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Group Update Method for Sparse Minimax Problems

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Abstract

A group update algorithm is presented for solving minimax problems with a finite number of functions, whose Hessians are sparse. The method uses the gradient evaluations as efficiently as possible by updating successively the elements in partitioning groups of the columns of every Hessian in the process of iterations. The chosen direction is determined directly by the nonzero elements of the Hessians in terms of partitioning groups. The local \(q\)-superlinear convergence of the method is proved, without requiring the imposition of a strict complementarity condition, and the \(r\)-convergence rate is estimated. Furthermore, two efficient methods handling nonconvex case are given. The global convergence of one method is proved, and the local \(q\)-superlinear convergence and \(r\)-convergence rate of another method are also proved or estimated by a novel technique. The robustness and efficiency of the algorithms are verified by numerical tests.

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Acknowledgments

This research was sponsored by the National Natural Science Foundation of China (No. 71090404, 71102070, 11171221, 71271138, 71202065, 71103199, 71371140, 91230103, 11171051), the Fundamental Research Funds for the Central Universities (No. DUT13LK04), Shanghai First-class Academic Discipline Projects (No. XTKX2012, S1201YL XK), the Innovation Program of Shanghai Municipal Education Commission (No. 14YZ088, 14YZ089), Programs of National Training Foundation of University of Shanghai for Science and Technology (No. 13XGM03), and the Innovation Fund Project for Graduate and Undergraduate Student of Shanghai (No. JWCXSL1302, SH2013054, XJ2014098). Authors are indebted to the reviewers and the editors for their constructive comments which greatly improved the contents and exposition of this paper.

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Authors and Affiliations

  1. Business School, University of Shanghai for Science & Technology, Shanghai, 200093, China

    Junxiang Li

  2. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

    Mingsong Cheng & Bo Yu

  3. School of Mathematics, Jilin University, Changchun, 130012, China

    Shuting Zhang

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  1. Junxiang Li
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  2. Mingsong Cheng
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  3. Bo Yu
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  4. Shuting Zhang
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Correspondence to Junxiang Li.

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Li, J., Cheng, M., Yu, B. et al. Group Update Method for Sparse Minimax Problems. J Optim Theory Appl 166, 257–277 (2015). https://doi.org/10.1007/s10957-014-0646-z

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  • DOI: https://doi.org/10.1007/s10957-014-0646-z

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