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56
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.
RSVM: Reduced support vector machines
 Data Mining Institute, Computer Sciences Department, University of Wisconsin
, 2001
"... Abstract An algorithm is proposed which generates a nonlinear kernelbased separating surface that requires as little as 1 % of a large dataset for its explicit evaluation. To generate this nonlinear surface, the entire dataset is used as a constraint in an optimization problem with very few variabl ..."
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Cited by 122 (16 self)
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Abstract An algorithm is proposed which generates a nonlinear kernelbased separating surface that requires as little as 1 % of a large dataset for its explicit evaluation. To generate this nonlinear surface, the entire dataset is used as a constraint in an optimization problem with very few variables corresponding to the 1%
A Smoothing Method For Mathematical Programs With Equilibrium Constraints
, 1996
"... The mathematical program with equilibrium constraints (MPEC) is an optimization problem with variational inequality constraints. MPEC problems include bilevel programming problems as a particular case and have a wide range of applications. MPEC problems with strongly monotone variational inequalitie ..."
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Cited by 52 (5 self)
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The mathematical program with equilibrium constraints (MPEC) is an optimization problem with variational inequality constraints. MPEC problems include bilevel programming problems as a particular case and have a wide range of applications. MPEC problems with strongly monotone variational inequalities are considered in this paper. They are transformed into an equivalent onelevel nonsmooth optimization problem. Then, a sequence of smooth, regular problems that progressively approximate the nonsmooth problem and that can be solved by standard available software for constrained optimization is introduced. It is shown that the solutions (stationary points) of the approximate problems converge to a solution (stationary point) of the original MPEC problem. Numerical results showing viability of the approach are reported.
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 look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities
, 2000
"... ..."
SSVM: A Smooth Support Vector Machine for Classification
 Computational Optimization and Applications
, 1999
"... Smoothing methods, extensively used for solving important mathematical programming problems and applications, are applied here to generate and solve an unconstrained smooth reformulation of the support vector machine for pattern classification using a completely arbitrary kernel. We term such re ..."
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Cited by 31 (0 self)
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Smoothing methods, extensively used for solving important mathematical programming problems and applications, are applied here to generate and solve an unconstrained smooth reformulation of the support vector machine for pattern classification using a completely arbitrary kernel. We term such reformulation a smooth support vec tor machine (SSVM). A fast NewtonArmijo algorithm for solving the SSVM converges globally and quadratically. Numerical results and comparisons are given to demonstrate the effectiveness and speed of the algorithm. On six publicly available datasets, tenfold cross validation correctness of SSVM was the highest compared with four other methods as well as the fastest. On larger problems, SSVM was compa rable or faster than SVM light [17], SOR [23] and SMO [27]. SSVM can also generate a highly nonlinear separating surface such as a checker board.
NonInterior Continuation Methods For Solving Semidefinite Complementarity Problems
 Math. Programming
, 1999
"... There recently has been much interest in noninterior continuation/smoothing methods for solving linear/nonlinear complementarity problems. We describe extensions of such methods to complementarity problems defined over the cone of blockdiagonal symmetric positive semidefinite real matrices. These ..."
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Cited by 29 (3 self)
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There recently has been much interest in noninterior continuation/smoothing methods for solving linear/nonlinear complementarity problems. We describe extensions of such methods to complementarity problems defined over the cone of blockdiagonal symmetric positive semidefinite real matrices. These extensions involve the ChenMangasarian class of smoothing functions and the smoothed FischerBurmeister function. Issues such as existence of Newton directions, boundedness of iterates, global convergence, and local superlinear convergence will be studied. Preliminary numerical experience on semidefinite linear programs is also reported. Key words. Semidefinite complementarity problem, smoothing function, noninterior continuation, global convergence, local superlinear convergence. 1 Introduction There recently has been much interest in semidefinite linear programs (SDLP) and, more generally, semidefinite linear complementarity problems (SDLCP), which are extensions of LP and LCP, respecti...
A Global and Local Superlinear ContinuationSmoothing Method for ... and Monotone NCP
 SIAM J. Optim
, 1997
"... We propose a continuation method for a class of nonlinear complementarity problems(NCPs), including the NCP with a P 0 and R 0 function and the monotone NCP with a feasible interior point. The continuation method is based on a class of ChenMangasarian smooth functions. Unlike many existing continua ..."
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Cited by 24 (6 self)
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We propose a continuation method for a class of nonlinear complementarity problems(NCPs), including the NCP with a P 0 and R 0 function and the monotone NCP with a feasible interior point. The continuation method is based on a class of ChenMangasarian smooth functions. Unlike many existing continuation methods, the method follows the noninterior smoothing paths, and as a result, an initial point can be easily constructed. In addition, we introduce a procedure to dynamically update the neighborhoods associated with the smoothing paths, so that the algorithm is both globally convergent and locally superlinearly convergent under suitable assumptions. Finally, a hybrid continuationsmoothing method is proposed and is shown to have the same convergence properties under weaker conditions. 1 Introduction Let F : R n ! R n be a continuously differentiable function. The nonlinear complementarity problem, denoted by NCP(F ), is to find a vector (x; y) 2 R n \Theta R n such that F (x)...
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 22 (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 ...
Analysis Of A NonInterior Continuation Method Based On ChenMangasarian Smoothing Functions For Complementarity Problems
 Reformuation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods
, 1998
"... Recently Chen and Mangasarian proposed a class of smoothing functions for linear/nonlinear programs and complementarity problems that unifies many previous proposals. Here we study a noninterior continuation method based on these functions in which, like interior pathfollowing methods, the iterate ..."
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Cited by 20 (3 self)
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Recently Chen and Mangasarian proposed a class of smoothing functions for linear/nonlinear programs and complementarity problems that unifies many previous proposals. Here we study a noninterior continuation method based on these functions in which, like interior pathfollowing methods, the iterates are maintained to lie in a neighborhood of some path and, at each iteration, one or two Newtontype steps are taken and then the smoothing parameter is decreased. We show that the method attains global convergence and linear convergence under conditions similar to those required for other methods. We also show that these conditions are in some sense necessary. By introducing an inexpensive activeset strategy in computing one of the Newton directions, we show that the method attains local superlinear convergence under conditions milder than those for other methods. The proof of this uses a local error bound on the distance from an iterate to a solution in terms of the smoothing parameter. ...