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Articles 1 - 6 of 6
Full-Text Articles in Physical Sciences and Mathematics
Second-Order Subdifferential Calculus With Applications To Tilt Stability In Optimization, Boris S. Mordukhovich, R T. Rockafellar
Second-Order Subdifferential Calculus With Applications To Tilt Stability In Optimization, Boris S. Mordukhovich, R T. Rockafellar
Mathematics Research Reports
The paper concerns the second-order generalized differentiation theory of variational analysis and new applications of this theory to some problems of constrained optimization in finitedimensional spaces. The main attention is paid to the so-called (full and partial) second-order subdifferentials of extended-real-valued functions, which are dual-type constructions generated by coderivatives of first-order sub differential mappings. We develop an extended second-order subdifferential calculus and analyze the basic second-order qualification condition ensuring the fulfillment of the principal secondorder chain rule for strongly and fully amenable compositions. The calculus results obtained in this way and computing the second-order subdifferentials for piecewise linear-quadratic functions and …
Directional Subdifferentials And Optimality Conditions, Ivan Ginchev, Boris S. Mordukhovich
Directional Subdifferentials And Optimality Conditions, Ivan Ginchev, Boris S. Mordukhovich
Mathematics Research Reports
This paper is devoted to the introduction and development of new dual-space constructions of generalized differentiation in variational analysis, which combine certain features of subdifferentials for nonsmooth functions (resp. normal cones to sets) and directional derivatives (resp. tangents). We derive some basic properties of these constructions and apply them to optimality conditions in problems of unconstrained and constrained optimization.
Rated Extremal Principles For Finite And Infinite Systems, Hung M. Phan, Boris S. Mordukhovich
Rated Extremal Principles For Finite And Infinite Systems, Hung M. Phan, Boris S. Mordukhovich
Mathematics Research Reports
In this paper we introduce new notions of local extremality for finite and infinite systems of closed sets and establish the corresponding extremal principles for them called here rated extremal principles. These developments are in the core geometric theory of variational analysis. We present their applications to calculus and optimality conditions for problems with infinitely many constraints.
Metric Regularity Of Mappings And Generalized Normals To Set Images, Boris S. Mordukhovich, Nguyen Mau Nam, Bingwu Wang
Metric Regularity Of Mappings And Generalized Normals To Set Images, Boris S. Mordukhovich, Nguyen Mau Nam, Bingwu Wang
Mathematics Research Reports
The primary goal of this paper is to study some notions of normals to nonconvex sets in finite-dimensional and infinite-dimensional spaces and their images under single-valued and set-valued mappings. The main motivation for our study comes from variational analysis and optimization, where the problems under consideration play a crucial role in many important aspects of generalized differential calculus and applications. Our major results provide precise equality formulas (sometimes just efficient upper estimates) allowing us to compute generalized normals in various senses to direct and inverse images of nonconvex sets under single-valued and set-valued mappings between Banach spaces. The main tools …
Generalized Differentiation Of Parameter-Dependent Sets And Mappings, Boris S. Mordukhovich, Bingwu Wang
Generalized Differentiation Of Parameter-Dependent Sets And Mappings, Boris S. Mordukhovich, Bingwu Wang
Mathematics Research Reports
The paper concerns new aspects of generalized differentiation theory that plays a crucial role in many areas of modern variational analysis, optimization, and their applications. In contrast to the majority of previous developments, we focus here on generalized differentiation of parameter-dependent objects (sets, set-valued mappings, and nonsmooth functions), which naturally appear, e.g., in parametric optimization and related topics. The basic generalized differential constructions needed in this case are different for those known in parameter-independent settings, while they still enjoy comprehensive calculus rules developed in this paper.
Optimal Control Of Nonconvex Differential Inclusions, Boris S. Mordukhovich
Optimal Control Of Nonconvex Differential Inclusions, Boris S. Mordukhovich
Mathematics Research Reports
The paper concerns new aspects of generalized differentiation theory that plays a crucial role in many areas of modern variational analysis, optimization, and their applications. In contrast to the majority of previous developments, we focus here on generalized differentiation of parameter-dependent objects (sets, set-valued mappings, and nonsmooth functions), which naturally appear, e.g., in parametric optimization and related topics. The basic generalized differential constructions needed in this case are different for those known in parameter-independent settings, while they still enjoy comprehensive calculus rules developed in this paper.