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Using negative curvature in solving nonlinear programs

Author

Listed:
  • Donald Goldfarb

    (Columbia University)

  • Cun Mu

    (Columbia University)

  • John Wright

    (Columbia University)

  • Chaoxu Zhou

    (Columbia University)

Abstract

Minimization methods that search along a curvilinear path composed of a non-ascent negative curvature direction in addition to the direction of steepest descent, dating back to the late 1970s, have been an effective approach to finding a stationary point of a function at which its Hessian is positive semidefinite. For constrained nonlinear programs arising from recent applications, the primary goal is to find a stationary point that satisfies the second-order necessary optimality conditions. Motivated by this, we generalize the approach of using negative curvature directions from unconstrained optimization to equality constrained problems and prove that our proposed negative curvature method is guaranteed to converge to a stationary point satisfying second-order necessary conditions.

Suggested Citation

  • Donald Goldfarb & Cun Mu & John Wright & Chaoxu Zhou, 2017. "Using negative curvature in solving nonlinear programs," Computational Optimization and Applications, Springer, vol. 68(3), pages 479-502, December.
    • Handle: RePEc:spr:coopap:v:68:y:2017:i:3:d:10.1007_s10589-017-9925-6
    DOI: 10.1007/s10589-017-9925-6
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    References listed on IDEAS

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    1. Gábor Pataki, 1998. "On the Rank of Extreme Matrices in Semidefinite Programs and the Multiplicity of Optimal Eigenvalues," Mathematics of Operations Research, INFORMS, vol. 23(2), pages 339-358, May.
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    Cited by:

    1. Renato Leone & Giovanni Fasano & Massimo Roma & Yaroslav D. Sergeyev, 2020. "Iterative Grossone-Based Computation of Negative Curvature Directions in Large-Scale Optimization," Journal of Optimization Theory and Applications, Springer, vol. 186(2), pages 554-589, August.
    2. Shun Arahata & Takayuki Okuno & Akiko Takeda, 2023. "Complexity analysis of interior-point methods for second-order stationary points of nonlinear semidefinite optimization problems," Computational Optimization and Applications, Springer, vol. 86(2), pages 555-598, November.

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