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Active-Set Projected Trust-Region Algorithm for Box-Constrained Nonsmooth Equations

Author

Listed:
  • L. Qi

    (Hong Kong Polytechnic University)

  • X. J. Tong

    (Changsha University of Science and Technology)

  • D. H. Li

    (Hunan University)

Abstract

In this paper, by means of an active-set strategy, we present a trust-region method for solving box-constrained nonsmooth equations. Nice properties of the proposed method include: (a) all iterates remain feasible; (b) the search direction, as adequate combination of the projected gradient direction and the trust-region direction, is an asymptotic Newton direction under mild conditions; (c) the subproblem of the proposed method, possessing the form of an unconstrained trust-region subproblem, can be solved by existing methods; (d) the subproblem of the proposed method is of reduced dimension, which is potentially cheaper when applied to solve large-scale problems. Under appropriate conditions, we establish global and local superlinear/quadratic convergence of the method. Preliminary numerical results are given.

Suggested Citation

  • L. Qi & X. J. Tong & D. H. Li, 2004. "Active-Set Projected Trust-Region Algorithm for Box-Constrained Nonsmooth Equations," Journal of Optimization Theory and Applications, Springer, vol. 120(3), pages 601-625, March.
    • Handle: RePEc:spr:joptap:v:120:y:2004:i:3:d:10.1023_b:jota.0000025712.43243.eb
    DOI: 10.1023/B:JOTA.0000025712.43243.eb
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    References listed on IDEAS

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    1. Liqun Qi, 1999. "Regular Pseudo-Smooth NCP and BVIP Functions and Globally and Quadratically Convergent Generalized Newton Methods for Complementarity and Variational Inequality Problems," Mathematics of Operations Research, INFORMS, vol. 24(2), pages 440-471, May.
    2. F. Facchinei & C. Kanzow, 1997. "On Unconstrained and Constrained Stationary Points of the Implicit Lagrangian," Journal of Optimization Theory and Applications, Springer, vol. 92(1), pages 99-115, January.
    3. Liqun Qi, 1993. "Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 227-244, February.
    4. A. Fischer, 1998. "New Constrained Optimization Reformulation of Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 97(1), pages 105-117, April.
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    Cited by:

    1. Li, Xiangli & Guo, Xiao, 2015. "Spectral residual methods with two new non-monotone line searches for large-scale nonlinear systems of equations," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 59-69.
    2. X. J. Tong & L. Qi, 2004. "On the Convergence of a Trust-Region Method for Solving Constrained Nonlinear Equations with Degenerate Solutions," Journal of Optimization Theory and Applications, Springer, vol. 123(1), pages 187-211, October.
    3. Leonardo Galli & Christian Kanzow & Marco Sciandrone, 2018. "A nonmonotone trust-region method for generalized Nash equilibrium and related problems with strong convergence properties," Computational Optimization and Applications, Springer, vol. 69(3), pages 629-652, April.
    4. Changyu Wang & Qian Liu & Cheng Ma, 2013. "Smoothing SQP algorithm for semismooth equations with box constraints," Computational Optimization and Applications, Springer, vol. 55(2), pages 399-425, June.
    5. Gonglin Yuan & Zengxin Wei & Zhongxing Wang, 2013. "Gradient trust region algorithm with limited memory BFGS update for nonsmooth convex minimization," Computational Optimization and Applications, Springer, vol. 54(1), pages 45-64, January.
    6. J. Chen & L. Qi, 2010. "Pseudotransient Continuation for Solving Systems of Nonsmooth Equations with Inequality Constraints," Journal of Optimization Theory and Applications, Springer, vol. 147(2), pages 223-242, November.

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