The Josephy–Newton Method for Semismooth Generalized Equations and Semismooth SQP for Optimization

While generalized equations with differentiable single-valued base mappings and the associated Josephy–Newton method have been studied extensively, the setting with semismooth base mapping had not been previously considered (apart from the two special cases of usual nonlinear equations and of Karush–Kuhn–Tucker optimality systems). We introduce for the general semismooth case appropriate notions of solution regularity and prove local convergence of the corresponding Josephy–Newton method. As an application, we immediately recover the known primal-dual local convergence properties of semismooth sequential quadratic programming algorithm (SQP), but also obtain some new results that complete the analysis of the SQP primal rate of convergence, including its quasi-Newton variant.

[1]  R. Rockafellar,et al.  Generalized linear-quadratic problems of deterministic and stochastic optimal control in discrete time , 1990 .

[2]  Houyuan Jiang,et al.  Semismooth Karush-Kuhn-Tucker Equations and Convergence Analysis of Newton and Quasi-Newton Methods for Solving these Equations , 1997, Math. Oper. Res..

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

[4]  Liqun Qi,et al.  Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations , 1993, Math. Oper. Res..

[5]  R. Tyrrell Rockafellar,et al.  Newton’s method for generalized equations: a sequential implicit function theorem , 2010, Math. Program..

[6]  Paul T. Boggs,et al.  Sequential Quadratic Programming , 1995, Acta Numerica.

[7]  Alexey F. Izmailov,et al.  Semismooth Newton method for the lifted reformulation of mathematical programs with complementarity constraints , 2012, Comput. Optim. Appl..

[8]  B. Mordukhovich.  Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis , 1994 .

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

[10]  N. Josephy.  Newton's Method for Generalized Equations. , 1979 .

[11]  J. F. Bonnans,et al.  Local analysis of Newton-type methods for variational inequalities and nonlinear programming , 1994 .

[12]  B. Kummer.  NEWTON's METHOD FOR NON-DIFFERENTIABLE FUNCTIONS , 1988, Advances in Mathematical Optimization.

[13]  Alexey F. Izmailov,et al.  The Theory of 2-Regularity for Mappings with Lipschitzian Deriatives and its Applications to Optimality Conditions , 2002, Math. Oper. Res..

[14]  Jean Charles Gilbert,et al.  Numerical Optimization: Theoretical and Practical Aspects , 2003 .

[15]  Alexey F. Izmailov,et al.  Sharp Primal Superlinear Convergence Results for Some Newtonian Methods for Constrained Optimization , 2010, SIAM J. Optim..

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

[17]  J. Frédéric Bonnans,et al.  Pseudopower expansion of solutions of generalized equations and constrained optimization problems , 1995, Math. Program..

[18]  Alexey F. Izmailov,et al.  A Truncated SQP Method Based on Inexact Interior-Point Solutions of Subproblems , 2010, SIAM J. Optim..

[19]  Liqun Qi,et al.  Superlinearly convergent approximate Newton methods for LC1 optimization problems , 1994, Math. Program..

[20]  B. Kummer.  Newton’s Method Based on Generalized Derivatives for Nonsmooth Functions: Convergence Analysis , 1992 .

[21]  R. Tyrrell Rockafellar,et al.  Computational schemes for large-scale problems in extended linear-quadratic programming , 1990, Math. Program..

[22]  Defeng Sun,et al.  Superlinear Convergence of Approximate Newton Methods for LC1 Optimization Problems without Strict Complementarity , 1995 .

[23]  Alexey F. Izmailov,et al.  Inexact Josephy–Newton framework for generalized equations and its applications to local analysis of Newtonian methods for constrained optimization , 2010, Comput. Optim. Appl..

[24]  Jong-Shi Pang,et al.  Nonsmooth Equations: Motivation and Algorithms , 1993, SIAM J. Optim..

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

[26]  Oliver Stein.  Lifting mathematical programs with complementarity constraints , 2012, Math. Program..

[27]  N. Josephy.  Quasi-Newton methods for generalized equations , 1979 .

[28]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[29]  D. Klatte,et al.  On second-order sufficient optimality conditions for c 1,1-optimization problems , 1988 .

[30]  Liqun Qi,et al.  A nonsmooth version of Newton's method , 1993, Math. Program..

[31]  José Mario Martínez,et al.  On Augmented Lagrangian Methods with General Lower-Level Constraints , 2007, SIAM J. Optim..

[32]  J. Frédéric Bonnans,et al.  Numerical Optimization: Theoretical and Practical Aspects (Universitext) , 2006 .