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A Semismooth Newton-Type Method for the Nearest Doubly Stochastic Matrix Problem

Author

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  • Hao Hu

    (School of Mathematical and Statistical Sciences, Clemson University, Clemson, South Carolina 29634; Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada)

  • Xinxin Li

    (School of Mathematics, Jilin University, Changchun 130012, China)

  • Haesol Im

    (Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada)

  • Henry Wolkowicz

    (Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada)

Abstract

We study a semismooth Newton-type method for the nearest doubly stochastic matrix problem where the nonsingularity of the Jacobian can fail. The optimality conditions for this problem are formulated as a system of strongly semismooth functions. We show that the nonsingularity of the Jacobian does not hold for this system. By exploiting the problem structure, we construct a modified two step semismooth Newton method that guarantees a nonsingular Jacobian matrix at each iteration, and that converges to the nearest doubly stochastic matrix quadratically.

Suggested Citation

  • Hao Hu & Xinxin Li & Haesol Im & Henry Wolkowicz, 2024. "A Semismooth Newton-Type Method for the Nearest Doubly Stochastic Matrix Problem," Mathematics of Operations Research, INFORMS, vol. 49(2), pages 729-751, May.
    • Handle: RePEc:inm:ormoor:v:49:y:2024:i:2:p:729-751
    DOI: 10.1287/moor.2023.1382
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    References listed on IDEAS

    as
    1. Naomi Graham & Hao Hu & Jiyoung Im & Xinxin Li & Henry Wolkowicz, 2022. "A Restricted Dual Peaceman-Rachford Splitting Method for a Strengthened DNN Relaxation for QAP," INFORMS Journal on Computing, INFORMS, vol. 34(4), pages 2125-2143, July.
    2. Defeng Sun & Jie Sun, 2002. "Semismooth Matrix-Valued Functions," Mathematics of Operations Research, INFORMS, vol. 27(1), pages 150-169, February.
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    Cited by:

    1. Woosuk L. Jung & David Torregrosa-Belén & Henry Wolkowicz, 2025. "The $$\omega $$ ω -condition number: applications to preconditioning and low rank generalized Jacobian updating," Computational Optimization and Applications, Springer, vol. 91(1), pages 235-282, May.

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