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12
Engineering and economic applications of complementarity problems
 SIAM REVIEW
, 1997
"... 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 c ..."
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Cited by 195 (24 self)
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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.
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 49 (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...
General InteriorPoint Maps and Existence of Weighted Paths for Nonlinear Semidefinite Complementarity Problems
, 1999
"... Extending the previous work of Monteiro and Pang (1998), this paper studies properties of fundamental maps that can be used to describe the central path of the monotone nonlinear complementarity problems over the cone of symmetric positive semidefinite matrices. Instead of focusing our attention on ..."
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Cited by 15 (4 self)
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Extending the previous work of Monteiro and Pang (1998), this paper studies properties of fundamental maps that can be used to describe the central path of the monotone nonlinear complementarity problems over the cone of symmetric positive semidefinite matrices. Instead of focusing our attention on a specific map as was done in the approach of Monteiro and Pang (1998), this paper considers a general form of a fundamental map and introduces conditions on the map that allow us to extend the main results of Monteiro and Pang (1998) to this general map. Each fundamental map leads to a family of "weighted" continuous trajectories which include the central trajectory as a special case. Hence, for complementarity problems over the cone of symmetric positive semidefinite matrices, the notion of weighted central path depends on the fundamental map used to represent the central path.
A Potential Reduction Newton Method for Constrained Equations
 SIAM Journal on Optimization
, 1997
"... Extending our previous work [11], this paper presents a general potential reduction Newton method for solving a constrained system of nonlinear equations. A main convergence result for the method is established. Specializations of the method to a convex semidefinite program and a monotone complement ..."
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Cited by 13 (4 self)
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Extending our previous work [11], this paper presents a general potential reduction Newton method for solving a constrained system of nonlinear equations. A main convergence result for the method is established. Specializations of the method to a convex semidefinite program and a monotone complementarity problem in symmetric matrices are discussed. Strong convergence results are established in these specializations. 1 Introduction In the paper [11], we have introduced the problem of solving a system of nonlinear equations subject to additional constraints on the variables, i.e., a constrained system of equations. We have demonstrated that constrained equations (CEs) provide a unifying framework for the study of complementarity problems of various types, including the standard nonlinear complementarity problem and the KarushKuhnTucker system of a variational inequality. Postulating a partitioning property of the CE, we have introduced an interior point potential reduction algorithm f...
An Active SetType Newton Method For Constrained Nonlinear Systems
 In Complementarity: Applications, Algorithms and Extensions (2001), Ferris M., Mangasarian O., Pang J.S., (Eds
, 1999
"... . We consider the problem of finding a solution of a nonlinear system of equations subject to some box constraints. To this end, we introduce a new active settype Newton method with global and local fast convergence properties. The method generates feasible iterates and has to solve only one linear ..."
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Cited by 6 (2 self)
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. We consider the problem of finding a solution of a nonlinear system of equations subject to some box constraints. To this end, we introduce a new active settype Newton method with global and local fast convergence properties. The method generates feasible iterates and has to solve only one linear system of equations at each iteration. Due to our active set strategy, this linear system is of reduced dimension. Key Words. Nonlinear equations, box constraints, Newton's method, active set strategy, projected gradient, global convergence, quadratic convergence. 1 The research of this author was partially supported by the DFG (Deutsche Forschungsgemeinschaft). 1 Introduction The problem we address in this paper is to find a solution of the constrained nonlinear system F (x) = 0; x 2 [l; u]; (1) where F : [l; u] ! IR n is a given function which is assumed to be continuously differentiable in an open set containing the box [l; u], and where l = (l 1 ; : : : ; l n ) T ; u = (u 1 ; ...
On the reformulation of nonlinear complementarity problems using the FischerBurmeister function
, 1999
"... A boundedlevelset result for a reformulation of the boxconstrained variational inequality problem proposed recently by Facchinei, Fischer and Kanzow is proved. An application of this result to the (unbounded) nonlinear complementarity problem is suggested. Keywords. Complementarity, unconstraine ..."
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Cited by 5 (4 self)
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A boundedlevelset result for a reformulation of the boxconstrained variational inequality problem proposed recently by Facchinei, Fischer and Kanzow is proved. An application of this result to the (unbounded) nonlinear complementarity problem is suggested. Keywords. Complementarity, unconstrained minimization, reformulation. AMS: 90C33, 90C30 1 Main results Given F : IR n ! IR n , F 2 C 1 (IR n ), the boxconstrained variational inequality problem (BVIP) consists on finding x 2\Omega such that hF (x); z \Gamma xi 0 for all z 2\Omega ; (1) where\Omega is the compact box \Omega = fx 2 IR n j ` x rg: (2) The Nonlinear Complementarity Problem (NCP) is the problem (1) when\Omega = fx 2 IR n j x 0g. Department of Computer Science and Statistics, University of the State of S. Paulo (UNESP), C.P. 136, CEP 15054000, S~ao Jos'e do Rio PretoSP, Brazil. This author was supported by FAPESP (Grants 96/15520 and 96/12503 0). Email: andreani@nimitz.dcce.ibilce.unesp...
Solving Complementarity Problems By Means of a New Smooth Constrained Nonlinear Solver
, 1999
"... Given F : IR n ! IR m and\Omega a closed and convex set, the problem of finding x 2 IR n such that x 2\Omega and F (x) = 0 is considered. For solving this problem an algorithm of InexactNewton type is defined. Global and local convergence proofs are presented. As a practical application, th ..."
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Cited by 5 (5 self)
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Given F : IR n ! IR m and\Omega a closed and convex set, the problem of finding x 2 IR n such that x 2\Omega and F (x) = 0 is considered. For solving this problem an algorithm of InexactNewton type is defined. Global and local convergence proofs are presented. As a practical application, the Horizontal Nonlinear Complementarity Problem is introduced. It is shown that the InexactNewton algorithm can be applied to this problem. Numerical experiments are performed and commented. Keywords. Nonlinear systems, InexactNewton method, global convergence, convex constraints, box constraints, complementarity. AMS: 65H10, 90C33, 90C30 Department of Applied Mathematics, IMECCUNICAMP, University of Campinas, CP 6065, 13081970 Campinas SP, Brazil . This author was supported by FAPESP (Grant 9037246, 93/024796), FINEP and FAEPUNICAMP. y Department of Mathematics, IMECCUNICAMP, University of Campinas, CP 6065, 13081970 Campinas SP, Brazil. This author was supported by FAPESP (Gr...
A survey of GNE computation methods : theory and algorithms. Working paper
, 2012
"... This paper deals with optimization methods solving the generalized Nash equilibrium problem (GNEP), which extends the standard Nash problem by allowing constraints. Two cases are considered: general GNEPs where constraint functions are individualized and jointly convex GNEPs where there is a common ..."
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Cited by 4 (1 self)
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This paper deals with optimization methods solving the generalized Nash equilibrium problem (GNEP), which extends the standard Nash problem by allowing constraints. Two cases are considered: general GNEPs where constraint functions are individualized and jointly convex GNEPs where there is a common constraint function. Most recent methods are benchmarked against new methods. Numerical illustrations are proposed with the same software for a fair benchmark.
Representation,analysis And Solution Of Conditional Models In An EquationBased Environment
, 1998
"... Process modeling is an important task in many process engineering activities. At the lowest level, process models are represented by a large set of variables and a large system of linear and nonlinear equations that relate them. The equationbased modeling approach has been demonstrated as effective ..."
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Cited by 1 (0 self)
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Process modeling is an important task in many process engineering activities. At the lowest level, process models are represented by a large set of variables and a large system of linear and nonlinear equations that relate them. The equationbased modeling approach has been demonstrated as effective in solving simulation, optimization, parameter estimation and data reconciliation problems. Even though many currently available equationbased modeling systems have been reported in the literature, most of them give little or no attention to conditional models. Conditional models exist when the equations defining a system depend on where the model solution lies. Examples of conditional models in chemical engineering are systems involving physicochemical discontinuities such as flow and phase transitions. This work investigates the setting up and solving of conditional models within an equationbased modeling environment. We first describe modeling tools for the efficient representation of ...
An Active Settype Newton method for . . .
 IN COMPLEMENTARITY: APPLICATIONS, ALGORITHMS AND EXTENSIONS (2001), FERRIS M., MANGASARIAN O., PANG J.S., (EDS
, 1999
"... We consider the problem of finding a solution of a nonlinear system of equations subject to some box constraints. To this end, we introduce a new active settype Newton method with global and local fast convergence properties. The method generates feasible iterates and has to solve only one linear s ..."
Abstract
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We consider the problem of finding a solution of a nonlinear system of equations subject to some box constraints. To this end, we introduce a new active settype Newton method with global and local fast convergence properties. The method generates feasible iterates and has to solve only one linear system of equations at each iteration. Due to our active set strategy, this linear system is of reduced dimension.