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28
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.
Newton's Method For Large BoundConstrained Optimization Problems
 SIAM JOURNAL ON OPTIMIZATION
, 1998
"... We analyze a trust region version of Newton's method for boundconstrained problems. Our approach relies on the geometry of the feasible set, not on the particular representation in terms of constraints. The convergence theory holds for linearlyconstrained problems, and yields global and super ..."
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Cited by 76 (4 self)
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We analyze a trust region version of Newton's method for boundconstrained problems. Our approach relies on the geometry of the feasible set, not on the particular representation in terms of constraints. The convergence theory holds for linearlyconstrained problems, and yields global and superlinear convergence without assuming neither strict complementarity nor linear independence of the active constraints. We also show that the convergence theory leads to an efficient implementation for large boundconstrained problems.
A New Merit Function For Nonlinear Complementarity Problems And A Related Algorithm
, 1997
"... We investigate the properties of a new merit function which allows us to reduce a nonlinear complementarity problem to an unconstrained global minimization one. Assuming that the complementarity problem is defined by a P 0 function we prove that every stationary point of the unconstrained problem ..."
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Cited by 73 (6 self)
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We investigate the properties of a new merit function which allows us to reduce a nonlinear complementarity problem to an unconstrained global minimization one. Assuming that the complementarity problem is defined by a P 0 function we prove that every stationary point of the unconstrained problem is a global solution; furthermore, if the complementarity problem is defined by a uniform P function, the level sets of the merit function are bounded. The properties of the new merit function are compared with those of the MangasarianSolodov's implicit Lagrangian and Fukushima's regularized gap function. We also introduce a new, simple, activeset local method for the solution of complementarity problems and show how this local algorithm can be made globally convergent by using the new merit function.
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 Comparison of Large Scale Mixed Complementarity Problem Solvers
 Computational Optimization and Applications
, 1997
"... . This paper provides a means for comparing various computer codes for solving large scale mixed complementarity problems. We discuss inadequacies in how solvers are currently compared, and present a testing environment that addresses these inadequacies. This testing environment consists of a librar ..."
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Cited by 25 (13 self)
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. This paper provides a means for comparing various computer codes for solving large scale mixed complementarity problems. We discuss inadequacies in how solvers are currently compared, and present a testing environment that addresses these inadequacies. This testing environment consists of a library of test problems, along with GAMS and MATLAB interfaces that allow these problems to be easily accessed. The environment is intended for use as a tool by other researchers to better understand both their algorithms and their implementations, and to direct research toward problem classes that are currently the most challenging. As an initial benchmark, eight different algorithm implementations for large scale mixed complementarity problems are briefly described and tested with default parameter settings using the new testing environment. Keywords: complementarity problems, variational inequalities, computation, algorithms 1. Introduction In recent years, a considerable number of new algori...
A linearly convergent derivativefree descent method for strongly monotone complementarity problems
 Computational Optimization and Applications
"... Abstract. We establish the first rate of convergence result for the class of derivativefree descent methods for solving complementarity problems. The algorithm considered here is based on the implicit Lagrangian reformulation [26, 35] of the nonlinear complementarity problem, and makes use of the d ..."
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Cited by 9 (6 self)
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Abstract. We establish the first rate of convergence result for the class of derivativefree descent methods for solving complementarity problems. The algorithm considered here is based on the implicit Lagrangian reformulation [26, 35] of the nonlinear complementarity problem, and makes use of the descent direction proposed in [42], but employs a different Armijotype linesearch rule. We show that in the strongly monotone case, the iterates generated by the method converge globally at a linear rate to the solution of the problem. Keywords: convergence complementarity problems, implicit Lagrangian, descent algorithms, derivativefree methods, linear 1.
A Comparison of Algorithms for Large Scale Mixed Complementarity Problems
 Computational Optimization and Applications
, 1995
"... . This paper provides a means for comparing various computer codes for solving large scale mixed complementarity problems. We discuss inadequacies in how algorithms are currently compared, and present a testing environment that partially solves these inadequacies. This testing environment consists o ..."
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Cited by 8 (2 self)
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. This paper provides a means for comparing various computer codes for solving large scale mixed complementarity problems. We discuss inadequacies in how algorithms are currently compared, and present a testing environment that partially solves these inadequacies. This testing environment consists of a library of test problems, along with GAMS and MATLAB interfaces that allow these problems to be easily accessed. Eight different algorithm implementations for large scale mixed complementarity problems are briefly described and tested with default parameter settings using the new testing environment. Keywords: complementarity problems, variational inequalities, computation, algorithms 1. Introduction In recent years, a considerable number of new algorithms have been developed for solving large scale mixed complementarity problems. Many of these algorithms appear very promising theoretically, but it is difficult to understand how well they will work in practice. Indeed, many of the pap...
A survey of some nonsmooth equations and smoothing Newton methods
 Progress in Optimization, volume 30 of Applied Optimization
, 1999
"... In this article we review and summarize recent developments on nonsmooth equations and smoothing Newton methods. Several new suggestions are presented. 1 ..."
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Cited by 7 (2 self)
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In this article we review and summarize recent developments on nonsmooth equations and smoothing Newton methods. Several new suggestions are presented. 1
Some optimization reformulations of the extended linear complementarity problem
 Comput. Optim. Appl
"... Abstract. We consider the extended linear complementarity problem (XLCP) introduced by Mangasarian and Pang [22], of which the horizontal and vertical linear complementarity problems are two special cases. We give some new sufficient conditions for every stationary point of the natural bilinear prog ..."
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Cited by 6 (2 self)
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Abstract. We consider the extended linear complementarity problem (XLCP) introduced by Mangasarian and Pang [22], of which the horizontal and vertical linear complementarity problems are two special cases. We give some new sufficient conditions for every stationary point of the natural bilinear program associated with XLCP to be a solution of XLCP. We further propose some unconstrained and bound constrained reformulations for XLCP, and study the properties of their stationary points under assumptions similar to those for the bilinear program.
The Traffic Equilibrium Problem with Nonadditive Costs and Its Monotone Mixed Complementarity Problem Formulation ∗
, 2005
"... Abstract. Various models of traffic equilibrium problems (TEPs) with nonadditive route costs have been proposed in the last decade. However, equilibria of those models are not easy to obtain because the variational inequality problems (VIPs) derived from those models are not monotone in general. In ..."
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Cited by 4 (0 self)
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Abstract. Various models of traffic equilibrium problems (TEPs) with nonadditive route costs have been proposed in the last decade. However, equilibria of those models are not easy to obtain because the variational inequality problems (VIPs) derived from those models are not monotone in general. In this paper, we consider a TEP whose route cost functions are nonadditive disutility functions of time (with money converted to time). We show that the TEP with the disutility functions can be reformulated as a monotone Mixed Complementarity Problem (MCP) under appropriate conditions. We then establish the existence and uniqueness results for an equilibrium of the TEP. Numerical experiments are carried out using various sample networks with different disutility functions for both the singlemode case and the case of two different transportation modes in the network.