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On Solving Large-Scale Finite Minimax Problems Using Exponential Smoothing

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  • E. Y. Pee

    (Naval Postgraduate School)

  • J. O. Royset

    (Naval Postgraduate School)

Abstract

This paper focuses on finite minimax problems with many functions, and their solution by means of exponential smoothing. We conduct run-time complexity and rate of convergence analysis of smoothing algorithms and compare them with those of SQP algorithms. We find that smoothing algorithms may have only sublinear rate of convergence, but as shown by our complexity results, their slow rate of convergence may be compensated by small computational work per iteration. We present two smoothing algorithms with active-set strategies and novel precision-parameter adjustment schemes. Numerical results indicate that the algorithms are competitive with other algorithms from the literature, and especially so when a large number of functions are nearly active at stationary points.

Suggested Citation

  • E. Y. Pee & J. O. Royset, 2011. "On Solving Large-Scale Finite Minimax Problems Using Exponential Smoothing," Journal of Optimization Theory and Applications, Springer, vol. 148(2), pages 390-421, February.
    • Handle: RePEc:spr:joptap:v:148:y:2011:i:2:d:10.1007_s10957-010-9759-1
    DOI: 10.1007/s10957-010-9759-1
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    References listed on IDEAS

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    1. E. Polak, 2008. "On the Convergence of the Pshenichnyi-Pironneau-Polak Minimax Algorithm with an Active Set Strategy," Journal of Optimization Theory and Applications, Springer, vol. 138(2), pages 305-309, August.
    2. Xiaoqiang Cai & Kok-Lay Teo & Xiaoqi Yang & Xun Yu Zhou, 2000. "Portfolio Optimization Under a Minimax Rule," Management Science, INFORMS, vol. 46(7), pages 957-972, July.
    3. Nesterov, Y., 1995. "Complexity estimates of some cutting plane methods based on the analytic barrier," LIDAM Reprints CORE 1167, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. K. A. Ariyawansa & P. L. Jiang, 2000. "On Complexity of the Translational-Cut Algorithm for Convex Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 107(2), pages 223-243, November.
    5. E. Obasanjo & G. Tzallas-Regas & B. Rustem, 2010. "An Interior-Point Algorithm for Nonlinear Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 144(2), pages 291-318, February.
    6. E. Polak & R. S. Womersley & H. X. Yin, 2008. "An Algorithm Based on Active Sets and Smoothing for Discretized Semi-Infinite Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 138(2), pages 311-328, August.
    7. E. Polak & J. O. Royset & R. S. Womersley, 2003. "Algorithms with Adaptive Smoothing for Finite Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 119(3), pages 459-484, December.
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    Cited by:

    1. Junxiang Li & Mingsong Cheng & Bo Yu & Shuting Zhang, 2015. "Group Update Method for Sparse Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 166(1), pages 257-277, July.
    2. Gonghao Duan & Ruiqing Niu, 2018. "Lake Area Analysis Using Exponential Smoothing Model and Long Time-Series Landsat Images in Wuhan, China," Sustainability, MDPI, vol. 10(1), pages 1-16, January.
    3. Yi Chen & David Gao, 2016. "Global solutions to nonconvex optimization of 4th-order polynomial and log-sum-exp functions," Journal of Global Optimization, Springer, vol. 64(3), pages 417-431, March.
    4. Ryan Alimo & Pooriya Beyhaghi & Thomas R. Bewley, 2020. "Delaunay-based derivative-free optimization via global surrogates. Part III: nonconvex constraints," Journal of Global Optimization, Springer, vol. 77(4), pages 743-776, August.
    5. J. O. Royset & E. Y. Pee, 2012. "Rate of Convergence Analysis of Discretization and Smoothing Algorithms for Semiinfinite Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 155(3), pages 855-882, December.
    6. Daehan Won & Hasan Manzour & Wanpracha Chaovalitwongse, 2020. "Convex Optimization for Group Feature Selection in Networked Data," INFORMS Journal on Computing, INFORMS, vol. 32(1), pages 182-198, January.
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    8. Johannes Royset, 2013. "On sample size control in sample average approximations for solving smooth stochastic programs," Computational Optimization and Applications, Springer, vol. 55(2), pages 265-309, June.

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