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27
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
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Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities
 Mathematics of Computation
, 1998
"... Abstract. The smoothing Newton method for solving a system of nonsmooth equations F (x) = 0, which may arise from the nonlinear complementarity problem, the variational inequality problem or other problems, can be regarded as a variant of the smoothing method. At the kth step, the nonsmooth functio ..."
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Cited by 34 (16 self)
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Abstract. The smoothing Newton method for solving a system of nonsmooth equations F (x) = 0, which may arise from the nonlinear complementarity problem, the variational inequality problem or other problems, can be regarded as a variant of the smoothing method. At the kth step, the nonsmooth function F is approximated by a smooth function f(·,εk), and the derivative of f(·,εk) at x k is used as the Newton iterative matrix. The merits of smoothing methods and smoothing Newton methods are global convergence and convenience in handling. In this paper, we show that the smoothing Newton method is also superlinearly convergent if F is semismooth at the solution and f satisfies a Jacobian consistency property. We show that most common smooth functions, such as the GabrielMoré function, have this property. As an application, we show that for box constrained variational inequalities if the involved function is P –uniform, the iteration sequence generated by the smoothing Newton method will converge to the unique solution of the problem globally and superlinearly (quadratically). 1.
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...
New NCPFunctions and Their Properties
, 1997
"... . Recently, Luo and Tseng proposed a class of merit functions for the nonlinear complementarity problem (NCP) and showed that it enjoys several interesting properties under some assumptions. In this paper, adopting a similar idea to Luo and Tseng's, we present new merit functions for the NCP, which ..."
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Cited by 25 (11 self)
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. Recently, Luo and Tseng proposed a class of merit functions for the nonlinear complementarity problem (NCP) and showed that it enjoys several interesting properties under some assumptions. In this paper, adopting a similar idea to Luo and Tseng's, we present new merit functions for the NCP, which can be decomposed into component functions. We show that these merit functions not only share many properties with the one proposed by Luo and Tseng but also enjoy additional favorable properties owing to their decomposable structure. In particular, we present fairly mild conditions under which these merit functions have bounded level sets. Key words: Nonlinear complementarity problem, NCPfunction, merit function, unconstrained optimization reformulation, error bound, bounded level sets. 3 The work of the second and third authors was supported in part by the Scienti c Research GrantinAid from the Ministry of Education, Science and Culture, Japan. The work of the second author was also su...
A Theoretical And Numerical Comparison Of Some Semismooth Algorithms For Complementarity Problems
, 1997
"... In this paper we introduce a general line search scheme which easily allows us to define and analyze known and new semismooth algorithms for the solution of nonlinear complementarity problems. We enucleate the basic assumptions that a search direction to be used in the general scheme has to enjoy i ..."
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Cited by 22 (3 self)
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In this paper we introduce a general line search scheme which easily allows us to define and analyze known and new semismooth algorithms for the solution of nonlinear complementarity problems. We enucleate the basic assumptions that a search direction to be used in the general scheme has to enjoy in order to guarantee global convergence, local superlinear/quadratic convergence or finite convergence. We examine in detail several different semismooth algorithms and compare their theoretical features and their practical behavior on a set of largescale problems.
A truly globally convergent Newtontype method for the monotone nonlinear complementarity problem
 SIAM Journal on Optimization
"... Abstract. The Josephy–Newton method for solving a nonlinear complementarity problem consists of solving, possibly inexactly, a sequence of linear complementarity problems. Under appropriate regularity assumptions, this method is known to be locally (superlinearly) convergent. To enlarge the domain o ..."
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Cited by 17 (15 self)
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Abstract. The Josephy–Newton method for solving a nonlinear complementarity problem consists of solving, possibly inexactly, a sequence of linear complementarity problems. Under appropriate regularity assumptions, this method is known to be locally (superlinearly) convergent. To enlarge the domain of convergence of the Newton method, some globalization strategy based on a chosen merit function is typically used. However, to ensure global convergence to a solution, some additional restrictive assumptions are needed. These assumptions imply boundedness of level sets of the merit function and often even (global) uniqueness of the solution. We present a new globalization strategy for monotone problems which is not based on any merit function. Our linesearch procedure utilizes the regularized Newton direction and the monotonicity structure of the problem to force global convergence by means of a (computationally explicit) projection step which reduces the distance to the solution set of the problem. The resulting algorithm is truly globally convergent in the sense that the subproblems are always solvable, and the whole sequence of iterates converges to a solution of the problem without any regularity assumptions. In fact, the solution set can even be unbounded. Each iteration of the new method has the same order of computational cost as an iteration of the damped Newton method. Under natural assumptions, the local superlinear rate of convergence is also achieved. Key words. nonlinear complementarity problem, Newton method, proximal point method, projection method, global convergence, superlinear convergence
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 New Merit Function and a Descent Method for Semidefinite Complementarity Problems
, 1997
"... Recently, Tseng extended several merit functions for the nonlinear complementarity problem to the semidefinite complementarity problem (SDCP) and investigated various properties of those functions. In this paper, we propose a new merit function for the SDCP based on the squared FischerBurmeister ..."
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Cited by 15 (3 self)
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Recently, Tseng extended several merit functions for the nonlinear complementarity problem to the semidefinite complementarity problem (SDCP) and investigated various properties of those functions. In this paper, we propose a new merit function for the SDCP based on the squared FischerBurmeister function and show that it has some favorable properties. Particularly, we give conditions under which the function provides a global error bound for the SDCP and conditions under which it has bounded level sets. We also present a derivativefree method for solving the SDCP and prove its global convergence under suitable assumptions.
A new unconstrained differentiable merit function for box constrained variational inequality problems and a damped GaussNewton method
 Applied Mathematics Report AMR 96/37, School of Mathematics, the University of New South
, 1996
"... Abstract. In this paper we propose a new unconstrained differentiable merit function f for box constrained variational inequality problems VIP(l, u, F). We study various desirable properties of this new merit function f and propose a Gauss–Newton method in which each step requires only the solution ..."
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Cited by 15 (8 self)
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Abstract. In this paper we propose a new unconstrained differentiable merit function f for box constrained variational inequality problems VIP(l, u, F). We study various desirable properties of this new merit function f and propose a Gauss–Newton method in which each step requires only the solution of a system of linear equations. Global and superlinear convergence results for VIP(l, u, F) are obtained. Key results are the boundedness of the level sets of the merit function for any uniform Pfunction and the superlinear convergence of the algorithm without a nondegeneracy assumption. Numerical experiments confirm the good theoretical properties of the method.