On the identification of degenerate indices in the nonlinear complementarity problem with the proximal point algorithm

Yamashita, Nobuo; Dan, Hiroshige; Fukushima, Masao

doi:10.1007/s10107-003-0455-x

Cited by 15 publications

(13 citation statements)

References 18 publications

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“…For monotone nonlinear complementarity problems, Yamashita, Dan, and Fukusmima [24] describe a technique for classifying indices (including degenerate indices) at the limit point of a proximal point algorithm. This threshold is defined similarly to the one in [10], while the classification test is similar to that of [18].…”

Section: Related Workmentioning

confidence: 99%

Active Set Identification in Nonlinear Programming

Oberlin

Wright

2006

SIAM J. Optim.

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Abstract. Techniques that identify the active constraints at a solution of a nonlinear programming problem from a point near the solution can be a useful adjunct to nonlinear programming algorithms. They have the potential to improve the local convergence behavior of these algorithms, and in the best case can reduce an inequality constrained problem to an equality constrained problem with the same solution. This paper describes several techniques that do not require good Lagrange multiplier estimates for the constraints to be available a priori, but depend only on function and first derivative information. Computational tests comparing the effectiveness of these techniques on a variety of test problems are described. Many tests involve degenerate cases, in which the constraint gradients are not linearly independent and/or strict complementarity does not hold.

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Section: Related Workmentioning

confidence: 99%

Active Set Identification in Nonlinear Programming

Oberlin

Wright

2006

SIAM J. Optim.

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“…Clearly, x * ∈ R n solves the NCP(F) if and only if it is a solution of the equation (2). The nature merit function for (2) is Ψ(x) := 1 2 ∥Φ(x)∥ 2 .…”

Section: Introductionmentioning

confidence: 99%

“…From both theoretical and practical point of view, the identification of the degenerate set β of the solution is very important. If the degenerate indices can be identified before exactly knowing x * , then we only have to solve a reduced form of the equation (2) in a small neighbor of x * . The original degenerate NCP(F) will also be transformed to a non-degenerate problem.…”

Section: Introductionmentioning

confidence: 99%

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A Semismooth Active-set Algorithm for Degenerate Nonlinear Complementarity Problems

Yu¹

2013

JSW

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We propose a semismooth active-set Newton algorithm for solving the nonlinear complementarity problems with degenerate solutions. This method introduces the active-set technique to identify the degenerate set. At each iteration, the search direction is obtained by two reduced linear systems. Instead of employing gradient steps as adjustments to guarantee the sufficient reduction of the merit function, the algorithm employs a Newton-type direction, which is more efficient than gradient direction, in the adjustment step. This method has globally convergence. When near the solution, the degenerate set will be identified correctly, and only one reduced linear system is solved at each iteration. Under some mild assumptions, locally superlinear convergence is obtained as well. Numerical experiments on MATLAB shows the efficiency of the method

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“…Active set identification plays an important role in optimization theory [1,2,5,6,25]. Accurate identification of active constraints is important from both theoretical and practical points of view.…”

Section: A Hybrid Algorithm For Mpccmentioning

confidence: 99%

Hybrid Approach with Active Set Identification for Mathematical Programs with Complementarity Constraints

Lin

Fukushima

2006

J Optim Theory Appl

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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.

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On the identification of degenerate indices in the nonlinear complementarity problem with the proximal point algorithm

Cited by 15 publications

References 18 publications

Active Set Identification in Nonlinear Programming

Active Set Identification in Nonlinear Programming

A Semismooth Active-set Algorithm for Degenerate Nonlinear Complementarity Problems

Hybrid Approach with Active Set Identification for Mathematical Programs with Complementarity Constraints

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