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An Algorithm Based on Active Sets and Smoothing for Discretized Semi-Infinite Minimax Problems

Author

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  • E. Polak

    (University of California)

  • R. S. Womersley

    (University of New South Wales)

  • H. X. Yin

    (Chinese Academy of Sciences)

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.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:138:y:2008:i:2:d:10.1007_s10957-008-9355-9
    DOI: 10.1007/s10957-008-9355-9
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    References listed on IDEAS

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    1. 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. 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.
    2. Zhengyong Zhou & Xiaoyang Dai, 2023. "An active set strategy to address the ill-conditioning of smoothing methods for solving finite linear minimax problems," Journal of Global Optimization, Springer, vol. 85(2), pages 421-439, February.
    3. 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.
    4. H. Chung & E. Polak & S. Sastry, 2010. "On the Use of Outer Approximations as an External Active Set Strategy," Journal of Optimization Theory and Applications, Springer, vol. 146(1), pages 51-75, July.
    5. Jin-bao Jian & Qing-juan Hu & Chun-ming Tang, 2014. "Superlinearly Convergent Norm-Relaxed SQP Method Based on Active Set Identification and New Line Search for Constrained Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 163(3), pages 859-883, December.
    6. 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.
    7. Jin-bao Jian & Xing-de Mo & Li-juan Qiu & Su-ming Yang & Fu-sheng Wang, 2014. "Simple Sequential Quadratically Constrained Quadratic Programming Feasible Algorithm with Active Identification Sets for Constrained Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 160(1), pages 158-188, January.

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