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Primal superlinear convergence of SQP methods in piecewise linear-quadratic composite optimization
TL;DR: In this article, the primal superlinear convergence of the quasi-Newton sequential quadratic programming (SQP) method for piecewise linear-quadratic composite optimization problems is analyzed.
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.
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11 May 2000
TL;DR: It is shown here how the model derived recently in [Bouchut-Boyaval, M3AS (23) 2013] can be modified for flows on rugous topographies varying around an inclined plane.
Abstract: Basic notation.- Introduction.- Background material.- Optimality conditions.- Basic perturbation theory.- Second order analysis of the optimal value and optimal solutions.- Optimal Control.- References.
2,067 citations
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975 citations
01 Jan 1981
TL;DR: A multifunction is polyhedral if its graph is the union of finitely many polyhedral convex sets as discussed by the authors, and it is shown how these properties can be applied to such areas as linear complementarity and parametric programming.
Abstract: A multifunction is polyhedral if its graph is the union of finitely many polyhedral convex sets. This paper points out some fairly strong continuity properties that such multifunctions satisfy, and it shows how these may be applied to such areas as linear complementarity and parametric programming.
531 citations
Book•
05 Jun 2009
TL;DR: In this paper, the authors define implicit functions defined implicitly by equations, and derive regularity properties of set-valued solution mappings through generalized derivatives, and apply them in Numerical Variational Analysis.
Abstract: Functions Defined Implicitly by Equations.- Implicit Function Theorems for Variational Problems.- Regularity properties of set-valued solution mappings.- Regularity Properties Through Generalized Derivatives.- Regularity in infinite dimensions.- Applications in Numerical Variational Analysis.
413 citations