Criticality of Lagrange multipliers in extended nonlinear optimization

Abstract: The paper is devoted to a comprehensive study of composite models in variational analysis and optimization the importance of which for numerous theoretical, algorithmic, and applied issues of operations research is difficult to overstate. The underlying theme of our study is a systematical replacement of conventional metric regularity and related requirements by much weaker metric subregulatity ones that lead us to significantly stronger and completely new results of first-order and second-order variational analysis and optimization. In this way we develop extended calculus rules for first-order and second-order generalized differential constructions with paying the main attention in second-order variational theory to the new and rather large class of fully subamenable compositions. Applications to optimization include deriving enhanced no-gap second order optimality conditions in constrained composite models, complete characterizations of the uniqueness of Lagrange multipliers and strong metric subregularity of KKT systems in parametric optimization, etc.

Abstract: This paper mainly concerns with the primal superlinear convergence of the quasi-Newton sequential quadratic programming (SQP) method for piecewise linear-quadratic composite optimization problems We show that the latter primal superlinear convergence can be justified under the noncriticality of Lagrange multipliers and a version of the Dennis-More condition Furthermore, we show that if we replace the noncriticality condition with the second-order sufficient condition, this primal superlinear convergence is equivalent with an appropriate version of the Dennis-More condition We also recover Bonnans’ result in (Appl Math Optim 29, 161–186, 1994) for the primal-dual superlinear of the basic SQP method for this class of composite problems under the second-order sufficient condition and the uniqueness of Lagrange multipliers To achieve these goals, we first obtain an extension of the reduction lemma for convex Piecewise linear-quadratic functions and then provide a comprehensive analysis of the noncriticality of Lagrange multipliers for composite problems We also establish certain primal estimates for KKT systems of composite problems, which play a significant role in our local convergence analysis of the quasi-Newton SQP method

Abstract: It is a well-known phenomenon that the presence of critical Lagrange multipliers in constrained optimization problems may cause a deterioration of the convergence speed of primal-dual Newton-type m...

Abstract: This paper mainly concerns with the primal superlinear convergence of the quasi-Newton sequential quadratic programming (SQP) method for piecewise linear-quadratic composite optimization problems. We show that the latter primal superlinear convergence can be justified under the noncriticality of Lagrange multipliers and a version of the Dennis-More condition. Furthermore, we show that if we replace the noncriticality condition with the second-order sufficient condition, this primal superlinear convergence is equivalent with an appropriate version of the Dennis-More condition. We also recover Bonnans' result in [1] for the primal-dual superlinear of the basic SQP method for this class of composite problems under the second-order sufficient condition and the uniqueness of Lagrange multipliers. To achieve these goals, we first obtain an extension of the reduction lemma for convex Piecewise linear-quadratic functions and then provide a comprehensive analysis of the noncriticality of Lagrange multipliers for composite problems. We also establish certain primal estimates for KKT systems of composite problems, which play a significant role in our local convergence analysis of the quasi-Newton SQP method.

Abstract: In this paper, we examine the properly twice epi-differentiability and compute the second order epi-subderivative of the indicator function to a class of sets including the finite union of parabolically derivable and parabolically regular sets. In this way, we provide no-gap second order optimality conditions for a disjunctive constrained problem. Moreover, we derive applications of our results to some types of disjunctive programs.

Abstract: We consider some basic properties of nonsmooth and set-valued mappings (multifunctions) connected with open and inverse mapping principles, distance estimates to the level sets (metric regularity), and a locally Lipschitzian behavior. These properties have many important applications to various problems in nonlinear analysis, optimization, control theory, etc., especially for studying sensitivity and stability questions with respect to perturbations of initial data and parameters. We establish interrelations between these properties and prove effective criteria for their fulfillment stated in terms of robust generalized derivatives for multifunctions and nonsmooth mappings

Abstract: Necessary and sufficient conditions are obtained for the Lipschitzian stability of local solutions to finite-dimensional parameterized optimization problems in a very general setting. Properties of prox-regularity of the essential objective function and positive definiteness of its coderivative Hessian are the keys to these results. A previous characterization of tilt stability arises as a special case.

Abstract: This paper is mainly devoted to the study of the so-called full Lipschitzian stability of local solutions to finite-dimensional parameterized problems of constrained optimization, which has been well recognized as a very important property from the viewpoints of both optimization theory and its applications. Based on second-order generalized differential tools of variational anal- ysis, we obtain necessary and sufficient conditions for fully stable local minimizers in general classes of constrained optimization problems, including problems of composite optimization, mathemati- cal programs with polyhedral constraints, as well as problems of extended and classical nonlinear programming with twice continuously differentiable data.

Abstract: This paper is devoted to studying the robust isolated calmness of the Karush--Kuhn--Tucker (KKT) solution mapping for a large class of interesting conic programming problems (including most commonly known ones arising from applications) at a locally optimal solution. Under the Robinson constraint qualification, we show that the KKT solution mapping is robustly isolated calm if and only if both the strict Robinson constraint qualification and the second order sufficient condition hold. This implies, among others, that at a locally optimal solution the second order sufficient condition is needed for the KKT solution mapping to have the Aubin property.