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Newton methods for solving two classes of nonsmooth equations. (English) Zbl 1068.65063
The author considers systems of nonsmooth equations which are formed by max-type functions or by smooth compositions of max-type functions. The modification of the Newton method, proposed by the author, is based on the new definition of the differential for the functions $$F:\mathbb R^n\rightarrow \mathbb R^n.$$ This method can be implemented more easily than previous ones because they do not require an element of the Clarke generalized Jacobian [cf. F. H. Clarke, Optimization and nonsmooth analysis (1983; Zbl 0582.49001)]. The $$Q$$-superlinear convergence is proved.
Reviewer: Jan Zítko (Praha)

##### MSC:
 65H10 Numerical computation of solutions to systems of equations
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##### References:
 [1] W. Chen, G. Chen and E. Feng: Variational principle with nonlinear complementarity for three dimensional contact problems and its numerical method. Sci. China Ser. A 39 (1996), 528-539. · Zbl 0862.73053 [2] X. Chen: A verification method for solutions of nonsmooth equations. Computing 58 (1997), 281-294. · Zbl 0882.65038 [3] F. H. Clarke: Optimization and Nonsmooth Analysis. John Wiley and Sons, New York, 1983. · Zbl 0582.49001 [4] V. F. Demyanov, G. E. Stavroulakis, L. N. Polyakova, and P. D. Panagiotopoulous: Quasidifferentiability and Nonsmooth Modelling in Mechanics, Engineering and Economic. Kluwer Academic Publishers, Dordercht, 1996. [5] F. Meng, Y. Gao and Z. Xia: Second-order directional derivatives for max-type functions. J. Dalian Univ. Tech. 38 (1998), 621-624. · Zbl 0923.90131 [6] M. Mifflin: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15 (1977), 959-972. · Zbl 0376.90081 [7] J. M. Ortega, W. C. Rheinboldt: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York, 1970. · Zbl 0241.65046 [8] L. Qi: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18 (1993), 227-244. · Zbl 0776.65037 [9] L. Qi, J. Sun: A nonsmooth version of Newton’s method. Math. Programming 58 (1993), 353-367. · Zbl 0780.90090 [10] D. Sun, J. Han: Newton and quasi-Newton methods for a class of nonsmooth equations and related problems. SIAM J. Optim. 7 (1997), 463-480. · Zbl 0872.90087
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