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A Hybrid Algorithm with Active Set Identification for Mathematical Programs with Complementarity Constraints
, 2002
"... We consider a mathematical program with complementarity constraints (MPCC). Our purpose is to develop methods that enable us to compute a solution or a point with some kind of stationarity to MPCC by solving a finite number of nonlinear programs. To this end, we first introduce an active set identif ..."
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Cited by 6 (2 self)
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We consider a mathematical program with complementarity constraints (MPCC). Our purpose is to develop methods that enable us to compute a solution or a point with some kind of stationarity to MPCC by solving a finite number of nonlinear programs. To this end, we first introduce an active set identification technique. Then, by applying this technique to a smoothing continuation method presented by Fukushima and Pang (1999), we propose a hybrid method for solving MPCC. Under reasonable assumptions, the hybrid algorithm is shown to possess a finite termination property. Numerical experience shows that the proposed approach is quite e#ective.
Hybrid approach with active set identification for mathematical programs with complementarity constraints
 J. of Optimization Theory and Applications
"... Abstract. We consider a mathematical program with complementarity constraints (MPCC). Our purpose is to develop methods that enable us to compute a solution or a point with some kind of stationarity to MPCC by solving a finite number of nonlinear programs. We apply an active set identification techn ..."
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Abstract. We consider a mathematical program with complementarity constraints (MPCC). Our purpose is to develop methods that enable us to compute a solution or a point with some kind of stationarity to MPCC by solving a finite number of nonlinear programs. We apply an active set identification technique to a smoothing continuation method (Fukushima and Pang, 1999) and propose a hybrid algorithm for solving MPCC. We also develop two kinds of modifications, one of which makes use of an index addition strategy and the other adopts an index subtraction strategy. We show that, under reasonable assumptions, all the proposed algorithms possess a finite termination property. Further discussions and computational results are given as well. Key Words. mathematical program with complementarity constraints, MPCCLICQ, weak secondorder necessary condition, (B, M, C) stationarity, asymptotically weak nondegeneracy, identification function. 1
Numerical results for a globalized activeset Newton method for mixed complementarity problems
 COMPUTATIONAL AND APPLIED MATHEMATICS
, 2004
"... We discuss a globalization scheme for a class of activeset Newton methods for solving the mixed complementarity problem (MCP), which was proposed by the authors in [3]. The attractive features of the local phase of the method are that it requires solving only one system of linear equations per iter ..."
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We discuss a globalization scheme for a class of activeset Newton methods for solving the mixed complementarity problem (MCP), which was proposed by the authors in [3]. The attractive features of the local phase of the method are that it requires solving only one system of linear equations per iteration, yet the local superlinear convergence is guaranteed under extremely mild assumptions, in particular weaker than the property of semistability of an MCP solution. Thus the local superlinear convergence conditions of the method are weaker than conditions required for the semismooth (generalized) Newton methods and also weaker than convergence conditions of the linearization (Josephy–Newton) method. Numerical experiments on some test problems are presented, including results on the MCPLIB collection for the globalized version.
Copyright © 2005 SBMAC
"... www.scielo.br/cam Numerical results for a globalized activeset Newton method for mixed complementarity problems ..."
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www.scielo.br/cam Numerical results for a globalized activeset Newton method for mixed complementarity problems
An LPNewton Method: . . . KKT Systems, and Nonisolated Solutions
, 2011
"... We define a new Newtontype method for the solution of constrained systems of equations and analyze in detail its properties. Under suitable conditions, that do not include differentiability or local uniqueness of solutions, the method converges locally quadratically to a solution of the system of ..."
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We define a new Newtontype method for the solution of constrained systems of equations and analyze in detail its properties. Under suitable conditions, that do not include differentiability or local uniqueness of solutions, the method converges locally quadratically to a solution of the system of equations, thus filling an important gap in the existing theory. The new algorithm improves on known methods and, when particularized to KKT systems deriving from optimality conditions for constrained optimization or variational inequalities, it has theoretical advantages even over methods specifically designed to solve such systems.
Hybrid Algorithms with Index Addition and Subtraction Strategies for Solving Mathematical Programs with Complementarity Constraints
, 2003
"... Recently, the authors have presented a hybrid approach for solving mathematical programs with complementarity constraints. This approach employs an activeset identification function and possesses a finite termination property under some appropriate conditions including the socalled asymptotically ..."
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Recently, the authors have presented a hybrid approach for solving mathematical programs with complementarity constraints. This approach employs an activeset identification function and possesses a finite termination property under some appropriate conditions including the socalled asymptotically weak nondegeneracy. In this paper, we continue this work and propose two modified methods, one of which makes use of an index addition strategy and the other utilizes the converse strategy. Both the methods do not require the abovementioned assumption and also have a finite termination property under very weak assumptions. Numerical experience shows that the proposed approaches are effective.
Convergence Of The Iterates of Descent Methods . . .
 SIAM J. OPTIM
, 2005
"... In the early eighties Lojasiewicz [Loj84] proved that a bounded solution of a gradient flow for an analytic cost function converges to a welldefined limit point. In this paper, we show that the iterates of numerical descent algorithms, for an analytic cost function, share this convergence proper ..."
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In the early eighties Lojasiewicz [Loj84] proved that a bounded solution of a gradient flow for an analytic cost function converges to a welldefined limit point. In this paper, we show that the iterates of numerical descent algorithms, for an analytic cost function, share this convergence property if they satisfy certain natural descent conditions. The results obtained are applicable to a broad class of optimization schemes and strengthen classical "weak convergence" results for descent methods to "strong limitpoint convergence" for a large class of cost functions of practical interest. The result does not require that the cost has isolated critical points, requires no assumptions on the convexity of the cost, nor any nondegeneracy conditions on the Hessian of the cost at critical points.
ON THE IDENTIFICATION OF ZERO VARIABLES IN AN INTERIORPOINT FRAMEWORK
, 1998
"... Abstract. We consider column sucient linear complementarity problems and study the problem of identifying those variables that are zero at a solution. To this end we propose a new, computationally inexpensive technique that is based on growth functions. We analyze in detail the theoretical propert ..."
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Abstract. We consider column sucient linear complementarity problems and study the problem of identifying those variables that are zero at a solution. To this end we propose a new, computationally inexpensive technique that is based on growth functions. We analyze in detail the theoretical properties of the identication technique and test it numerically. The identication technique is particularly suited to interiorpoint methods but can be applied to a wider class of methods. Key words. Linear complementarity problem, column sucient matrix, identication of zero variables, growth function, indicator function, interiorpoint method. AMS subject classications. 90C05, 90C33, 65K05