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89
A Semismooth Equation Approach To The Solution Of Nonlinear Complementarity Problems
, 1995
"... In this paper we present a new algorithm for the solution of nonlinear complementarity problems. The algorithm is based on a semismooth equation reformulation of the complementarity problem. We exploit the recent extension of Newton's method to semismooth systems of equations and the fact that the n ..."
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Cited by 79 (9 self)
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In this paper we present a new algorithm for the solution of nonlinear complementarity problems. The algorithm is based on a semismooth equation reformulation of the complementarity problem. We exploit the recent extension of Newton's method to semismooth systems of equations and the fact that the natural merit function associated to the equation reformulation is continuously differentiable to develop an algorithm whose global and quadratic convergence properties can be established under very mild assumptions. Other interesting features of the new algorithm are an extreme simplicity along with a low computational burden per iteration. We include numerical tests which show the viability of the approach.
A New Merit Function For Nonlinear Complementarity Problems And A Related Algorithm
, 1997
"... We investigate the properties of a new merit function which allows us to reduce a nonlinear complementarity problem to an unconstrained global minimization one. Assuming that the complementarity problem is defined by a P 0 function we prove that every stationary point of the unconstrained problem ..."
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Cited by 73 (6 self)
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We investigate the properties of a new merit function which allows us to reduce a nonlinear complementarity problem to an unconstrained global minimization one. Assuming that the complementarity problem is defined by a P 0 function we prove that every stationary point of the unconstrained problem is a global solution; furthermore, if the complementarity problem is defined by a uniform P function, the level sets of the merit function are bounded. The properties of the new merit function are compared with those of the MangasarianSolodov's implicit Lagrangian and Fukushima's regularized gap function. We also introduce a new, simple, activeset local method for the solution of complementarity problems and show how this local algorithm can be made globally convergent by using the new merit function.
T.S.: Interior point methods for massive support vector machines
 Data Mining Institute, Computer Sciences Department, University of Wisconsin
, 2000
"... Abstract. We investigate the use of interiorpoint methods for solving quadratic programming problems with a small number of linear constraints, where the quadratic term consists of a lowrank update to a positive semidefinite matrix. Several formulations of the support vector machine fit into this ..."
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Cited by 43 (1 self)
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Abstract. We investigate the use of interiorpoint methods for solving quadratic programming problems with a small number of linear constraints, where the quadratic term consists of a lowrank update to a positive semidefinite matrix. Several formulations of the support vector machine fit into this category. An interesting feature of these particular problems is the volume of data, which can lead to quadratic programs with between 10 and 100 million variables and, if written explicitly, a dense Q matrix. Our code is based on OOQP, an objectoriented interiorpoint code, with the linear algebra specialized for the support vector machine application. For the targeted massive problems, all of the data is stored out of core and we overlap computation and input/output to reduce overhead. Results are reported for several linear support vector machine formulations demonstrating that the method is reliable and scalable. Key words. support vector machine, interiorpoint method, linear algebra AMS subject classifications. 90C51, 90C20, 62H30 PII. S1052623400374379 1. Introduction. Interiorpoint methods [30] are frequently used to solve large convex quadratic and linear programs for two reasons. First, the number of iterations
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.
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 41 (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
"... ..."
A New Nonsmooth Equations Approach To Nonlinear Complementarity Problems
, 1997
"... Based on Fischer's function, a new nonsmooth equations approach is presented for solving nonlinear complementarity problems. Under some suitable assumptions, a local and Qquadratic convergence result is established for the generalized Newton method applied to the system of nonsmooth equations, whic ..."
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Cited by 39 (6 self)
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Based on Fischer's function, a new nonsmooth equations approach is presented for solving nonlinear complementarity problems. Under some suitable assumptions, a local and Qquadratic convergence result is established for the generalized Newton method applied to the system of nonsmooth equations, which is a reformulation of nonlinear complementarity problems. To globalize the generalized Newton method, a hybrid method combining the generalized Newton method with the steepest descent method is proposed. Global and Qquadratic convergence is established for this hybrid method. Some numerical results are also reported.
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.
A Semismooth Newton Method For Variational Inequalities: Theoretical Results And Preliminary Numerical Experience
, 1997
"... Variational inequalities over sets defined by systems of equalities and inequalities are considered. A continuously differentiable merit function is proposed whose unconstrained minima coincide with the KKTpoints of the variational inequality. A detailed study of its properties is carried out showi ..."
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Cited by 33 (10 self)
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Variational inequalities over sets defined by systems of equalities and inequalities are considered. A continuously differentiable merit function is proposed whose unconstrained minima coincide with the KKTpoints of the variational inequality. A detailed study of its properties is carried out showing that under mild assumptions this reformulation possesses many desirable features. A simple algorithm is proposed for which it is possible to prove global convergence and a fast local convergence rate. Preliminary numerical results showing viability of the approach are reported.
Semismooth Newton methods for operator equations in function spaces
, 2000
"... We develop a semismoothness concept for nonsmooth superposition operators in function spaces. The considered class of operators includes NCPfunctionbased reformulations of infinitedimensional nonlinear complementarity problems, and thus covers a very comprehensive class of applications. Our resul ..."
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Cited by 29 (3 self)
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We develop a semismoothness concept for nonsmooth superposition operators in function spaces. The considered class of operators includes NCPfunctionbased reformulations of infinitedimensional nonlinear complementarity problems, and thus covers a very comprehensive class of applications. Our results generalize semismoothness and fforder semismoothness from finitedimensional spaces to a Banach space setting. Hereby, a new generalized differential is used that can be seen as an extension of Qi's finitedimensional Csubdifferential to our infinitedimensional framework. We apply these semismoothness results to develop a Newtonlike method for nonsmooth operator equations and prove its local qsuperlinear convergence to regular solutions. If the underlying operator is fforder semismoothness, convergence of qorder 1 + ff is proved. We also establish the semismoothness of composite operators and develop corresponding chain rules. The developed theory is accompanied by illustrating e...