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598
Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems
 Mathematical Programming
, 1992
"... Abstract Whether or not the general asymmetric variational inequality problem can be formulated as a differentiable optimization problem has been an open question. This paper gives an affirmative answer to this question. We provide a new optimization problem formulation of the variational inequalit ..."
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Cited by 121 (12 self)
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Abstract Whether or not the general asymmetric variational inequality problem can be formulated as a differentiable optimization problem has been an open question. This paper gives an affirmative answer to this question. We provide a new optimization problem formulation of the variational inequality problem and show that its objective function is continuously differentiable whenever the mapping involved in the latter problem is continuously differentiable. We also show that under appropriate assumptions on the latter mapping, any stationary point of the optimization problem is a global optimal solution, and hence solves the variational inequality problem. We discuss descent methods for solving the equivalent optimization problem and comment on systems of nonlinear equations and nonlinear complementarity problems.
Error bounds for monotone approximation schemes for HamiltonJacobiBellman equations
, 2007
"... We obtain nonsymmetric upper and lower bounds on the rate of convergence of general monotone approximation/numerical schemes for parabolic HamiltonJacobiBellman equations by introducing a new notion of consistency. Our results are robust and general – they improve and extend earlier results by Kry ..."
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Cited by 83 (10 self)
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We obtain nonsymmetric upper and lower bounds on the rate of convergence of general monotone approximation/numerical schemes for parabolic HamiltonJacobiBellman equations by introducing a new notion of consistency. Our results are robust and general – they improve and extend earlier results by Krylov, Barles, and Jakobsen. We apply our general results to various schemes including Crank–Nicholson type finite difference schemes, splitting methods, and the classical approximation by piecewise constant controls. In the first two cases our results are new, and in the last two cases the results are obtained by a new method which we develop here.
Image Sequence Analysis via Partial Differential Equations
, 1999
"... This article deals with the problem of restoring and motion segmenting noisy image sequences with a static background. Usually, motion segmentation and image restoration are considered separately in image sequence restoration. Moreover, motion segmentation is often noise sensitive. In this article, ..."
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Cited by 53 (3 self)
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This article deals with the problem of restoring and motion segmenting noisy image sequences with a static background. Usually, motion segmentation and image restoration are considered separately in image sequence restoration. Moreover, motion segmentation is often noise sensitive. In this article, the motion segmentation and the image restoration parts are performed in a coupled way, allowing the motion segmentation part to positively influence the restoration part and viceversa. This is the key of our approach that allows to deal simultaneously with the problem of restoration and motion segmentation. To this end, we propose a theoretically justified optimization problem that permits to take into account both requirements. The model is theoretically justified. Existence and unicity are proved in the space of bounded variations. A suitable numerical scheme based on half quadratic minimization is then proposed and its convergence and stability demonstrated. Experimental results obtaine...
A Supply Chain Network Equilibrium Model with . . .
, 2002
"... In this paper, we develop a supply chain network model consisting of manufacturers and retailers in which the demands associated with the retail outlets are random. We model the optimizing behavior of the various decisionmakers, derive the equilibrium conditions, and establish the finitedimensiona ..."
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Cited by 52 (30 self)
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In this paper, we develop a supply chain network model consisting of manufacturers and retailers in which the demands associated with the retail outlets are random. We model the optimizing behavior of the various decisionmakers, derive the equilibrium conditions, and establish the finitedimensional variational inequality formulation. We provide qualitative properties of the equilibrium pattern in terms of existence and uniqueness results and also establish conditions under which the proposed computational procedure is guaranteed to converge. Finally, we illustrate the model through several numerical examples for which the equilibrium prices and product shipments are computed. This is the first supply chain network equilibrium model with random demands for which modeling, qualitative analysis, and computational results have been obtained.
Monotone Multigrid Methods for Elliptic Variational Inequalities I
 I. Numer. Math
, 1993
"... . We derive fast solvers for discrete elliptic variational inequalities of the first kind (obstacle problems) as resulting from the approximation of related continuous problems by piecewise linear finite elements. Using basic ideas of successive subspace correction, we modify wellknown relaxation ..."
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Cited by 51 (13 self)
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. We derive fast solvers for discrete elliptic variational inequalities of the first kind (obstacle problems) as resulting from the approximation of related continuous problems by piecewise linear finite elements. Using basic ideas of successive subspace correction, we modify wellknown relaxation methods by extending the set of search directions. Extended underrelaxations are called monotone multigrid methods, if they are quasioptimal in a certain sense. By construction, all monotone multigrid methods are globally convergent. We take a closer look at two natural variants, the standard monotone multigrid method and a truncated version. For the considered model problems, the asymptotic convergence rates resulting from the standard approach suffer from insufficient coarsegrid transport, while the truncated monotone multigrid method provides the same efficiency as in the unconstrained case. Key words: obstacle problems, adaptive finite element methods, multigrid methods AMS (MOS) subje...
A SignChanging Solution For A Superlinear Dirichlet Problem, II
 Rocky Mountain J. Math
"... In previous jointwork of A. Castro, J. Cossio and J. M. Neuberger (see [2]), it was shown that a superlinear Dirichlet problem has at least three nontrivial solutions when the derivative of the nonlinearity at zero is less than the first eigenvalue of ;\Delta with Dirichlet boundry condition. One ..."
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Cited by 49 (11 self)
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In previous jointwork of A. Castro, J. Cossio and J. M. Neuberger (see [2]), it was shown that a superlinear Dirichlet problem has at least three nontrivial solutions when the derivative of the nonlinearity at zero is less than the first eigenvalue of ;\Delta with Dirichlet boundry condition. One of these solutions changes sign exactlyonce and the other two are of one sign. In this paper we show that when this derivativeisbetween the kth and k+1st eigenvalues there still exists a solution whichchanges sign at most k times. In particular, when k = 1 the signchanging exactlyonce solution persists although onesign solutions no longer exist. 1 Introduction. Let \Omega be a smooth bounded region in R N , \Delta the Laplacian operator, and f 2 C 1 (R# R) such that f(0) = 0. In this paper we study the boundary value problem ( \Deltau + f(u)=0 in\Omega u =0 in @\Omega : (1) We assume that there exist constants A?0andp 2 (1# N+2 N;2 ) suchthatjf 0 (u)jA(juj ...
Semismooth Newton methods for operator equations in function spaces
, 2000
"... We develop a semismoothness concept for nonsmooth superposition operators in function spaces. The considered class of operators includes NCPfunctionbased reformulations of infinitedimensional nonlinear complementarity problems, and thus covers a very comprehensive class of applications. Our resul ..."
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Cited by 49 (3 self)
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We develop a semismoothness concept for nonsmooth superposition operators in function spaces. The considered class of operators includes NCPfunctionbased reformulations of infinitedimensional nonlinear complementarity problems, and thus covers a very comprehensive class of applications. Our results generalize semismoothness and fforder semismoothness from finitedimensional spaces to a Banach space setting. Hereby, a new generalized differential is used that can be seen as an extension of Qi's finitedimensional Csubdifferential to our infinitedimensional framework. We apply these semismoothness results to develop a Newtonlike method for nonsmooth operator equations and prove its local qsuperlinear convergence to regular solutions. If the underlying operator is fforder semismoothness, convergence of qorder 1 + ff is proved. We also establish the semismoothness of composite operators and develop corresponding chain rules. The developed theory is accompanied by illustrating e...
A Posteriori Error Estimates for Variable TimeStep Discretizations of Nonlinear Evolution Equations
"... We study the backward Euler method with variable timesteps for abstract evolution equations in Hilbert spaces. Exploiting convexity of the underlying potential or the anglebounded condition, thereby assuming no further regularity, we derive novel a posteriori estimates of the discretization error ..."
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Cited by 46 (9 self)
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We study the backward Euler method with variable timesteps for abstract evolution equations in Hilbert spaces. Exploiting convexity of the underlying potential or the anglebounded condition, thereby assuming no further regularity, we derive novel a posteriori estimates of the discretization error in terms of computable quantities related to the amount of energy dissipation or monotonicity residual. These estimators solely depend on the discrete solution and data and impose no constraints between consecutive timesteps. We also prove that they converge to zero with an optimal rate with respect to the regularity of the solution. We apply the abstract results to a number of concrete strongly nonlinear problems of parabolic type with degenerate or singular character.
Adaptive Multilevel  Methods for Obstacle Problems
, 1992
"... We consider the discretization of obstacle problems for second order elliptic differential operators by piecewise linear finite elements. Assuming that the discrete problems are reduced to a sequence of linear problems by suitable active set strategies, the linear problems are solved iteratively by ..."
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Cited by 44 (8 self)
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We consider the discretization of obstacle problems for second order elliptic differential operators by piecewise linear finite elements. Assuming that the discrete problems are reduced to a sequence of linear problems by suitable active set strategies, the linear problems are solved iteratively by preconditioned cgiterations. The proposed preconditioners are treated theoretically as abstract additive Schwarz methods and are implemented as truncated hierarchical basis preconditioners. To allow for local mesh refinement we derive semilocal and local a posteriori error estimates, providing lower and upper estimates for the discretization error. The theoretical results are illustrated by numerical computations.