Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
nonsmooth equations; Newton method; approximate Newton method; max-type function; composite function; convergence
nonsmooth equations; Newton method; approximate Newton method; max-type function; composite function; convergence
Summary:
The paper is devoted to two systems of nonsmooth equations. One is the system of equations of max-type functions and the other is the system of equations of smooth compositions of max-type functions. The Newton and approximate Newton methods for these two systems are proposed. The Q-superlinear convergence of the Newton methods and the Q-linear convergence of the approximate Newton methods are established. The present methods can be more easily implemented than the previous ones, since they do not require an element of Clarke generalized Jacobian, of B-differential, or of b-differential, at each iteration point.
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. MR 1409809
[2] X. Chen: A verification method for solutions of nonsmooth equations. Computing 58 (1997), 281–294. DOI 10.1007/BF02684394 | MR 1457114 | Zbl 0882.65038
[3] F. H. Clarke: Optimization and Nonsmooth Analysis. John Wiley and Sons, New York, 1983. MR 0709590 | 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. MR 1409141
[5] F. Meng, Y. Gao and Z. Xia: Second-order directional derivatives for max-type functions. J. Dalian Univ. Tech. 38 (1998), 621–624. (Chinese) MR 1667292
[6] M. Mifflin: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15 (1977), 959–972. DOI 10.1137/0315061 | MR 0461556 | Zbl 0376.90081
[7] J. M. Ortega, W. C. Rheinboldt: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York, 1970. MR 0273810
[8] L. Qi: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18 (1993), 227–244. DOI 10.1287/moor.18.1.227 | MR 1250115 | Zbl 0776.65037
[9] L. Qi, J. Sun: A nonsmooth version of Newton’s method. Math. Programming 58 (1993), 353–367. DOI 10.1007/BF01581275 | MR 1216791
[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. DOI 10.1137/S1052623494274970 | MR 1443629
Search
Partner of
