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The Lagrangian Globalization Method for Nonsmooth Constrained Equations |
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| Authors: | Xiaojiao Tong Liqun Qi Yu-Fei Yang |
| Institution: | (1) Institute of Mathematics, Changsha University of Science and Technology, Changsha, China;(2) Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong;(3) Institute of Applied Mathematics, Hunan University, Changsha, China;(4) Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong |
| 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. 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. |
| Keywords: | constrained nonsmooth equations lagrangian globalization method stationary point global convergence |
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