An Accelerated Smoothing Newton Method with Cubic Convergence for Weighted Complementarity Problems

Tang, Jingyong; Zhou, Jinchuan; Zhang, Hongchao

doi:10.1007/s10957-022-02152-6

Cited by 3 publications

(6 citation statements)

References 37 publications

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“…In our tests, we set n = 2m, δ = 10 −6 , ρ = 0.6, γ = 0.01, l = 0.1, c = 0.01, σ 1 = σ 2 = 0.001, ξ 0 = 0.1, and η k = 1/2 k+2 . Furthermore, we use the algorithm in [23], denoted as ANM_Tang, and that in [24], denoted as TSN_Liu, and compare them to SNQ_L. We utilize the same parameters for ANM_Tang and TSN_Liu as those in [23,24], respectively.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…The two-step Newton approach for solving nonlinear equations has recently been used in an attempt to solve WCPs. Tang et al [23] present a smoothing Newton approach to solve WCPs with local cubic convergence rates. The approach accelerates the convergence rate by adding an approximate Newton step, as the sequence of iterations is close to the solution, raising the local convergence rate from second-order to third-order.…”

Section: Introductionmentioning

confidence: 99%

“…The algorithm computes two Newton equations directly to obtain the next iteration point, in contrast to the algorithm in [23]. If the value of the objective function meets a certain descent criterion, the algorithm takes the iteration point produced by the two Newton directions directly as the next iteration point; otherwise, the step size is confirmed via a derivative-free line search to find the next iteration point.…”

mentioning

confidence: 99%

“…In Step 2, the additional term χ k = min{1, ξ 4 k } is added to Newton Equations ( 11) and ( 12), in contrast to the current smoothing Newton methods [17,23,30]. The property of local biquadratic convergence of Algorithm 1 depends on this particular perturbation term.…”

mentioning

confidence: 99%

“…The main distinction between Algorithm 1 and the accelerated algorithm in [23] as two-step Newton algorithms is that Algorithm 1 employs two Newton directions from the beginning, whereas the accelerated algorithm in [23] begins with one Newton direction and adds a second Newton direction when certain conditions are met. We also note that Algorithm 1 is able to solve the WCP better than the accelerated method in the following section of numerical experiments.…”

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confidence: 99%

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A Two-Step Newton Algorithm for the Weighted Complementarity Problem with Local Biquadratic Convergence

Liu,

Zhang

2023

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We discuss the weighted complementarity problem, extending the nonlinear complementarity problem on Rn. In contrast to the NCP, many equilibrium problems in science, engineering, and economics can be transformed into WCPs for more efficient methods. Smoothing Newton algorithms, known for their at least locally superlinear convergence properties, have been widely applied to solve WCPs. We suggest a two-step Newton approach with a local biquadratic order convergence rate to solve the WCP. The new method needs to calculate two Newton equations at each iteration. We also insert a new term, which is of crucial importance for the local biquadratic convergence properties when solving the Newton equation. We demonstrate that the solution to the WCP is the accumulation point of the iterative sequence produced by the approach. We further demonstrate that the algorithm possesses local biquadratic convergence properties. Numerical results indicate the method to be practical and efficient.

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