2,570 research outputs found

    Mathematical programs with complementarity constraints: convergence properties of a smoothing method

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    In this paper, optimization problems PP with complementarity constraints are considered. Characterizations for local minimizers xˉ\bar{x} of PP of Orders 1 and 2 are presented. We analyze a parametric smoothing approach for solving these programs in which PP is replaced by a perturbed problem PτP_{\tau} depending on a (small) parameter τ\tau. We are interested in the convergence behavior of the feasible set Fτ\cal{F}_{\tau} and the convergence of the solutions xˉτ\bar{x}_{\tau} of PτP_{\tau} for τ0.\tau\to 0. In particular, it is shown that, under generic assumptions, the solutions xˉτ\bar{x}_{\tau} are unique and converge to a solution xˉ\bar{x} of PP with a rate O(τ)\cal{O}(\sqrt{\tau}). Moreover, the convergence for the Hausdorff distance d(Fτd(\cal{F}_{\tau}, F)\cal{F}) between the feasible sets of PτP_{\tau} and PP is of order O(τ)\cal{O}(\sqrt{\tau})

    Entropic Regularization Approach for Mathematical Programs with Equilibrium Constraints

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    A new smoothing approach based on entropic perturbationis proposed for solving mathematical programs withequilibrium constraints. Some of the desirableproperties of the smoothing function are shown. Theviability of the proposed approach is supported by acomputationalstudy on a set of well-known test problems.mathematical programs with equilibrium constraints;entropic regularization;smoothing approach

    Entropic regularization approach for mathematical programs with equilibrium constraints

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    A new smoothing approach based on entropic perturbation is proposed for solving mathematical programs with equilibrium constraints. Some of the desirable properties of the smoothing function are shown. The viability of the proposed approach is supported by a computational study on a set of well-known test problems.Entropic regularization;Smoothing approach;Mathematical programs with equilibrium constraints

    Recursive Approximation of the High Dimensional max Function

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    An alternative smoothing method for the high dimensional max functionhas been studied. The proposed method is a recursive extension of thetwo dimensional smoothing functions. In order to analyze the proposedmethod, a theoretical framework related to smoothing methods has beendiscussed. Moreover, we support our discussion by considering someapplication areas. This is followed by a comparison with analternative well-known smoothing method.n dimensional max function;recursive approximation;smoothing methods;vertical linear complementarity (VLCP)

    Optimization viewpoint on Kalman smoothing, with applications to robust and sparse estimation

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    In this paper, we present the optimization formulation of the Kalman filtering and smoothing problems, and use this perspective to develop a variety of extensions and applications. We first formulate classic Kalman smoothing as a least squares problem, highlight special structure, and show that the classic filtering and smoothing algorithms are equivalent to a particular algorithm for solving this problem. Once this equivalence is established, we present extensions of Kalman smoothing to systems with nonlinear process and measurement models, systems with linear and nonlinear inequality constraints, systems with outliers in the measurements or sudden changes in the state, and systems where the sparsity of the state sequence must be accounted for. All extensions preserve the computational efficiency of the classic algorithms, and most of the extensions are illustrated with numerical examples, which are part of an open source Kalman smoothing Matlab/Octave package.Comment: 46 pages, 11 figure

    An interior-point method for mpecs based on strictly feasible relaxations.

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    An interior-point method for solving mathematical programs with equilibrium constraints (MPECs) is proposed. At each iteration of the algorithm, a single primaldual step is computed from each subproblem of a sequence. Each subproblem is defined as a relaxation of the MPEC with a nonempty strictly feasible region. In contrast to previous approaches, the proposed relaxation scheme preserves the nonempty strict feasibility of each subproblem even in the limit. Local and superlinear convergence of the algorithm is proved even with a less restrictive strict complementarity condition than the standard one. Moreover, mechanisms for inducing global convergence in practice are proposed. Numerical results on the MacMPEC test problem set demonstrate the fast-local convergence properties of the algorithm

    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
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