Results 1  10
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23
A Penalized FischerBurmeister NcpFunction: Theoretical Investigation And Numerical Results
, 1997
"... We introduce a new NCPfunction that reformulates a nonlinear complementarity problem as a system of semismooth equations \Phi(x) = 0. The new NCPfunction possesses all the nice properties of the FischerBurmeister function for local convergence. In addition, its natural merit function \Psi(x) = ..."
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Cited by 43 (12 self)
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We introduce a new NCPfunction that reformulates a nonlinear complementarity problem as a system of semismooth equations \Phi(x) = 0. The new NCPfunction possesses all the nice properties of the FischerBurmeister function for local convergence. In addition, its natural merit function \Psi(x) = 1 2 \Phi(x) T \Phi(x) has all the nice features of the KanzowYamashitaFukushima merit function for global convergence. In particular, the merit function has bounded level sets for a monotone complementarity problem with a strictly feasible point. This property allows the existing semismooth Newton methods to solve this important class of complementarity problems without additional assumptions. We investigate the properties of a semismooth Newtontype method based on the new NCPfunction and apply the method to a large class of complementarity problems. The numerical results indicate that the new algorithm is extremely promising.
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.
Smoothing Methods and Semismooth Methods for Nondifferentiable Operator Equations
 SIAM J. Numer. Anal
, 1999
"... . We consider superlinearly convergent analogues of Newton methods for nondifferentiable operator equations in function spaces. The superlinear convergence analysis of semismooth methods for nondifferentiable equations described by defining a locally Lipschitzian operator in R n is based on Radema ..."
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Cited by 25 (1 self)
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. We consider superlinearly convergent analogues of Newton methods for nondifferentiable operator equations in function spaces. The superlinear convergence analysis of semismooth methods for nondifferentiable equations described by defining a locally Lipschitzian operator in R n is based on Rademacher's theorem which does not hold in function spaces. We introduce a concept of slant differentiability and use it to study superlinear convergence of smoothing methods and semismooth methods in a unified framework. We show that a function is slantly differentiable at a point if and only if it is Lipschitz continuous at that point. An application to the Dirichlet problems for a simple class of nonsmooth elliptic partial differential equations is discussed. Key words. Smoothing methods, semismooth methods, superlinear convergence, nondifferentiable operator equation, nonsmooth elliptic partial differential equations. AMS subject classifications. 65J15, 65H10, 65J20. 1. Introduction. This p...
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)...
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. ...
A Global Linear and Local Quadratic Continuation Smoothing Method for Variational Inequalities with Box Constraints
, 1997
"... In this paper, we propose a continuation method for box constrained variational inequality problems. The continuation method is based on the class of GabrielMor'e smooth functions and has the following attractive features: It can start from any point; It has a simple and natural neighborhood defini ..."
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Cited by 19 (3 self)
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In this paper, we propose a continuation method for box constrained variational inequality problems. The continuation method is based on the class of GabrielMor'e smooth functions and has the following attractive features: It can start from any point; It has a simple and natural neighborhood definition; It solves only one approximate Newton equation at each iteration; It converges globally linearly and locally quadratically under nondegeneracy assumption at the solution point and other suitable assumptions. A hybrid method is also presented, which is shown to preserve the above convergence properties without the nondegeneracy assumption at the solution point. In particular, the hybrid method converges finitely for affine problems. 1 Introduction Let F : R n ! R n be a continuously differentiable function. Let l 2 fR [ \Gamma1g n and u 2 fR [1g n such that l ! u. The variational inequality problem (VIP) with box constraints, denoted by VIP(l; u; F ), is to find x 2 [l; u] such...
Regularization of P 0 functions in Box Variational Inequality Problems
, 1997
"... In two recent papers, Facchinei [7] and Facchinei and Kanzow [8] have shown that for a continuously differentiable P 0 function f , the nonlinear complementarity problem NCP(f " ) corresponding to the regularization f " (x) := f(x) + "x has a unique solution for every " ? 0, that dist (x("); SOL(f) ..."
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Cited by 14 (0 self)
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In two recent papers, Facchinei [7] and Facchinei and Kanzow [8] have shown that for a continuously differentiable P 0 function f , the nonlinear complementarity problem NCP(f " ) corresponding to the regularization f " (x) := f(x) + "x has a unique solution for every " ? 0, that dist (x("); SOL(f)) ! 0 as " ! 0 when the solution set SOL(f) of NCP(f) is nonempty and bounded, and NCP(f) is stable if and only if the solution set is nonempty and bounded. They prove these results via the the Fischer function and the Mountain Pass Theorem. In this paper, we generalize these NCP results to a Box Variational Inequality Problem corresponding to a continuous P 0 function where the regularization is described by an integral. We also describe an upper semicontinuity property of the inverse of a weakly univalent function and study its consequences. Research supported by the National Science Foundation Grant CCR9307685 1 Introduction Consider a continuous function f : R n ! R n and a re...
Complementarity And Related Problems: A Survey
, 1998
"... This survey gives an introduction to some of the recent developments in the field of complementarity and related problems. After presenting two typical examples and the basic existence and uniqueness results, we focus on some new trends for solving nonlinear complementarity problems. Extensions to ..."
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Cited by 14 (0 self)
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This survey gives an introduction to some of the recent developments in the field of complementarity and related problems. After presenting two typical examples and the basic existence and uniqueness results, we focus on some new trends for solving nonlinear complementarity problems. Extensions to mixed complementarity problems, variational inequalities and mathematical programs with equilibrium constraints are also discussed.
Smoothing Functions and a Smoothing Newton Method for Complementarity and Variational Inequality Problems
 JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
, 1998
"... In this paper, we discuss smoothing approximations of nonsmooth functions arising from complementarity and variational inequality problems. We present some new results which are essential in designing Newtontype methods. We introduce several new classes of smoothing functions for nonlinear compleme ..."
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Cited by 13 (3 self)
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In this paper, we discuss smoothing approximations of nonsmooth functions arising from complementarity and variational inequality problems. We present some new results which are essential in designing Newtontype methods. We introduce several new classes of smoothing functions for nonlinear complementarity problems and order complementarity problems. In particular, in the first time some computable smoothing functions for variational inequality problems with general constraints are introduced. Then we propose a new version of smoothing Newton methods and establish its global and superlinear (quadratical) convergence under conditions weaker than those in the literature.