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The Lagrangian Globalization Method for Nonsmooth Constrained Equations

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Abstract

The difficulty suffered in optimization-based algorithms for the solution of nonlinear equations lies in that the traditional methods for solving the optimization problem have been mainly concerned with finding a stationary point or a local minimizer of the underlying optimization problem, which is not necessarily a solution of the equations. One method to overcome this difficulty is the Lagrangian globalization (LG for simplicity) method. This paper extends the LG method to nonsmooth equations with bound constraints. The absolute system of equations is introduced. A so-called Projected Generalized-Gradient Direction (PGGD) is constructed and proved to be a descent direction of the reformulated nonsmooth optimization problem. This projected approach keeps the feasibility of the iterates. The convergence of the new algorithm is established by specializing the PGGD. Numerical tests are given.

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Authors and Affiliations

  1. Institute of Mathematics, Changsha University of Science and Technology, Changsha, China

    Xiaojiao Tong

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

    Liqun Qi

  3. Institute of Applied Mathematics, Hunan University, Changsha, China

    Yu-Fei Yang

  4. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

    Yu-Fei Yang

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  1. Xiaojiao Tong
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  2. Liqun Qi
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  3. Yu-Fei Yang
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Correspondence to Xiaojiao Tong or Yu-Fei Yang.

Additional information

This author's work was done when she was visiting The Hong Kong Polytechnic University.

His work is also supported by the Research Grant Council of Hong Kong.

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Tong, X., Qi, L. & Yang, YF. The Lagrangian Globalization Method for Nonsmooth Constrained Equations. Comput Optim Applic 33, 89–109 (2006). https://doi.org/10.1007/s10589-005-5960-9

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  • DOI: https://doi.org/10.1007/s10589-005-5960-9

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