Abstract
This paper extends the Lagrangian globalization (LG) method to the nonsmooth equation \(\Phi (x) = 0\) arising from a nonlinear complementarity problem (NCP) and presents a descent algorithm for the LG phase. The aim of this paper is not to present a new method for solving the NCP, but to find \({\hat x}\) such that \(\left\| {\Phi (\hat x)} \right\| < \left\| {\Phi (\bar x)} \right\|\) when the NCP has a solution and \({\bar x}\) is a stationary point but not a solution.
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CHEN, X., QI, L., and SUN, D., Global and Superlinear Convergence of the Smoothing Newton Method and Its Application to General Box-Constrained Variational Inequalities, Mathematics of Computation, Vol. 67, pp. 519-540, 1998.
DE LUCA, T., FACCHINEI, F., and KANZOW, C., A Semismooth Equation Approach to the Solution of Nonlinear Complementarity Problems, Mathematical Programming, Vol. 75, pp. 407-439, 1996.
FISCHER, A., Solution of Monotone Complementarity Problems with Locally Lipschitzian Functions, Mathematical Programming, Vol. 76, pp. 513-532, 1997.
KANZOW, C., and PIEPER, H., Jacobian Smoothing Methods for Nonlinear Complementarity Problems, SIAM Journal on Optimization, Vol. 9, pp. 342-373, 1999.
MORé, J. J., Global Methods for Nonlinear Complementarity Problems, Mathematics of Operations Research, Vol. 21, pp. 589-614, 1996.
YAMASHITA, N., and FUKUSHIMA, M., Modified Newton Methods for Solving a Semismooth Reformulation of Monotone Complementarity Problems, Mathematical Programming, Vol. 76, pp. 469-491, 1997.
YANG, Y. F., and LI, D. H., A Globally and Superlinearly Convergent Method for Monotone Complementarity Problems with Nonsmooth Functions, Chinese Advances in Mathematics, Vol. 6, pp. 555-557, 1998.
NAZARETH, J. L., Lagrangian Globalization: Solving Nonlinear Equations via Constrained Optimization, The Mathematics of Numerical Analysis, Edited by J. Renegar, M. Shub, and S. Smale, Lectures in Applied Mathematics, American Mathematical Society, Providence, Rhode Island, Vol. 32, pp. 533-542, 1996.
NAZARETH, J. L., and QI, L., Globalization of Newton's Methods for Solving Nonlinear Equations, Numerical Linear Algebra with Applications, Vol. 3, pp. 239-249, 1996.
CLARKE, F. H., Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, NY, 1983.
QI, L., and SUN, J., A Nonsmooth Version of Newton's Method, Mathematical Programming, Vol. 58, pp. 353-367, 1993.
POLAK, E., MAYNE, D. Q., and WARDI, Y., On the Extension of Constrained Optimization Algorithms from Differentiable to Nondifferentiable Problems, SIAM Journal on Control and Optimization, Vol. 21, pp. 179-204, 1983.
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Chen, X., Qi, L. & Yang, Y.F. Lagrangian Globalization Methods for Nonlinear Complementarity Problems. Journal of Optimization Theory and Applications 112, 77–95 (2002). https://doi.org/10.1023/A:1013092412197
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DOI: https://doi.org/10.1023/A:1013092412197