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A derivative-free approximate gradient sampling algorithm for finite minimax problems

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  • W. Hare
  • J. Nutini

Abstract

In this paper we present a derivative-free optimization algorithm for finite minimax problems. The algorithm calculates an approximate gradient for each of the active functions of the finite max function and uses these to generate an approximate subdifferential. The negative projection of 0 onto this set is used as a descent direction in an Armijo-like line search. We also present a robust version of the algorithm, which uses the ‘almost active’ functions of the finite max function in the calculation of the approximate subdifferential. Convergence results are presented for both algorithms, showing that either f(x k )→−∞ or every cluster point is a Clarke stationary point. Theoretical and numerical results are presented for three specific approximate gradients: the simplex gradient, the centered simplex gradient and the Gupal estimate of the gradient of the Steklov averaged function. A performance comparison is made between the regular and robust algorithms, the three approximate gradients, and a regular and robust stopping condition. Copyright Springer Science+Business Media New York 2013

Suggested Citation

  • W. Hare & J. Nutini, 2013. "A derivative-free approximate gradient sampling algorithm for finite minimax problems," Computational Optimization and Applications, Springer, vol. 56(1), pages 1-38, September.
    • Handle: RePEc:spr:coopap:v:56:y:2013:i:1:p:1-38
    DOI: 10.1007/s10589-013-9547-6
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    References listed on IDEAS

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    1. J. V. Burke & A. S. Lewis & M. L. Overton, 2002. "Approximating Subdifferentials by Random Sampling of Gradients," Mathematics of Operations Research, INFORMS, vol. 27(3), pages 567-584, 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. 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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    1. Chad Davis & Warren Hare, 2013. "Exploiting Known Structures to Approximate Normal Cones," Mathematics of Operations Research, INFORMS, vol. 38(4), pages 665-681, November.
    2. W. Hare & C. Sagastizábal & M. Solodov, 2016. "A proximal bundle method for nonsmooth nonconvex functions with inexact information," Computational Optimization and Applications, Springer, vol. 63(1), pages 1-28, January.
    3. M. V. Dolgopolik, 2018. "A convergence analysis of the method of codifferential descent," Computational Optimization and Applications, Springer, vol. 71(3), pages 879-913, December.
    4. 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.
    5. Boukouvala, Fani & Misener, Ruth & Floudas, Christodoulos A., 2016. "Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization, CDFO," European Journal of Operational Research, Elsevier, vol. 252(3), pages 701-727.
    6. Angelo Ciccazzo & Vittorio Latorre & Giampaolo Liuzzi & Stefano Lucidi & Francesco Rinaldi, 2015. "Derivative-Free Robust Optimization for Circuit Design," Journal of Optimization Theory and Applications, Springer, vol. 164(3), pages 842-861, March.

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