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69
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 155 (25 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 143 (19 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 65 (6 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 45 (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
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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 37 (2 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 33 (4 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 P0 and R0 NCP or monotone NCP
 SIAM on Journal Optimization
, 1999
"... ..."
Smoothing Functions for SecondOrderCone Complementarity Problems
, 2000
"... Smoothing functions have been much studied in the solution of optimization and complementarity problems with nonnegativity constraints. In this paper, we extend smoothing functions to problems where the nonnegative orthant is replaced by the direct product of secondorder cones. These smoothing func ..."
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Cited by 27 (3 self)
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Smoothing functions have been much studied in the solution of optimization and complementarity problems with nonnegativity constraints. In this paper, we extend smoothing functions to problems where the nonnegative orthant is replaced by the direct product of secondorder cones. These smoothing functions include the ChenMangasarian class and the smoothed FischerBurmeister function. We study the Lipschitzian and dierential properties of these functions and, in particular, we derive computable formulas for these functions and their Jacobians. These properties and formulas can then be used to develop and analyze noninterior continuation methods for solving the corresponding optimization and complementarity problems. In particular, we establish existence and uniqueness of the Newton direction when the underlying mapping is monotone. This research is supported by a GrantinAid for Scientic Research (B) from the Ministry of Education, Science, Sports and Culture of Japan. The third ...
Secondorder cone programming relaxation of sensor network localization
 SIAM J. Optimization
, 2007
"... Abstract. The sensor network localization problem has been much studied. Recently Biswas and Ye proposed a semidefinite programming (SDP) relaxation of this problem which has various nice properties and for which a number of solution methods have been proposed. Here, we study a secondorder cone pro ..."
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Cited by 24 (2 self)
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Abstract. The sensor network localization problem has been much studied. Recently Biswas and Ye proposed a semidefinite programming (SDP) relaxation of this problem which has various nice properties and for which a number of solution methods have been proposed. Here, we study a secondorder cone programming (SOCP) relaxation of this problem, motivated by its simpler structure and its potential to be solved faster than SDP. We show that the SOCP relaxation, though weaker than the SDP relaxation, has nice properties that make it useful as a problem preprocessor. In particular, sensors that are uniquely positioned among interior solutions of the SOCP relaxation are accurate up to the square root of the distance error. Thus, these sensors, which are easily identified, are accurately positioned. In our numerical simulation, the interior solution found can accurately position up to 80–90 % of the sensors. We also propose a smoothing coordinate gradient descent method for finding an interior solution that is faster than an interiorpoint method. Key words. sensor network localization, semidefinite program, secondorder cone program, approximation algorithm, error bound