Sensitivity Analysis of a Stationary Point Set Map Under Total Perturbations. Part 1: Lipschitzian Stability

By applying some theorems of Levy and Mordukhovich (Math Program 99:311–327, 2004) and other related results, we estimate the Fréchet coderivative and the Mordukhovich coderivative of the stationary point set map of a smooth parametric optimization problem with one smooth functional constraint under total perturbations. From the obtained formulas, we derive necessary and sufficient conditions for the local Lipschitz-like property of the stationary point set map. This leads us to new insights into the preceding deep investigations of Levy and Mordukhovich in the above-cited paper and of Qui (J Optim Theory Appl 161:398–429, 2014, J Glob Optim 65:615–635, 2016).

[1]  Nguyen Dong Yen,et al.  A Class of Linear Generalized Equations , 2014, SIAM J. Optim..

[2]  Jean-Alexis Célia,et al.  A variant of Newton's method for generalized equations , 2005 .

[3]  D. Klatte.  Book review: Implicit Functions and Solution Mappings:A View from Variational Analysis. Second Edition. By A. L. Dontchev and R. T. Rockafellar. Springer, New York, 2014 , 2015 .

[4]  Nguyen Mau Nam,et al.  Coderivatives of normal cone mappings and Lipschitzian stability of parametric variational inequalities , 2010 .

[5]  B. S. Mordukhovich,et al.  Full Lipschitzian and Hölderian Stability in Optimization with Applications to Mathematical Programming and Optimal Control , 2014, SIAM J. Optim..

[6]  Asen L. Dontchev,et al.  UNIFORM CONVERGENCE OF THE NEWTON METHOD FOR AUBIN CONTINUOUS MAPS , 1996 .

[7]  Nghia Thaian Tran.  Full Stability In Optimization , 2013 .

[8]  Boris S. Mordukhovich,et al.  Local Monotonicity and Full Stability for Parametric Variational Systems , 2016, SIAM J. Optim..

[9]  B. Mordukhovich,et al.  Coderivatives in parametric optimization , 2004, Math. Program..

[10]  R. Rockafellar,et al.  Implicit Functions and Solution Mappings: A View from Variational Analysis , 2009 .

[11]  Boris Polyak,et al.  B.S. Mordukhovich. Variational Analysis and Generalized Differentiation. I. Basic Theory, II. Applications , 2009 .

[12]  Lei Guo,et al.  Stability Analysis for Parametric Mathematical Programs with Geometric Constraints and Its Applications , 2012, SIAM J. Optim..

[14]  Jen-Chih Yao,et al.  Point-based sufficient conditions for metric regularity of implicit multifunctions , 2009 .

[15]  Stephen M. Robinson,et al.  Variational Inequalities over Perturbed Polyhedral Convex Sets , 2008, Math. Oper. Res..

[16]  Nguyen Thanh Qui.  Linearly perturbed polyhedral normal cone mappings and applications , 2011 .

[17]  Nguyen Thanh Qui.  New results on linearly perturbed polyhedral normal cone mappings , 2011 .

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

[19]  S. M. Robinson.  Stability Theory for Systems of Inequalities, Part II: Differentiable Nonlinear Systems , 1976 .

[20]  N. D. Yen,et al.  Fréchet and normal coderivatives of implicit multifunctions , 2011 .

[21]  Nguyen Thanh Qui.  Nonlinear Perturbations of Polyhedral Normal Cone Mappings and Affine Variational Inequalities , 2012 .

[22]  Boris S. Mordukhovich,et al.  Robinson Stability of Parametric Constraint Systems via Variational Analysis , 2016, SIAM J. Optim..

[23]  Nguyen Thanh Qui.  Coderivatives of implicit multifunctions and stability of variational systems , 2016, J. Glob. Optim..

[24]  Boris S. Mordukhovich,et al.  Full Stability in Finite-Dimensional Optimization , 2015, Math. Oper. Res..

[25]  Boris S. Mordukhovich,et al.  Second-Order Analysis of Polyhedral Systems in Finite and Infinite Dimensions with Applications to Robust Stability of Variational Inequalities , 2010, SIAM J. Optim..

[26]  R. Rockafellar,et al.  Implicit Functions and Solution Mappings , 2009 .

[27]  W. Rudin.  Principles of mathematical analysis , 1964 .

[28]  S. M. Robinson,et al.  Solution Continuity in Monotone Affine Variational Inequalities , 2007, SIAM J. Optim..

[29]  Wen Song,et al.  Linearly perturbed generalized polyhedral normal cone mappings and applications , 2016 .

[30]  J. Borwein,et al.  Verifiable necessary and sufficient conditions for openness and regularity of set-valued and single-valued maps , 1988 .

[31]  B. Mordukhovich.  Variational analysis and generalized differentiation , 2006 .

[32]  B. Mordukhovich.  Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions , 1993 .

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

[34]  Boris S. Mordukhovich,et al.  Characterizations of Full Stability in Constrained Optimization , 2013, SIAM J. Optim..

[35]  Nguyen Thanh Qui.  Generalized Differentiation of a Class of Normal Cone Operators , 2014, J. Optim. Theory Appl..

[36]  Boris S. Mordukhovich,et al.  Second-Order Subdifferential Calculus with Applications to Tilt Stability in Optimization , 2011, SIAM J. Optim..

[37]  C. Jean-Alexis.  A variant of Newton's method for generalized equations , 2006 .

[38]  J. Penot.  Metric regularity, openness and Lipschitzian behavior of multifunctions , 1989 .

[39]  Duong Thi Kim Huyen,et al.  Coderivatives and the Solution Map of a Linear Constraint System , 2016, SIAM J. Optim..

[40]  Nguyen Thi Quynh Trang.  A note on Lipschitzian stability of variational inequalities over perturbed polyhedral convex sets , 2016, Optim. Lett..