Results 1  10
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14
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
"... ..."
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...
Global and Local Superlinear Convergence Analysis of NewtonType Methods for Semismooth Equations with Smooth Least Squares
 in Reformulation  Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods
, 1998
"... : The local superlinear convergence of the generalized Newton method for solving systems of nonsmooth equations has been proved by Qi and Sun under the semismooth condition and nonsingularity of the generalized Jacobian at the solution. Unlike the Newton method for systems of smooth equations, globa ..."
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Cited by 16 (0 self)
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: The local superlinear convergence of the generalized Newton method for solving systems of nonsmooth equations has been proved by Qi and Sun under the semismooth condition and nonsingularity of the generalized Jacobian at the solution. Unlike the Newton method for systems of smooth equations, globalization of the generalized Newton method seems difficult to achieve in general. However, we show that global convergence analysis of various traditional Newtontype methods for systems of smooth equations can be extended to systems of nonsmooth equations with semismooth operators whose least squares objective is smooth. The value of these methods is demonstrated from their applications to various semismooth equation reformulations of nonlinear complementarity and related problems. AMS (MOS) Subject Classifications. 90C30, 90C33. Key Words. Nonsmooth equation, semismooth operator, Newton's method, GaussNewton method, global convergence, superlinear convergence, complementarity problem. 1 ...
A Regularization Newton Method for Solving Nonlinear Complementarity Problems
 APPL. MATH. OPTIM
, 1998
"... In this paper we construct a regularization Newton method for solving the nonlinear complementarity problem (NCP(F )) and analyze its convergence properties under the assumption that F is a P 0 function. We prove that every accumulation point of the sequence of iterates is a solution of NCP(F ) an ..."
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Cited by 15 (7 self)
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In this paper we construct a regularization Newton method for solving the nonlinear complementarity problem (NCP(F )) and analyze its convergence properties under the assumption that F is a P 0 function. We prove that every accumulation point of the sequence of iterates is a solution of NCP(F ) and that the sequence of iterates is bounded if the solution set of NCP(F ) is nonempty and bounded. Moreover, if F is a monotone and Lipschitz continuous function, we prove that the sequence of iterates is bounded if and only if the solution set of NCP(F ) is nonempty by setting t = 1 2 , where t 2 [ 1 2 ; 1] is a parameter. If NCP(F ) has a locally unique solution and satisfies a nonsingularity condition, then the convergence rate is superlinear (quadratic) without strict complementarity conditions. At each step, we only solve a linear system of equations. Numerical results are provided and further applications to other problems are discussed.
On NCPfunctions
 Computational Optimization and Applications
, 1999
"... Abstract. In this paper we reformulate several NCPfunctions for the nonlinear complementarity problem (NCP) from their merit function forms and study some important properties of these NCPfunctions. We point out that some of these NCPfunctions have all the nice properties investigated by Chen, Ch ..."
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Cited by 12 (2 self)
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Abstract. In this paper we reformulate several NCPfunctions for the nonlinear complementarity problem (NCP) from their merit function forms and study some important properties of these NCPfunctions. We point out that some of these NCPfunctions have all the nice properties investigated by Chen, Chen and Kanzow [2] for a modified FischerBurmeister function, while some other NCPfunctions may lose one or several of these properties. We also provide a modified normal map and a smoothing technique to overcome the limitation of these NCPfunctions. A numerical comparison for the behaviour of various NCPfunctions is provided.
A survey of some nonsmooth equations and smoothing Newton methods
 Progress in Optimization, volume 30 of Applied Optimization
, 1999
"... In this article we review and summarize recent developments on nonsmooth equations and smoothing Newton methods. Several new suggestions are presented. 1 ..."
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Cited by 7 (2 self)
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In this article we review and summarize recent developments on nonsmooth equations and smoothing Newton methods. Several new suggestions are presented. 1
Smoothing Newton and QuasiNewton Methods for Mixed Complementarity Problems
 Comput. Optim. Appl
, 1999
"... The mixed complementarity problem can be reformulated as a nonsmooth equation by using the median operator. In this paper, we first study some useful properties of this reformulation and then derive the ChenHarkerKanzowSmale smoothing function for the mixed complementarity problem. On the basis o ..."
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Cited by 6 (3 self)
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The mixed complementarity problem can be reformulated as a nonsmooth equation by using the median operator. In this paper, we first study some useful properties of this reformulation and then derive the ChenHarkerKanzowSmale smoothing function for the mixed complementarity problem. On the basis of this smoothing function, we present a smoothing Newton method for solving the mixed complementarity problem. The smoothing Newton method converges globally if the problem involves a differentiable P 0 function. Under suitable conditions, the method exhibits a quadratic convergence property. We also present a smoothing Broydenlike method based on the same smoothing function. Under appropriate conditions, the method converges globally and superlinearly. Key words: Mixed complementarity problem, smoothing function, Newton's method, quasiNewton method 1 Present address (available until October, 1999): Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto Unive...
Nonsmooth Equations and Smoothing Newton Methods
 Applied Mathematics Report AMR 98/10 , School of Mathematics, the University of New South
, 1998
"... In this article we review and summarize recent developments on nonsmooth equations and smoothing Newton methods. Several new suggestions are presented. 1 Introduction Suppose that H : ! n ! ! n is locally Lipschitz but not necessarily continuously differentiable. To solve H(x) = 0 (1.1) has be ..."
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Cited by 3 (2 self)
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In this article we review and summarize recent developments on nonsmooth equations and smoothing Newton methods. Several new suggestions are presented. 1 Introduction Suppose that H : ! n ! ! n is locally Lipschitz but not necessarily continuously differentiable. To solve H(x) = 0 (1.1) has become one of most active research directions in mathematical programming. The early study of nonsmooth equations can be traced back to [Eav71, Man75, Man76]. The system of nonsmooth equations arises from many applications. Pang and Qi [PaQ93] reviewed eight problems in the studies of complementarity problems, variational inequality problems and optimization problems, which can be reformulated as systems of nonsmooth equations. In this paper, we review recent developments of algorithms for solving nonsmooth equations. Section 2 is devoted to semismooth Newton methods and Section 3 discusses smoothing Newton methods. We make several final remarks in Section 4. 2 Semismooth Newton methods 2.1 L...