Programming. Athena Press, Cambridge (1995)
Polak, E.: Optimization: Algorithms and Consistent Approximations. Springer, New York (1997)
Polak, E., Royset, J.O., Womersley, R.S.: Algorithms with adaptive smoothing for finite min-max problems. J. Optim. Theory Appl. 119(3), 459–484 (2003)
Kort, B.W., Bertsekas, D.P.: A new penalty function for constrained minimization. In: Proceedings 1972 IEEE Conference on Decision and Control, New Orleans, December 1972
Bertsekas, D.P.: Nondifferentiable optimization via approximation. In: Balinski, M.L., Wolfe, P. (eds.) Mathematical Programming Study 3, Nondifferentiable Optimization, pp. 1–15. North-Holland, Amsterdam (1975)
Bertsekas, D.P.: Approximation procedures based on the method of multipliers. J. Optim. Theory Appl. 23, 487–510 (1977)
Li, X.S.: An entropy-based aggregate method for minimax optimization. Eng. Optim. 14, 277–285 (1997)
Peng, J., Lin, Z.: A non-interior continuation method for generalized linear complementarity problems. Math. Program. 86, 533–563 (1999)
Xu, S.: Smoothing method for minimax problems. Comput. Optim. Appl. 20(3), 267–279 (2001)
Li, X.S.: An aggregate function method for nonlinear programming. Sci. China (A) 34, 1467–1473 (1991)
Bertsekas, D.P.: Minmax methods based on approximation. In: Proceeding of the 1976 Johns Hopkins Conference on Information Sciences and Systems, Baltimore, 1976
Bertsekas, D.P.: Lagrange Multiplier Methods in Constrained Optimization. Academic, New York (1982), Chap. 3; republished by Athena Scientific, Cambridge (1996)
Womersley, R.S., Sloan, I.H.: How good can polynomial interpolation on the sphere be? Adv. Comput. Math. 14, 195–226 (2001)
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Polak, E., Womersley, R.S. & Yin, H.X. An Algorithm Based on Active Sets and Smoothing for Discretized Semi-Infinite Minimax Problems. J Optim Theory Appl 138, 311–328 (2008). https://doi.org/10.1007/s10957-008-9355-9
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DOI: https://doi.org/10.1007/s10957-008-9355-9