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22
Engineering and economic applications of complementarity problems
 SIAM Review
, 1997
"... Abstract. This paper gives an extensive documentation of applications of finitedimensional nonlinear complementarity problems in engineering and equilibrium modeling. For most applications, we describe the problem briefly, state the defining equations of the model, and give functional expressions f ..."
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Cited by 127 (24 self)
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Abstract. This paper gives an extensive documentation of applications of finitedimensional nonlinear complementarity problems in engineering and equilibrium modeling. For most applications, we describe the problem briefly, state the defining equations of the model, and give functional expressions for the complementarity formulations. The goal of this documentation is threefold: (i) to summarize the essential applications of the nonlinear complementarity problem known to date, (ii) to provide a basis for the continued research on the nonlinear complementarity problem, and (iii) to supply a broad collection of realistic complementarity problems for use in algorithmic experimentation and other studies.
A Modified ForwardBackward Splitting Method For Maximal Monotone Mappings
 SIAM J. Control Optim
, 1998
"... We consider the forwardbackward splitting method for finding a zero of the sum of two maximal monotone mappings. This method is known to converge when the inverse of the forward mapping is strongly monotone. We propose a modification to this method, in the spirit of the extragradient method for mon ..."
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Cited by 49 (0 self)
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We consider the forwardbackward splitting method for finding a zero of the sum of two maximal monotone mappings. This method is known to converge when the inverse of the forward mapping is strongly monotone. We propose a modification to this method, in the spirit of the extragradient method for monotone variational inequalities, under which the method converges assuming only the forward mapping is monotone and (Lipschitz) continuous on some closed convex subset of its domain. The modification entails an additional forward step and a projection step at each iteration. Applications of the modified method to decomposition in convex programming and monotone variational inequalities are discussed.
Algorithms For Complementarity Problems And Generalized Equations
, 1995
"... Recent improvements in the capabilities of complementarity solvers have led to an increased interest in using the complementarity problem framework to address practical problems arising in mathematical programming, economics, engineering, and the sciences. As a result, increasingly more difficult pr ..."
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Cited by 41 (5 self)
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Recent improvements in the capabilities of complementarity solvers have led to an increased interest in using the complementarity problem framework to address practical problems arising in mathematical programming, economics, engineering, and the sciences. As a result, increasingly more difficult problems are being proposed that exceed the capabilities of even the best algorithms currently available. There is, therefore, an immediate need to improve the capabilities of complementarity solvers. This thesis addresses this need in two significant ways. First, the thesis proposes and develops a proximal perturbation strategy that enhances the robustness of Newtonbased complementarity solvers. This strategy enables algorithms to reliably find solutions even for problems whose natural merit functions have strict local minima that are not solutions. Based upon this strategy, three new algorithms are proposed for solving nonlinear mixed complementarity problems that represent a significant improvement in robustness over previous algorithms. These algorithms have local Qquadratic convergence behavior, yet depend only on a pseudomonotonicity assumption to achieve global convergence from arbitrary starting points. Using the MCPLIB and GAMSLIB test libraries, we perform extensive computational tests that demonstrate the effectiveness of these algorithms on realistic problems. Second, the thesis extends some previously existing algorithms to solve more general problem classes. Specifically, the NE/SQP method of Pang & Gabriel (1993), the semismooth equations approach of De Luca, Facchinei & Kanz...
A new projection method for variational inequality problems
 SIAM J. Control Optim
, 1999
"... Abstract. We propose a new projection algorithm for solving the variational inequality problem, where the underlying function is continuous and satisfies a certain generalized monotonicity assumption (e.g., it can be pseudomonotone). The method is simple and admits a nice geometric interpretation. I ..."
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Cited by 20 (11 self)
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Abstract. We propose a new projection algorithm for solving the variational inequality problem, where the underlying function is continuous and satisfies a certain generalized monotonicity assumption (e.g., it can be pseudomonotone). The method is simple and admits a nice geometric interpretation. It consists of two steps. First, we construct an appropriate hyperplane which strictly separates the current iterate from the solutions of the problem. This procedure requires a single projection onto the feasible set and employs an Armijotype linesearch along a feasible direction. Then the next iterate is obtained as the projection of the current iterate onto the intersection of the feasible set with the halfspace containing the solution set. Thus, in contrast with most other projectiontype methods, only two projection operations per iteration are needed. The method is shown to be globally convergent to a solution of the variational inequality problem under minimal assumptions. Preliminary computational experience is also reported. Key words. variational inequalities, projection methods, pseudomonotone maps
Complementarity And Related Problems: A Survey
, 1998
"... This survey gives an introduction to some of the recent developments in the field of complementarity and related problems. After presenting two typical examples and the basic existence and uniqueness results, we focus on some new trends for solving nonlinear complementarity problems. Extensions to ..."
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Cited by 14 (0 self)
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This survey gives an introduction to some of the recent developments in the field of complementarity and related problems. After presenting two typical examples and the basic existence and uniqueness results, we focus on some new trends for solving nonlinear complementarity problems. Extensions to mixed complementarity problems, variational inequalities and mathematical programs with equilibrium constraints are also discussed.
Convergence analysis of perturbed feasible descent methods
 JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
, 1997
"... We develop a general approach to convergence analysis of feasible descent methods in the presence of perturbations. The important novel feature of our analysis is that perturbations need not tend to zero in the limit. In that case, standard convergence analysis techniques are not applicable. There ..."
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Cited by 7 (2 self)
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We develop a general approach to convergence analysis of feasible descent methods in the presence of perturbations. The important novel feature of our analysis is that perturbations need not tend to zero in the limit. In that case, standard convergence analysis techniques are not applicable. Therefore, a new approach is needed. We show that, in the presence of perturbations, a certain eapproximate solution can be obtained, where e depends linearly on the level of perturbations. Applications to the gradient projection, proximal minimization, extragradient and incremental gradient algorithms are described.
A new version of extragradient method for variational inequality problems
 Computers Math. Appl
"... AbstractIn this paper, we propose a new version of extragradient method for the variational inequality problem. The method uses a new searching direction which differs from any one in existing projectiontype methods, and is of a better stepsize rule. Under a certain generalized monotonicity condit ..."
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Cited by 6 (2 self)
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AbstractIn this paper, we propose a new version of extragradient method for the variational inequality problem. The method uses a new searching direction which differs from any one in existing projectiontype methods, and is of a better stepsize rule. Under a certain generalized monotonicity condition, it is proved to be globally convergent. @ 2001 Eisevier Science Ltd. All rights reserved. KeywordsVsriationai inequality, Extragradient method, Convergence. 1.
Complementarity Problems
 J. Comput. Appl. Math
, 2000
"... This paper provides an introduction to complementarity problems, with an emphasis on applications and solution algorithms. Various forms of complementarity problems are described along with a few sample applications, which provide a sense of what types of problems can be addressed eectively with ..."
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Cited by 6 (0 self)
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This paper provides an introduction to complementarity problems, with an emphasis on applications and solution algorithms. Various forms of complementarity problems are described along with a few sample applications, which provide a sense of what types of problems can be addressed eectively with complementarity problems. The most important algorithms are presented along with a discussion of when they can be used eectively. We also provide a brief introduction to the study of matrix classes and their relation to linear complementarity problems. Finally, we provide a brief summary of current research trends. Key words: complementarity problems,variational inequalities, matrix classes 1 Introduction The distinguishing feature of a complementarity problem is the set of complementarity conditions. Each of these conditions requires that the product of two or more nonnegative quantities should be zero. (Here, each quantity is either a decision variable, or a function of the decisi...
On a PrimalDual Analytic Center Cutting Plane Method for Variational Inequalities
 Computational Optimization and Applications
, 1998
"... . We present an algorithm for variational inequalities V I(F ; Y ) that uses a primaldual version of the Analytic Center Cutting Plane Method. The pointtoset mapping F is assumed to be monotone, or pseudomonotone. Each computation of a new analytic center requires at most four Newton iterations, ..."
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Cited by 5 (2 self)
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. We present an algorithm for variational inequalities V I(F ; Y ) that uses a primaldual version of the Analytic Center Cutting Plane Method. The pointtoset mapping F is assumed to be monotone, or pseudomonotone. Each computation of a new analytic center requires at most four Newton iterations, in theory, and in practice one or sometimes two. Linear equalities that may be included in the definition of the set Y are taken explicitly into account. We report numerical experiments on several wellknown variational inequality problems as well as on one where the functional results from the solution of large subproblems. The method is robust and competitive with algorithms which use the same information as this one. Keywords: variational inequalities; analytic center; cutting plane method; monotone mappings; interior points methods; Newton's method; primaldual. * Research supported in part by McGill University Fellowships ** Research supported by NSERC grant OPG0004152 and by the FCAR...
A Class Of Globally Convergent Algorithms For Pseudomonotone Variational Inequalities
, 2001
"... We describe a fairly broad class of algorithms for solving variational inequalities, global convergence of which is based on the strategy of generating a hyperplane separating the current iterate from the solution set. The methods are shown to converge under very mild assumptions. Specifically, the ..."
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Cited by 4 (4 self)
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We describe a fairly broad class of algorithms for solving variational inequalities, global convergence of which is based on the strategy of generating a hyperplane separating the current iterate from the solution set. The methods are shown to converge under very mild assumptions. Specifically, the problem mapping is only assumed to be continuous and pseudomonotone with respect to at least one solution. The strategy to obtain (super)linear rate of convergence is also discussed. The algorithms in this class di#er in the tools which are used to construct the separating hyperplane. Our general scheme subsumes an extragradienttype projection method, a globally and locally superlinearly convergent JosephyNewtontype method, a certain minimizationbased method, and a splitting technique. 1 INTRODUCTION Given a function F : R n # R n and a set C # R n , the classical variational inequality problem [1, 3, 8, 9, 5], abbreviated VIP(F, C), is to find a point x such that x # C, #F ...