Results 1 
7 of
7
A NOTE ON UPPER LIPSCHITZ STABILITY, ERROR BOUNDS, AND CRITICAL MULTIPLIERS FOR LIPSCHITZCONTINUOUS KKT SYSTEMS
, 2012
"... We prove a new local upper Lipschitz stability result and the associated local error bound for solutions of parametric Karush–Kuhn–Tucker systems corresponding to variational problems with Lipschitzian base mappings and constraints possessing Lipschitzian derivatives, and without any constraint qual ..."
Abstract

Cited by 8 (5 self)
 Add to MetaCart
(Show Context)
We prove a new local upper Lipschitz stability result and the associated local error bound for solutions of parametric Karush–Kuhn–Tucker systems corresponding to variational problems with Lipschitzian base mappings and constraints possessing Lipschitzian derivatives, and without any constraint qualifications. This property is equivalent to the appropriately extended to this nonsmooth setting notion of noncriticality of the Lagrange multiplier associated to the primal solution, which is weaker than secondorder sufficiency. All this extends several results previously known only for optimization problems with twice differentiable data, or assuming some constraint qualifications. In addition, our results are obtained in the more general variational setting.
THE JOSEPHY–NEWTON METHOD FOR SEMISMOOTH GENERALIZED EQUATIONS AND SEMISMOOTH SQP FOR OPTIMIZATION
, 2011
"... While generalized equations with differentiable singlevalued 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 ..."
Abstract

Cited by 4 (4 self)
 Add to MetaCart
(Show Context)
While generalized equations with differentiable singlevalued 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 KarushKuhnTucker 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 primaldual local convergence properties of semismooth SQP, but also obtain some new results that complete the analysis of the SQP primal rate of convergence, including its quasiNewton variant. Key words: generalized equation, Bdifferential, generalized Jacobian, BDregularity, CDregularity,
SEMISMOOTH SQP METHOD FOR EQUALITYCONSTRAINED OPTIMIZATION PROBLEMS WITH AN APPLICATION TO THE LIFTED REFORMULATION OF MATHEMATICAL PROGRAMS WITH COMPLEMENTARITY CONSTRAINTS
, 2010
"... We consider the sequential quadratic programming algorithm (SQP) applied to equalityconstrained optimization problems, where the problem data is differentiable with Lipschitzcontinuous first derivatives. For this setting, DennisMoré type analysis of primal superlinear convergence is presented. Our ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
We consider the sequential quadratic programming algorithm (SQP) applied to equalityconstrained optimization problems, where the problem data is differentiable with Lipschitzcontinuous first derivatives. For this setting, DennisMoré type analysis of primal superlinear convergence is presented. Our main motivation is a special modification of SQP tailored to the structure of the lifted reformulation of mathematical programs with complementarity constraints (MPCC). For this problem, we propose a special positive definite modification of the matrices in the generalized Hessian, which is suitable for globalization of SQP based on the penalty function, and at the same time can be expected to satisfy our general DennisMoré type conditions, thus preserving local superlinear convergence. (Standard quasiNewton updates in the SQP framework require twice differentiability of the problem data at the solution for superlinear convergence.) Preliminary numerical results comparing a number of quasiNewton versions of semismooth SQP applied to MPCC are also reported. Key words: sequential quadratic programming, semismoothness, Bdifferential, BDregularity, semismooth Newton method, secondorder sufficiency, mathematical programs with complementarity constraints.
MULTIPLIER METHODS FOR OPTIMIZATION PROBLEMS WITH LIPSCHITZIAN DERIVATIVES
"... ABSTRACT In this study we obtain sharp local convergence results for the augmented Lagrangian method and for the linearly constrained Lagrangian method. Both of these two methods gave rise to successful software. The augmented Lagrangian method is used by LANCELOT [7], ALGENCAN Specifically, we ex ..."
Abstract
 Add to MetaCart
(Show Context)
ABSTRACT In this study we obtain sharp local convergence results for the augmented Lagrangian method and for the linearly constrained Lagrangian method. Both of these two methods gave rise to successful software. The augmented Lagrangian method is used by LANCELOT [7], ALGENCAN Specifically, we extend the sharpest known theorems about local convergence of the two methods Concerning the linearly constrained Lagrangian method, we also improve the result from [6] with respect to the rate of convergence, replacing the superlinear convergence rate estimate by the quadratic one. In our analysis, the augmented Lagrangian method and the linearly constrained Lagrangian method are viewed as special cases of an abstract Newtonian iterative framework that we develop and study first. We believe that this framework may serve as a convenient tool for local convergence analysis of many other algorithms as well.
A Smooth Newton Method for Nonlinear Programming Problems with Inequality Constraints
"... The paper presents a reformulation of the KarushKuhnTucker (KKT) system associated nonlinear programming problem into an equivalent system of smooth equations. Classical Newton method is applied to solve the system of equations. The superlinear convergence of the primal sequence, generated by prop ..."
Abstract
 Add to MetaCart
(Show Context)
The paper presents a reformulation of the KarushKuhnTucker (KKT) system associated nonlinear programming problem into an equivalent system of smooth equations. Classical Newton method is applied to solve the system of equations. The superlinear convergence of the primal sequence, generated by proposed method, is proved. The preliminary numerical results with a problems test set are presented.
PRONEX–Optimization, and by FAPERJ.
, 2013
"... Local convergence of the method of multipliers for variational and optimization problems under the noncriticality assumption ..."
Abstract
 Add to MetaCart
(Show Context)
Local convergence of the method of multipliers for variational and optimization problems under the noncriticality assumption