248 research outputs found

    Nonsmooth Analysis

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    This survey of nonsmooth analysis sets out to prove an inverse function theorem for set-valued maps. The inverse function theorem for the more usual smooth maps plays a very important role in the solution of many problems in pure and applied analysis, and we can expect such an adaptation of this theorem also to be of great value. For example, it can be used to solve convex minimization problems and to prove the Lipschitz behavior of its solutions when the natural parameters vary--a very important problem in marginal theory in economics

    On Inverse Function Theorems for Set-Valued Maps

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    We prove several equivalent versions of the inverse function theorem: an inverse function theorem for smooth maps on closed subsets, one for set-valued maps, a generalized implicit function theorem for set-valued maps. We provide applications to optimization theory and local controllability of differential inclusions

    Generalized differentiation with positively homogeneous maps: Applications in set-valued analysis and metric regularity

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    We propose a new concept of generalized differentiation of set-valued maps that captures the first order information. This concept encompasses the standard notions of Frechet differentiability, strict differentiability, calmness and Lipschitz continuity in single-valued maps, and the Aubin property and Lipschitz continuity in set-valued maps. We present calculus rules, sharpen the relationship between the Aubin property and coderivatives, and study how metric regularity and open covering can be refined to have a directional property similar to our concept of generalized differentiation. Finally, we discuss the relationship between the robust form of generalization differentiation and its one sided counterpart.Comment: This submission corrects errors from the previous version after referees' comments. Changes are in Proposition 2.4, Proposition 4.12, and Sections 7 and

    Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems

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    The present article presents a summarizing view at differential-algebraic equations (DAEs) and analyzes how new application fields and corresponding mathematical models lead to innovations both in theory and in numerical analysis for this problem class. Recent numerical methods for nonsmooth dynamical systems subject to unilateral contact and friction illustrate the topicality of this development.Comment: Preprint of Book Chapte

    Controllability and Observability of Control Systems under Uncertainty

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    This report surveys the results of nonlinear systems theory (controllability and observability) obtained at IIASA during the last three summers. Classical methods based on differential geometry require some regularity and fail as soon as state-dependent constraints are brought to bear on the controls, or uncertainty and disturbances are involved in the system. Since these important features appear in most realistic control problems, new methods had to be devised, which encompass the classical ones, and allow the presence of a priori feedback into the control systems. This is now possible thanks to new tools, in the development of which IIASA played an important role: differential inclusions and set-valued analysis

    First order optimality condition for constrained set-valued optimization

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    A constrained optimization problem with set-valued data is considered. Different kind of solutions are defined for such a problem. We recall weak minimizer, efficient minimizer and proper minimizer. The latter are defined in a way that embrace also the case when the ordering cone is not pointed. Moreover we present the new concept of isolated minimizer for set-valued optimization. These notions are investigated and appear when establishing first-order necessary and sufficient optimality conditions derived in terms of a Dini type derivative for set-valued maps. The case of convex (along rays) data is considered when studying sufficient optimality conditions for weak minimizers. Key words: Vector optimization, Set-valued optimization, First-order optimality conditions.

    Differential Calculus of Set-Valued Maps. An Update

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    IIASA has played a crucial role in the development of the "graphical approach" to the differential calculus of set-valued maps, around J.-P. Aubin, H. Frankowska, R.T. Rockafellar and allowed to make contacts with Soviet and eastern European mathematicians (C. Olech, B. Pschenichnyiy, E. Polovinkin, V. Tihomirov, etc.) who were following analogous approaches. Since 1981, they and their collaborators developed this calculus and applied it to a variety of problems, in mathematical programming (Kuhn-Tucker rules, sensitivity of solutions and Lagrange multipliers), in nonsmooth analysis (Inverse Functions Theorems, local uniqueness), in control theory (controllability of systems with feedbacks, Pontryagin's Maximum Principle, Hamilton-Jacobi-Bellman equations, observability and other issues), in viability theory (regulation of systems, heavy trajectories), etc. The first version of this survey appeared at IIASA in 1982, and constituted the seventh chapter of the book "Applied Nonlinear Analysis" published in 1984 by I. Ekeland and the author. Since then, many other results have been motivated by the successful applications of this calculus, and, maybe unfortunately, other concepts (such as the concept of intermediate tangent cone and derivatives introduced and used by H. Frankowska). Infinite-dimensional problems such as control problems or the more classical problems of calculus of variations require the use of adequate adaptations of the same main idea, as well as more technical assumptions. The time and the place (IIASA) were ripe to update the exposition of this differential calculus. The Russian translation of "Applied Nonlinear Analysis" triggered this revised version
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