Newton’s method with feasible inexact projections for solving constrained generalized equations
- 118 Downloads
This paper aims to address a new version of Newton’s method for solving constrained generalized equations. This method can be seen as a combination of the classical Newton’s method applied to generalized equations with a procedure to obtain a feasible inexact projection. Using the contraction mapping principle, we establish a local analysis of the proposed method under appropriate assumptions, namely metric regularity or strong metric regularity and Lipschitz continuity. Metric regularity is assumed to guarantee that the method generates a sequence that converges to a solution. Under strong metric regularity, we show the uniqueness of the solution in a suitable neighborhood, and that all sequences starting in this neighborhood converge to this solution. We also require the assumption of Lipschitz continuity to establish a linear or superlinear convergence rate for the method.
KeywordsConstrained generalized equations Newton’s method Feasible inexact projection Lipschitz continuity Metric regularity Strong metric regularity Local convergence
Mathematics Subject Classification65K15 49M15 90C30
The authors would like to thank the anonymous referees for their constructive comments, which have helped to substantially improve the presentation of the paper. In particular, we thank one of the referees for drawing our attention to the use of general procedures to find a feasible inexact projection, rather than a specific one as we were using in our first version. The authors were supported in part by CNPq Grants 305158/2014-7 and 302473/2017-3, FAPEG/GO and CAPES.
- 12.Dontchev, A.L.: Local analysis of a Newton-type method based on partial linearization. In: The Mathematics of Numerical Analysis (Park City, UT, 1995), Lectures in Applied Mathematics, vol. 32, pp. 295–306. American Mathematical Society, Providence (1996)Google Scholar
- 24.Gould, N.I.M., Toint, P.L.: Numerical methods for large-scale non-convex quadratic programming. In: Trends in Industrial and Applied Mathematics (Amritsar, 2001), Applied Optimization, vol. 72, pp. 149–179. Kluwer Academic Publications, Dordrecht (2002)Google Scholar
- 27.Josephy, N.H.: Newton’s method for generalized equations and the pies energy model. Ph.D. thesis, Department of Industrial Engineering, University of Wisconsin–Madison (1979)Google Scholar