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29
Semismooth Matrix Valued Functions
, 1999
"... . Matrix valued functions play an important role in the development of algorithms for semidefinite programming problems. This paper studies generalized di#erential properties of such functions related to nonsmoothsmoothing Newton methods. The first part of this paper discusses basic properties such ..."
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Cited by 44 (22 self)
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. Matrix valued functions play an important role in the development of algorithms for semidefinite programming problems. This paper studies generalized di#erential properties of such functions related to nonsmoothsmoothing Newton methods. The first part of this paper discusses basic properties such as the generalized derivative, Rademacher's theorem, Bderivative, directional derivative, and semismoothness. The second part shows that the matrix absolutevalue function, the matrix semidefiniteprojection function, and the matrix projective residual function are strongly semismooth. Keywords: Matrix functions, Newton's method, nonsmooth optimization, semidefinite programming. AMS subject classification: 65K05, 90C25, 90C33. 1 The research was partially supported by the Australian Research Council and grant RP3972073 of National University of Singapore. 2 Email: sun@maths.unsw.edu.au. 3 Fax: (65)7792621 Email: jsun@nus.edu.sg. 1 Introduction Let Mmn and M pq be the spaces ...
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...
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)...
On HomotopySmoothing Methods for Variational Inequalities
"... A variational inequality problem with a mapping g : ! n ! ! n and lower and upper bounds on variables can be reformulated as a system of nonsmooth equations F (x) = 0 in ! n . Recently, several homotopy methods, such as interiorpoint and smoothing methods, have been employed to solve the prob ..."
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Cited by 23 (5 self)
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A variational inequality problem with a mapping g : ! n ! ! n and lower and upper bounds on variables can be reformulated as a system of nonsmooth equations F (x) = 0 in ! n . Recently, several homotopy methods, such as interiorpoint and smoothing methods, have been employed to solve the problem. All of these methods use parametric functions and construct perturbed equations to approximate the problem. The solution to the perturbed system constitutes a smooth trajectory leading to the solution of the original variational inequality problem. The methods generate iterates to follow the trajectory. Among these methods ChenMangasarian and GabrielMor'e proposed a class of smooth functions to approximate F . In this paper, we study several properties of the trajectory defined by solutions of these smooth systems. We propose a homotopysmoothing method for solving the variational inequality problem, and show that the method converges globally and superlinearly under mild conditions. ...
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. ...
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 20 (2 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 ...
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...
A NonInterior PredictorCorrector PathFollowing Method for LCP
 Mathematical Programming
, 1997
"... In a previous work the authors introduced a noninterior predictorcorrector path following algorithm for the monotone linear complementarity problem. The method uses ChenHarkerKanzowSmale smoothing techniques to track the central path and employs a refined notion for the neighborhood of the ..."
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Cited by 17 (1 self)
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In a previous work the authors introduced a noninterior predictorcorrector path following algorithm for the monotone linear complementarity problem. The method uses ChenHarkerKanzowSmale smoothing techniques to track the central path and employs a refined notion for the neighborhood of the central path to obtain the boundedness of the iterates under the assumption of monotonicity and the existence of a feasible interior point. With these assumptions, the method is shown to be both globally linearly convergent and locally quadratically convergent. In this paper it is shown that this basic approach is still valid without the monotonicity assumption and regardless of the choice of norm in the definition of the neighborhood of the central path. Furthermore, it is shown that the method can be modified so that only one system of linear equations need to be solved at each iteration without sacrificing either the global or local convergence behavior of the method. The local behavior o...