On Noncritical Solutions of Complementarity Systems

The present paper is devoted to characterize local Lipschitzian error bounds for a system of nonsmooth equations $$\displaystyle \begin{aligned} F(u)=0, \end{aligned} $$ where \(F:\mathbb {R}^n\to \mathbb {R}^m\) is a (at least) locally Lipschitz continuous function whose further properties will be specified later on.

[1]  B. Mordukhovich.  Generalized Differential Calculus for Nonsmooth and Set-Valued Mappings , 1994 .

[2]  A. Ioffe.  Variational Analysis of Regular Mappings , 2017 .

[3]  Francisco Facchinei,et al.  An LP-Newton method: nonsmooth equations, KKT systems, and nonisolated solutions , 2013, Mathematical Programming.

[4]  R. Tyrrell Rockafellar,et al.  Sensitivity analysis for nonsmooth generalized equations , 1992, Math. Program..

[5]  A. Ioffe,et al.  Theory of extremal problems , 1979 .

[6]  Andreas Fischer,et al.  GENERALIZED NASH EQUILIBRIUM PROBLEMS - RECENT ADVANCES AND CHALLENGES , 2014 .

[7]  Andreas Fischer,et al.  A Levenberg-Marquardt algorithm for unconstrained multicriteria optimization , 2008, Oper. Res. Lett..

[8]  Ya-Xiang Yuan,et al.  On the Quadratic Convergence of the Levenberg-Marquardt Method without Nonsingularity Assumption , 2005, Computing.

[9]  S. M. Robinson.  Some continuity properties of polyhedral multifunctions , 1981 .

[10]  R. Tyrrell Rockafellar,et al.  Variational Analysis , 1998, Grundlehren der mathematischen Wissenschaften.

[11]  William W. Hager,et al.  Stability in the presence of degeneracy and error estimation , 1999, Math. Program..

[12]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[13]  Alexey F. Izmailov,et al.  Critical solutions of nonlinear equations: stability issues , 2018, Math. Program..

[14]  Stephen J. Wright.  Superlinear Convergence of a Stabilized SQP Method to a Degenerate Solution , 1998, Comput. Optim. Appl..

[15]  Alexey F. Izmailov,et al.  A note on upper Lipschitz stability, error bounds, and critical multipliers for Lipschitz-continuous KKT systems , 2013, Math. Program..

[16]  M. Fukushima,et al.  On the Rate of Convergence of the Levenberg-Marquardt Method , 2001 .

[17]  Francisco Facchinei,et al.  Generalized Nash equilibrium problems and Newton methods , 2008, Math. Program..

[18]  Andreas Fischer,et al.  A unified local convergence analysis of inexact constrained Levenberg–Marquardt methods , 2011, Optimization Letters.

[19]  A. Fischer,et al.  Constrained Lipschitzian Error Bounds and Noncritical Solutions of Constrained Equations , 2020, Set-Valued and Variational Analysis.

[20]  A. Fischer,et al.  A special complementarity function revisited , 2019 .

[21]  Andreas Fischer,et al.  Convergence conditions for Newton-type methods applied to complementarity systems with nonisolated solutions , 2016, Comput. Optim. Appl..

[22]  A. Ioffe.  Variational Analysis of Regular Mappings: Theory and Applications , 2017 .

[23]  V. Jeyakumar,et al.  Solution Point Characterizations and Convergence Analysis of a Descent Algorithm for Nonsmooth Continuous Complementarity Problems , 2001 .

[24]  Jong-Shi Pang,et al.  Error bounds in mathematical programming , 1997, Math. Program..

[25]  D. Klatte.  Nonsmooth equations in optimization , 2002 .

[26]  C. Kanzow.  Levenberg-Marquardt methods for constrained nonlinear equations with strong local convergence properties , 2004 .

[27]  Mohamed A. Tawhid,et al.  On Two Applications of H-Differentiability to Optimization and Complementarity Problems , 2000, Comput. Optim. Appl..

[28]  Francisco Facchinei,et al.  A family of Newton methods for nonsmooth constrained systems with nonisolated solutions , 2013, Math. Methods Oper. Res..

[29]  V. Jeyakumar,et al.  Approximate Jacobian Matrices for Nonsmooth Continuous Maps and C 1 -Optimization , 1998 .

[30]  Alexey F. Izmailov,et al.  Stabilized SQP revisited , 2012, Math. Program..

[31]  Andrei Dmitruk,et al.  LYUSTERNIK'S THEOREM AND THE THEORY OF EXTREMA , 1980 .

[32]  Mikhail V. Solodov,et al.  Stabilized sequential quadratic programming for optimization and a stabilized Newton-type method for variational problems , 2010, Math. Program..

[33]  Andreas Fischer,et al.  Local behavior of an iterative framework for generalized equations with nonisolated solutions , 2002, Math. Program..

[34]  M. Fukushima,et al.  Erratum to Levenberg-Marquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints , 2005 .

[35]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .