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Nonsingularity in second-order cone programming via the smoothing metric projector

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Abstract

Based on the differential properties of the smoothing metric projector onto the second-order cone, we prove that, for a locally optimal solution to a nonlinear second-order cone programming problem, the nonsingularity of the Clarke’s generalized Jacobian of the smoothing Karush-Kuhn-Tucker system, constructed by the smoothing metric projector, is equivalent to the strong second-order sufficient condition and constraint nondegeneracy, which is in turn equivalent to the strong regularity of the Karush-Kuhn-Tucker point. Moreover, this nonsingularity property guarantees the quadratic convergence of the corresponding smoothing Newton method for solving a Karush-Kuhn-Tucker point. Interestingly, the analysis does not need the strict complementarity condition.

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Authors and Affiliations

  1. College of Information Sciences and Engineering, Shandong Agricultural University, Tai’an, 271018, China

    Yun Wang

  2. Department of Applied Mathematics, Dalian University of Technology, Dalian, 116024, China

    LiWei Zhang

Authors
  1. Yun Wang
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  2. LiWei Zhang
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Correspondence to Yun Wang.

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Wang, Y., Zhang, L. Nonsingularity in second-order cone programming via the smoothing metric projector. Sci. China Math. 53, 1025–1038 (2010). https://doi.org/10.1007/s11425-009-0207-3

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  • DOI: https://doi.org/10.1007/s11425-009-0207-3

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