Lipschitzian Stability of Newton's Method for Variational Inclusions

We present an overview to Lipschitz-type properties of mappings associated with solutions of optimization problems including variational inequalities and mathematical programs. We show that these properties are inherited in various ways by the mapping acting from parameters of the problem and the starting point to the set of sequences generated by Newton’s method. Some new insights into convergence of Newton’s/SQP method are also presented.

[1]  Stephen M. Robinson,et al.  Strongly Regular Generalized Equations , 1980, Math. Oper. Res..

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

[3]  Diethard Klatte,et al.  Strong stability in nonlinear programming revisited , 1999, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[4]  Diethard Klatte.  Upper Lipschitz behavior of solutions to perturbed. C1,1 programs , 2000, Math. Program..

[5]  R. Tyrrell Rockafellar,et al.  Characterizations of Strong Regularity for Variational Inequalities over Polyhedral Convex Sets , 1996, SIAM J. Optim..

[6]  S. M. Robinson.  Generalized equations and their solutions, Part I: Basic theory , 1979 .

[7]  Jean-Pierre Aubin,et al.  Lipschitz Behavior of Solutions to Convex Minimization Problems , 1984, Math. Oper. Res..

[8]  R. Rockafellar.  Proto-Differentiability of Set-Valued Mappings and its Applications in Optimization☆ , 1989 .

[9]  R. Tyrrell Rockafellar,et al.  Extended Nonlinear Programming , 2000 .

[10]  Huifu Xu,et al.  Set-valued approximations and Newton’s methods , 1999, Math. Program..

[11]  Asen L. Dontchev,et al.  Characterizations of Lipschitz Stability in Optimization , 1995 .

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

[13]  William W. Hager,et al.  Uniform Convergence and Mesh Independence of Newton's Method for Discretized Variational Problems , 2000, SIAM J. Control. Optim..

[14]  A. Ioffe.  Metric regularity and subdifferential calculus , 2000 .

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

[16]  Adam B. Levy,et al.  Stability of Locally Optimal Solutions , 1999, SIAM J. Optim..

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

[18]  Adam B. Levy,et al.  Implicit multifunction theorems for the sensitivity analysis of variational conditions , 1996, Math. Program..

[19]  R. Cominetti.  Metric regularity, tangent sets, and second-order optimality conditions , 1990 .

[20]  R. Tyrrell Rockafellar,et al.  Tilt Stability of a Local Minimum , 1998, SIAM J. Optim..

[21]  James Renegar,et al.  Linear programming, complexity theory and elementary functional analysis , 1995, Math. Program..