×

zbMATH — the first resource for mathematics

Nonmonotone stabilization methods for nonlinear equations. (English) Zbl 0803.65070
A new globalization criterion is defined for solution methods of nonlinear equations \(H(x) = 0\), where \(H:\mathbb{R}^ n \to\mathbb{R}^ n\) is a given function. The authors assume that there exists a locally Lipschitzian merit function \(M\) with the property that \(M(x) \geq 0\) \(\forall x \in\mathbb{R}^ n\), \(M(x) = 0\) if and only if \(H(x) = 0\).
A nonmonotone stabilization algorithm is described and general conditions are given which are required for such a technique to give global convergence. These conditions are formulated in terms of a merit function, an auxiliary function, and the directions determined by a particular algorithm.
The authors prove the general convergence result under these assumptions without specifying the particular merit function, the auxiliary function or the direction, but only the conditions which they must satisfy. The described conditions are so weak that almost all the merit functions and auxiliary functions in the literature satisfy the given conditions. Some examples are presented.

MSC:
65H10 Numerical computation of solutions to systems of equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Burdakov, O. P.,Some Globally Convergent Modifications of Newton’s Method for Solving Systems of Nonlinear Equations, Soviet Mathematics Doklady, Vol. 22, pp. 376–379, 1980. · Zbl 0471.65029
[2] Stoer, J., andBulirsch, R.,Introduction to Numerical Analysis. Springer-Verlag, New York, New York, 1980. · Zbl 0423.65002
[3] Polak, E.,On the Global Stabilization of Locally Convergent Algorithms, Automatica, Vol. 12, pp. 337–349, 1976. · Zbl 0335.49023
[4] Han, S. P., Pang, J. S., andRangaraj, N.,Globally Convergent Newton Methods for Nonsmooth Equations, Mathematics of Operations Research, Vol. 17, pp. 586–607, 1992. · Zbl 0777.90057
[5] Harker, P. T., andXiao B.,Newton’s Method for the Nonlinear Complementarity Problem: A B-Differentiable Equation Approach, Mathematical Programming, Vol. 48, pp. 339–358, 1990. · Zbl 0724.90071
[6] Pang, J. S., Han, S. P., andRangaraj, N.,Minimization of Locally Lipschitzian Functions, SIAM Journal on Optimization, Vol. 1, pp. 57–82, 1991. · Zbl 0752.90070
[7] Grippo, L., Lampariello, F., andLucidi, S.,A Nonmonotone Line Search Technique for Newton’s Method, SIAM Journal on Numerical Analysis, Vol. 23, pp. 707–716, 1986. · Zbl 0616.65067
[8] Grippo, L., Lampariello, F., andLucidi, S.,A Class of Nonmonotone Stabilization Methods in Unconstrained Optimization, Numerische Mathematik, Vol. 59, pp. 779–805, 1991. · Zbl 0739.90062
[9] Dennis, J. E., andSchnabel, R. B.,Numerical Methods for Unconstrained Optimizations and Nonlinear Equations, Prentice-Hall, Englewood Cliffs, New Jersey, 1983.
[10] Shor, N. Z.,Minimization Methods for Nondifferentiable Functions, Springer-Verlag, Berlin, Germany, 1985. · Zbl 0561.90058
[11] Zowe, J.,Nondifferentiable Optimization, Computational Mathematical Programming, Edited by K. Schittkowski, Springer-Verlag, Berlin, Germany, pp. 321–356, 1985. · Zbl 0581.90072
[12] Bertsekas, D. P.,Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, New York, 1982. · Zbl 0572.90067
[13] Ortega, J. M., andRheinboldt, W. C.,Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, New York, 1970. · Zbl 0241.65046
[14] Clarke, F. H.,Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, New York, 1983. · Zbl 0582.49001
[15] Pshenichny, B. N., andDanilin, Yu. M.,Numerical Methods in Extremal Problems, MIR Publishers, Moscow, Russia, 1978.
[16] Kiwiel, K. C.,Methods of Descent for Nondifferentiable Optimization, Lecture Notes in Mathematics, Springer-Verlag, Berlin, Germany, Vol. 1113, 1985. · Zbl 0561.90059
[17] Ferris, M. C., andLucidi, S.,Globally Convergent Methods for Nonlinear Equations, Technical Report 1030, Computer Sciences Department, University of Wisconsin, 1991.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.