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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.
A unified approach to interior point algorithms for linear complementarity problems,
 Lecture Notes in Computer Science 538, SpringerVerlag
, 1991
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On the formulation and theory of Newton interior point methods for nonlinear programming
 JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
, 1996
"... In this work, we first study in detail the formulation of the primaldual interiorpoint method for linear programming. We show that, contrary to popular belief, it cannot be viewed.as adamped Newton method applied to the KarushKuhnTucker conditions for the logarithmic barrier function problem. N ..."
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Cited by 116 (5 self)
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In this work, we first study in detail the formulation of the primaldual interiorpoint method for linear programming. We show that, contrary to popular belief, it cannot be viewed.as adamped Newton method applied to the KarushKuhnTucker conditions for the logarithmic barrier function problem. Next, we extend the formulation to general nonlinear programming, and then validate this extension by demonstrating that this algorithm can be implemented so that it is locally and Qquadratically convergent under only the standard Newton method assumptions. We also establish a global convergence theory for this algorithm and include promising numerical experimentation.
TimeStepping for ThreeDimensional Rigid Body Dynamics
, 1998
"... This paper considers a wide number of timestepping methods, and discusses their implications for convergence theory and the nature of the limiting solutions. ..."
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Cited by 62 (20 self)
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This paper considers a wide number of timestepping methods, and discusses their implications for convergence theory and the nature of the limiting solutions.
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...
A global and local superlinear continuationsmoothing method for P0 + R0 and monotone NCP
 SIAM Journal Optimization
, 1997
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Global Methods For Nonlinear Complementarity Problems
 MATH. OPER. RES
, 1994
"... Global methods for nonlinear complementarity problems formulate the problem as a system of nonsmooth nonlinear equations approach, or use continuation to trace a path defined by a smooth system of nonlinear equations. We formulate the nonlinear complementarity problem as a boundconstrained nonlinea ..."
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Cited by 31 (1 self)
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Global methods for nonlinear complementarity problems formulate the problem as a system of nonsmooth nonlinear equations approach, or use continuation to trace a path defined by a smooth system of nonlinear equations. We formulate the nonlinear complementarity problem as a boundconstrained nonlinear least squares problem. Algorithms based on this formulation are applicable to general nonlinear complementarity problems, can be started from any nonnegative starting point, and each iteration only requires the solution of systems of linear equations. Convergence to a solution of the nonlinear complementarity problem is guaranteed under reasonable regularity assumptions. The converge rate is Qlinear, Qsuperlinear, or Qquadratic, depending on the tolerances used to solve the subproblems.
On HomotopySmoothing Methods for Variational Inequalities
"... A variational inequality problem with a mapping g : ! n ! ! n and lower and upper bounds on variables can be reformulated as a system of nonsmooth equations F (x) = 0 in ! n . Recently, several homotopy methods, such as interiorpoint and smoothing methods, have been employed to solve the prob ..."
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Cited by 26 (5 self)
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A variational inequality problem with a mapping g : ! n ! ! n and lower and upper bounds on variables can be reformulated as a system of nonsmooth equations F (x) = 0 in ! n . Recently, several homotopy methods, such as interiorpoint and smoothing methods, have been employed to solve the problem. All of these methods use parametric functions and construct perturbed equations to approximate the problem. The solution to the perturbed system constitutes a smooth trajectory leading to the solution of the original variational inequality problem. The methods generate iterates to follow the trajectory. Among these methods ChenMangasarian and GabrielMor'e proposed a class of smooth functions to approximate F . In this paper, we study several properties of the trajectory defined by solutions of these smooth systems. We propose a homotopysmoothing method for solving the variational inequality problem, and show that the method converges globally and superlinearly under mild conditions. ...
A Global Linear and Local Quadratic Noninterior Continuation Method For Nonlinear Complementarity Problems Based on ChenMangasarian Smoothing Functions
, 1997
"... A noninterior continuation method is proposed for nonlinear complementarity problems. The method improves the noninterior continuation methods recently studied by Burke and Xu [1] and Xu [29]. Our definition of neighborhood for the central path is simpler and more natural. In addition, our continu ..."
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Cited by 25 (2 self)
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A noninterior continuation method is proposed for nonlinear complementarity problems. The method improves the noninterior continuation methods recently studied by Burke and Xu [1] and Xu [29]. Our definition of neighborhood for the central path is simpler and more natural. In addition, our continuation method is based on a broader class of smooth functions introduced by Chen and Mangasarian [7]. The method is shown to be globally linearly and locally quadratically convergent under suitable assumptions. 1 Introduction Let F : R n ! R n be a continuously differentiable function. The nonlinear complementarity problem (NCP) is to find (x; y) 2 R n \Theta R n such that F (x) \Gamma y = 0; (1) x 0; y 0; x T y = 0: (2) Numerous methods have been developed to solve the NCP, for a comprehensive survey see [13, 23]. In this paper, we are interested in developing a noninterior continuation method for the NCP and analyzing its rate of convergence. Department of Management and ...