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Variational Analysis of Composite Models with Applications to Continuous Optimization
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In this article, a comprehensive study of composite models in variational analysis and optimization is presented, with the main attention paid to the new and rather large class of fully subamenable compositions, and the underlying theme of the study is a systematical replacement of conventional metric regularity and related requirements by much weaker metric subregulatity ones.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.read more
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Parabolic Regularity in Geometric Variational Analysis
Abstract: The paper is mainly devoted to systematic developments and applications of geometric aspects of second-order variational analysis that are revolved around the concept of parabolic regularity of sets. This concept has been known in variational analysis for more than two decades while being largely underinvestigated. We discover here that parabolic regularity is the key to derive new calculus rules and computation formulas for major second-order generalized differential constructions of variational analysis in connection with some properties of sets that go back to classical differential geometry and geometric measure theory. The established results of second-order variational analysis and generalized differentiation, being married to the developed calculus of parabolic regularity, allow us to obtain novel applications to both qualitative and quantitative/numerical aspects of constrained optimization including second-order optimality conditions, augmented Lagrangians, etc. under weak constraint qualifications.
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Generalized Newton Algorithms for Tilt-Stable Minimizers in Nonsmooth Optimization
TL;DR: Two versions of the generalized Newton method are developed to compute not merely arbitrary local minimizers of nonsmooth optimization problems but just those, which possess an important stability property known as tilt stability, which are based on graphical derivatives of the latter.
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A Generalized Newton Method for Subgradient Systems
TL;DR: In this article, a Newton-type algorithm is proposed to solve subdifferential inclusions defined by subgradients of extended-real-valued prox-regular functions, which can be efficiently computed for broad classes of extended real-valued functions.
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Augmented Lagrangian Method for Second-Order Cone Programs under Second-Order Sufficiency
TL;DR: This paper addresses problems of second-order cone programming important in optimization theory and applications by formulate the corresponding version ofsecond-order sufficiency and use it to establish the uniform second- order growth condition for the augmented Lagrangian.
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Local Convergence Analysis of Augmented Lagrangian Methods for Piecewise Linear-Quadratic Composite Optimization Problems
TL;DR: In this paper, the second-order sufficient condition for local optimality has been shown to justify linear convergence of the primal-dual sequence generated by the augmented Lagrangian method for piecewise linear-quadratic composite optimization problems.
References
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Optimality Conditions for Disjunctive Programs Based on Generalized Differentiation with Application to Mathematical Programs with Equilibrium Constraints
TL;DR: Based on the concepts of directional subregularity and their characterization by means of objects from generalized differentiation, the new stationarity concept of extended M- stationarity is obtained, which turns out to be an equivalent dual characterization of B-stationarity.
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Generalized second derivatives of convex functions and saddle functions
TL;DR: In this article, the second-order epi-derivatives of extended-real-valued functions are applied to convex functions on Rin and shown to be closely tied to proto-differentiation of the corresponding subgradient multifunctions.
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Second-order growth, tilt stability, and metric regularity of the subdifferential
TL;DR: In this paper, the authors established new relationships between second-order growth conditions on functions, the basic properties of metric regularity and subregularity of the limiting subdifferential, tilt-stability of local minimizers, and positive-definiteness/semidefiniteness properties of the second order Hessian.
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On Computation of Generalized Derivatives of the Normal-Cone Mapping and their Applications
Helmut Gfrerer,Jiří V. Outrata +1 more
TL;DR: A characterization of the isolated calmness property of the mentioned solution map is obtained and strong stationarity conditions for an MPEC with control constraints are derived.
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Quantitative convergence analysis of iterated expansive, set-valued mappings
TL;DR: In this paper, the authors develop a framework for quantitative convergence analysis of Picard iterations of expansive set-valued fixed point mappings, and prove local linear convergence of nonconvex cyclic projections for inconsistent (and consistent) feasibility problems.