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A local minimax characterization for computing multiple nonsmooth saddle critical points

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Abstract

This paper is concerned with characterizations of nonsmooth saddle critical points for numerical algorithm design. Most characterizations for nonsmooth saddle critical points in the literature focus on existence issue and are converted to solve global minimax problems. Thus they are not helpful for numerical algorithm design. Inspired by the results on computational theory and methods for finding multiple smooth saddle critical points in [14, 15, 19, 21, 23], a local minimax characterization for multiple nonsmooth saddle critical points in either a Hilbert space or a reflexive Banach space is established in this paper to provide a mathematical justification for numerical algorithm design. A local minimax algorithm for computing multiple nonsmooth saddle critical points is presented by its flow chart.

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Author notes

  1. Xudong Yao

    Present address: Department of Mathematics, University of Connecticut, Storrs, CT, USA

Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, College Station, TX

    Xudong Yao

  2. Department of Mathematics, Texas A&M University, College Station, TX, 77843-3368, USA

    Jianxin Zhou

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  1. Xudong Yao
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  2. Jianxin Zhou
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Correspondence to Jianxin Zhou.

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Dedicated to Terry Rockafellar on his 70th birthday

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Yao, X., Zhou, J. A local minimax characterization for computing multiple nonsmooth saddle critical points. Math. Program. 104, 749–760 (2005). https://doi.org/10.1007/s10107-005-0636-x

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  • DOI: https://doi.org/10.1007/s10107-005-0636-x

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