A Non-Interior-Point Continuation Method for Linear Complementarity Problems

Chen, Bintong; Harker, Patrick T.

doi:10.1137/0614081

Cited by 219 publications

(108 citation statements)

References 9 publications

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“…Therefore the de nitions ofp(x; ) given by (3) and (4) Thenp(x; ) is strictly increasing and strictly convex in x for a xed > 0. (4) Proof (1) By equation (6) in the proof of last proposition, the conclusion follows. Now we have a systematic way for generating a class of smooth plus functions.…”

Section: A Class Of Smoothing Functionsmentioning

confidence: 71%

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A class of smoothing functions for nonlinear and mixed complementarity problems

Chen

Mangasarian

1996

Comput Optim Applic

436

260

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We propose a class of parametric smooth functions that approximate the fundamental plus function, (x) + =maxf0; xg, by twice integrating a probability density function. This leads to classes of smooth parametric nonlinear equation approximations of nonlinear and mixed complementarity problems (NCPs and MCPs). For any solvable NCP or MCP, existence of an arbitrarily accurate solution to the smooth nonlinear equation as well as the NCP or MCP, is established for su ciently large value of a smoothing parameter . Newton-based algorithms are proposed for the smooth problem. For strongly monotone NCPs, global convergence and local quadratic convergence are established. For solvable monotone NCPs, each accumulation point of the proposed algorithms solves the smooth problem. Exact solutions of our smooth nonlinear equation for various values of the parameter , generate an interior path, which is di erent from the central path for interior point method. Computational results for 52 test problems compare favorably with those for another Newton-based method.

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Section: A Class Of Smoothing Functionsmentioning

confidence: 71%

“…Let the density function d(x) satisfy (A1)-(A3) and D 2 = 0, and let letp(x; ) be de ned by (4). If x solves the nonlinear equation (10) The unique solution is (1; 0).…”

Section: Relation To the Interior Point Methodsmentioning

confidence: 99%

A class of smoothing functions for nonlinear and mixed complementarity problems

Chen

Mangasarian

1996

Comput Optim Applic

436

260

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“…We make use of a trust region algorithm (see also [34]) to solve the constrained equilibrium problem. For the linear case a perturbed method was proposed by Chen and Harker [35] where the orthogonality condition was relaxed. With smoothing methods this condition is also relaxed.…”

Section: Introduction To Cohesive Traction-separation Lawsmentioning

confidence: 99%

Quasi‐static crack propagation in plane and plate structures using set‐valued traction‐separation laws

Areias

Rabczuk

2007

Numerical Meth Engineering

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SUMMARYWe introduce a numerical technique to model set-valued traction-separation laws in plate bending and also plane crack propagation problems. By use of recent developments in thin (Kirchhoff-Love) shell models and the extended finite element method, a complete and accurate algorithm for the cohesive law is presented and is used to determine the crack path. The cohesive law includes softening and unloading to origin, adhesion and contact. Pure debonding and contact are obtained as particular (degenerate) cases. A smooth root finding algorithm (based on the trust region method) is adopted. A step-driven algorithm is described with a smoothed law which can be made arbitrarily close to the exact non smooth law. In the examples shown the results were found to be step-size insensitive and accurate. In addition, the method provides the crack advance law, extracted from the cohesive law and the absence of stress singularity at the tip.

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“…Certainly the worst-case external load depends on x. [14] ipCHKS(Y,l,P) Figure 2(b), where the width of each member is proportional to its cross-sectional area. Figure 3 It is observed from Figure 4 As a moderately large example, consider a 5-story 5-bay plane frame structure shown in Figure 5, where VV = 4m and H = 3m.…”

mentioning

confidence: 99%