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https://digitalcommons.wayne.edu/math_reports
Recent documents in Mathematics Research Reportsen-usFri, 08 Dec 2023 01:20:49 PST3600Classical and Motivic Adams-Novikov Charts
https://digitalcommons.wayne.edu/math_reports/95
https://digitalcommons.wayne.edu/math_reports/95Wed, 17 Dec 2014 06:01:35 PST
This document contains large-format Adams-Novikov charts that compute the classical 2-complete stable homotopy groups. The charts are essentially complete through the 59-stem. We believe that these are the most accurate and extensive charts of their kind. We also include a motivic Adams-Novikov E∞ chart.
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Daniel C. IsaksenClassical and Motivic Adams Charts
https://digitalcommons.wayne.edu/math_reports/94
https://digitalcommons.wayne.edu/math_reports/94Wed, 17 Dec 2014 06:01:34 PST
This document contains large-format Adams charts that compute 2-complete stable homotopy groups, both in the classical context and in the motivic context over C. The charts are essentially complete through the 59-stem and contain partial results to the 70-stem. In the classical context, we believe that these are the most accurate charts of their kind. We also include Adams charts for the motivic homotopy groups of the cofiber of τ.
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Daniel C. IsaksenSecond-Order Subdifferential Calculus With Applications to Tilt Stability in Optimization
https://digitalcommons.wayne.edu/math_reports/93
https://digitalcommons.wayne.edu/math_reports/93Tue, 09 Sep 2014 08:12:32 PDT
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 their major specifications are applied then to the study of tilt stability of local minimizers for important classes of problems in constrained optimization that include, in particular, problems of nonlinear programming and certain classes of extended nonlinear programs described in composite terms.
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Boris S. Mordukhovich et al.Sensitivity Analysis For Two-Level Value Functions With Applications to Bilevel Programming
https://digitalcommons.wayne.edu/math_reports/92
https://digitalcommons.wayne.edu/math_reports/92Tue, 09 Sep 2014 08:12:30 PDT
This paper contributes to a deeper understanding of the link between a now conventional framework in hierarchical optimization spread under the name of the optimistic bilevel problem and its initial more difficult formulation that we call here the original optimistic bilevel optimization problem. It follows from this research that, although the process of deriving necessary optimality conditions for the latter problem is more involved, the conditions themselves do not to a large extent differ from those known for the conventional problem. It has been already well recognized in the literature that for optimality conditions of the usual optimistic bilevel program appropriate coderivative constructions for the set-valued solution map of the lower-level problem could be used, while it is shown in this paper that for the original optimistic formulation we have to go a step further to require and justify a certain Lipschitz-like property of this map. This occurs to be related to the local Lipschitz continuity of the optimal value function of an optimization problem constrained by solutions to another optimization problem; this function is labeled here as the two level value function. More generally, we conduct a detailed sensitivity analysis for value functions of mathematical programs with extended complementarity constraints. The results obtained in this vein are applied to the two-level value function and then to the original optimistic formulation of the bilevel optimization problem, for which we derive verifiable stationarity conditions of various types entirely in terms of the initial data.
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S Dempe et al.Variational Analysis of Marginal Functions with Applications to Bilevel Programming
https://digitalcommons.wayne.edu/math_reports/91
https://digitalcommons.wayne.edu/math_reports/91Tue, 09 Sep 2014 08:12:29 PDT
This paper pursues a twofold goal. First to derive new results on generalized differentiation in variational analysis focusing mainly on a broad class of intrinsically nondifferentiable marginal/value functions. Then the results established in this direction apply to deriving necessary optimality conditions for the optimistic version of bilevel programs that occupy a remarkable place in optimization theory and its various applications. We obtain new sets of optimality conditions in both smooth and smooth settings of finite-dimensional and infinite-dimensional spaces.
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Boris S. Mordukhovich et al.Several Approaches for the Derivation of Stationary Conditions for Elliptic MPECs with Upper-Level Control Constraints
https://digitalcommons.wayne.edu/math_reports/90
https://digitalcommons.wayne.edu/math_reports/90Tue, 09 Sep 2014 08:12:27 PDT
The derivation of multiplier-based optimality conditions for elliptic mathematical programs with equilibrium constraints (MPEC) is essential for the characterization of solutions and development of numerical methods. Though much can be said for broad classes of elliptic MPECs in both polyhedric and non-polyhedric settings, the calculation becomes significantly more complicated when additional constraints are imposed on the control. In this paper we develop three derivation methods for constrained MPEC problems: via concepts from variational analysis, via penalization of the control constraints, and via penalization of the lower-level problem with the subsequent regularization of the resulting nonsmoothness. The developed methods and obtained results are then compared and contrasted.
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M Hintermüller et al.Directional Subdifferentials and Optimality Conditions
https://digitalcommons.wayne.edu/math_reports/89
https://digitalcommons.wayne.edu/math_reports/89Tue, 09 Sep 2014 08:12:25 PDT
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.
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Ivan Ginchev et al.Applications of Variational Analysis to a Generalized Heron Problem
https://digitalcommons.wayne.edu/math_reports/88
https://digitalcommons.wayne.edu/math_reports/88Tue, 09 Sep 2014 08:12:24 PDT
This paper is a continuation of our ongoing efforts to solve a number of geometric problems and their extensions by using advanced tools of variational analysis and generalized differentiation. Here we propose and study, from both qualitative and numerical viewpoints, the following optimal location problem as well as its further extensions: on a given nonempty subset of a Banach space, find a point such that the sum of the distances from it to n given nonempty subsets of this space is minimal. This is a generalized version of the classical Heron problem: on a given straight line, find a point C such that the sum of the distances from C to the given points A and B is minimal. We show that the advanced variational techniques allow us to completely solve optimal location problems of this type in some important settings.
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Boris S. Mordukhovich et al.Constraint Qualifications and Optimality Conditions for Nonconvex Semi-Infinite and Infinite Programs
https://digitalcommons.wayne.edu/math_reports/87
https://digitalcommons.wayne.edu/math_reports/87Tue, 09 Sep 2014 08:12:23 PDT
The paper concerns the study of new classes of nonlinear and nonconvex optimization problems of the so-called infinite programming that are generally defined on infinite-dimensional spaces of decision variables and contain infinitely many of equality and inequality constraints with arbitrary (may not be compact) index sets. These problems reduce to semi-infinite programs in the case of finite-dimensional spaces of decision variables. We extend the classical Mangasarian-Fromovitz and Farkas-Minkowski constraint qualifications to such infinite and semi-infinite programs. The new qualification conditions are used for efficient computing the appropriate normal cones to sets of feasible solutions for these programs by employing advanced tools of variational analysis and generalized differentiation. In the further development we derive first-order necessary optimality conditions for infinite and semi-infinite programs, which are new in both finite-dimensional and infinite-dimensional settings.
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Boris S. Mordukhovich et al.Rated Extremal Principles for Finite and Infinite Systems
https://digitalcommons.wayne.edu/math_reports/86
https://digitalcommons.wayne.edu/math_reports/86Tue, 09 Sep 2014 08:12:21 PDT
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.
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Hung M. Phan et al.Quantitative Stability of Linear Infinite Inequality Systems Under Block Perturbations With Applications to Convex Systems
https://digitalcommons.wayne.edu/math_reports/85
https://digitalcommons.wayne.edu/math_reports/85Tue, 09 Sep 2014 08:12:19 PDT
The original motivation for this paper was to provide an efficient quantitative analysis of convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional (resp. finite-dimensional) Banach spaces and that are indexed by an arbitrary fixed set J. Parameter perturbations on the right-hand side of the inequalities are required to be merely bounded, and thus the natural parameter space is loo(J). Our basic strategy consists of linearizing the parameterized convex system via splitting convex inequalities into linear ones by using the Fenchel-Legendre conjugate. This approach yields that arbitrary bounded right-hand side perturbations of the convex system turn on constant-by-blocks perturbations in the linearized system. Based on advanced variational analysis, we derive a precise formula for computing the exact Lipschitzian bound of the feasible solution map of block-perturbed linear systems, which involves only the system's data, and then show that this exact bound agrees with the coderivative norm of the aforementioned mapping. In this way we extend to the convex setting the results of [3] developed for arbitrary perturbations with no block structure in the linear framework under the boundedness assumption on the system's coefficients. The latter boundedness assumption is removed in this paper when the decision space is reflexive. The last section provides the aimed application to the convex case.
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M J. Cánovas et al.Complete Characterizations of Local Weak Sharp Minima With Applications to Semi-Infinite Optimization and Complementarity
https://digitalcommons.wayne.edu/math_reports/84
https://digitalcommons.wayne.edu/math_reports/84Tue, 09 Sep 2014 08:12:17 PDT
In this paper we identify a favorable class of nonsmooth functions for which local weak sharp minima can be completely characterized in terms of normal cones and subdifferentials, or tangent cones and subderivatives, or their mixture in finite-dimensional spaces. The results obtained not only significantly extend previous ones in the literature, but also allow us to provide new types of criteria for local weak sharpness. Applications of the developed theory are given to semi-infinite programming and to semi-infinite complementarity problems.
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Boris S. Mordukhovich et al.Tangential Extremal Principles for Finite and Infinite Systems of Sets, II: Applications to Semi-Infinite and Multiobjective Optimization
https://digitalcommons.wayne.edu/math_reports/83
https://digitalcommons.wayne.edu/math_reports/83Tue, 09 Sep 2014 08:12:16 PDT
This paper contains selected applications of the new tangential extremal principles and related results developed in [20] to calculus rules for infinite intersections of sets and optimality conditions for problems of semi-infinite programming and multiobjective optimization with countable constraints.
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Boris S. Mordukhovich et al.Tangential Extremal Principles for Finite and Infinite Systems of Sets, I: Basic Theory
https://digitalcommons.wayne.edu/math_reports/82
https://digitalcommons.wayne.edu/math_reports/82Tue, 09 Sep 2014 08:12:14 PDT
In this paper we develop new extremal principles in variational analysis that deal with finite and infinite systems of convex and nonconvex sets. The results obtained, unified under the name of tangential extremal principles, combine primal and dual approaches to the study of variational systems being in fact first extremal principles applied to infinite systems of sets. The first part of the paper concerns the basic theory of tangential extremal principles while the second part presents applications to problems of semi-infinite programming and multiobjective optimization.
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Boris S. Mordukhovich et al.Quantitative Stability and Optimality Conditions in Convex Semi-Infinite and Infinite Programming
https://digitalcommons.wayne.edu/math_reports/81
https://digitalcommons.wayne.edu/math_reports/81Tue, 09 Sep 2014 08:12:12 PDT
This paper concerns parameterized convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional Banach (resp. finite-dimensional) spaces and that are indexed by an arbitrary fixed set T. Parameter perturbations on the right-hand side of the inequalities are measurable and bounded, and thus the natural parameter space is loo(T). Based on advanced variational analysis, we derive a precise formula for computing the exact Lipschitzian bound of the feasible solution map, which involves only the system data, and then show that this exact bound agrees with the coderivative norm of the aforementioned mapping. On one hand, in this way we extend to the convex setting the results of [4) developed in the linear framework under the boundedness assumption on the system coefficients. On the other hand, in the case when the decision space is reflexive, we succeed to remove this boundedness assumption in the general convex case, establishing therefore results new even for linear infinite and semi-infinite systems. The last part of the paper provides verifiable necessary optimality conditions for infinite and semi-infinite programs with convex inequality constraints and general nonsmooth and nonconvex objectives. In this way we extend the corresponding results of [5) obtained for programs with linear infinite inequality constraints.
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M J. Cánovas et al.Solving a Generalized Heron Problem by Means of Convex Analysis
https://digitalcommons.wayne.edu/math_reports/80
https://digitalcommons.wayne.edu/math_reports/80Tue, 09 Sep 2014 08:12:10 PDT
The classical Heron problem states: on a given straight line in the plane, find a point C such that the sum of the distances from C to the given points A and B is minimal. This problem can be solved using standard geometry or differential calculus. In the light of modern convex analysis, we are able to investigate more general versions of this problem. In this paper we propose and solve the following problem: on a given nonempty closed convex subset of IR!, find a point such that the sum of the distances from that point to n given nonempty closed convex subsets of JR• is minimal.
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Boris S. Mordukhovich et al.Lipchitzian Stability of Parametric Variational Inequalities Over Generalized Polyhedra in Banach Spaces
https://digitalcommons.wayne.edu/math_reports/79
https://digitalcommons.wayne.edu/math_reports/79Tue, 09 Sep 2014 08:12:09 PDT
This paper concerns the study of solution maps to parameterized variational inequalities over generalized polyhedra in reflexive Banach spaces. It has been recognized that generalized polyhedral sets are significantly different from the usual convex polyhedra in infinite dimensions and play an important role in various applications to optimization, particularly to generalized linear programming. Our main goal is to fully characterize robust Lipschitzian stability of the aforementioned solutions maps entirely via their initial data. This is done on the base of the coderivative criterion in variational analysis via efficient calculations of the coderivative and related objects for the systems under consideration. The case of generalized polyhedra is essentially more involved in comparison with usual convex polyhedral sets and requires developing elaborated techniques and new proofs of variational analysis.
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Liqun Ban et al.Generalized Newton's Method Based on Graphical Derivatives
https://digitalcommons.wayne.edu/math_reports/78
https://digitalcommons.wayne.edu/math_reports/78Tue, 09 Sep 2014 08:12:07 PDT
This paper concerns developing a numerical method of the Newton type to solve systems of nonlinear equations described by nonsmooth continuous functions. We propose and justify a new generalized Newton algorithm based on graphical derivatives, which have never been used to derive a Newton-type method for solving nonsmooth equations. Based on advanced techniques of variational analysis and generalized differentiation, we establish the well-posedness of the algorithm, its local superlinear convergence, and its global convergence of the Kantorovich type. Our convergence results hold with no semismoothness assumption, which is illustrated by examples. The algorithm and main results obtained in the paper are compared with well-recognized semismooth and B-differentiable versions of Newton's method for nonsmooth Lipschitzian equations.
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T Hoheisel et al.Extended Second Welfare Theorem for Nonconvex Economies with Infinite Commodities and Public Goods
https://digitalcommons.wayne.edu/math_reports/77
https://digitalcommons.wayne.edu/math_reports/77Tue, 09 Sep 2014 08:12:05 PDT
This paper is devoted to the study of nonconvex models of welfare economics with public goods and infinite-dimensional commodity spaces. Our main attention is paid to new extensions of the fundamental second welfare theorem to the models under consideration. Based on advanced tools of variational analysis and generalized differentiation, we establish appropriate approximate and exact versions of the extended second welfare theorem for Pareto, weak Pareto, and strong Pareto optimal allocations in both marginal price and decentralized price forms.
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Aychiluhim Habte et al.On Directionally Dependent Subdifferentials
https://digitalcommons.wayne.edu/math_reports/76
https://digitalcommons.wayne.edu/math_reports/76Tue, 09 Sep 2014 08:12:04 PDT
In this paper directionally contextual concepts of variational analysis, based on dual-space constructions similar to those in [4, 5], are introduced and studied. As an illustration of their usefulness, necessary and also sufficient optimality conditions in terms of directioual subdifferentials are established, and it is shown that they can be effective in the situations where known optimality conditions in terms of nondirectional subdifferentials fail.
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Ivan Ginchev et al.