A New Smoothing Technique for Mathematical Programs With Equilibrium Constraints

ZHU Zhi-bin; LUO Zhi-jun; ZENG Ji-wen

Volume 28 Issue 10

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Article Navigation > Applied Mathematics and Mechanics > 2007 > 28(10): 1253-1260

ZHU Zhi-bin, LUO Zhi-jun, ZENG Ji-wen. A New Smoothing Technique for Mathematical Programs With Equilibrium Constraints[J]. Applied Mathematics and Mechanics, 2007, 28(10): 1253-1260.

Citation:

ZHU Zhi-bin, LUO Zhi-jun, ZENG Ji-wen. A New Smoothing Technique for Mathematical Programs With Equilibrium Constraints[J]. Applied Mathematics and Mechanics, 2007, 28(10): 1253-1260.

Citation:

ZHU Zhi-bin, LUO Zhi-jun, ZENG Ji-wen. A New Smoothing Technique for Mathematical Programs With Equilibrium Constraints[J]. Applied Mathematics and Mechanics, 2007, 28(10): 1253-1260.

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A New Smoothing Technique for Mathematical Programs With Equilibrium Constraints

1.
Department of Computational Science and Mathematics, Guilin University of Electronic Technology, Guilin 541004, P. R. China;

Received Date: 2006-07-05
Rev Recd Date: 2007-07-25
Publish Date: 2007-10-15

Abstract

Abstract

A kind of mathematical programs with equilibrium constraints(MPEC) is studied.By using the idea of successive approximation,a smoothing nonlinear programming,which is equivalent to the MPEC problem,was proposed.Thereby,it is ensured that some classical optimization methods can be applied for the MPEC problem.In the end,two algorithm models were proposed with the detailed analysis of the global convergence.

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References(9)

References

[1]	Outrata J, Zowe J.A numerical approach to optimization problems with variational inequality constraints[J].Mathematical Programming,1995,68(1):105-130.
[2]	Luo Z Q,Pang J S,Ralph D,et al.Exact penalization and stationarity conditions of mathematical programs with equilibrium constraints[J].Mathematical Programming,1996,75(1)：19-76.
[3]	Outrata J, Kocvare M,Zowe J.Nonsmooth Approach to Optimization Problems With Equilibrium Consraints[M].Netherlands: Kluwer Academic Publishers, 1998.
[4]	Facchinei F, Jiang H Y,Qi L.A smoothing method for mathematical programs with equilibjum constraints[J].Mathematical Programming,1999,85(1)：107-134. doi: 10.1007/s101070050048
[5]	李飞,徐成贤.求解带均衡约束数学规划问题的一个连续化方法[J].计算数学,2004,26(1)：3-12.
[6]	Fukushima M, Luo Z Q,Pang J S.A globally convergent sequential quadratic programming algorithm for mathematical programs with linear complementarity constraints[J].Computational Optimization and Applications,1998,10(1)：5-34. doi: 10.1023/A:1018359900133
[7]	Qi L, Chen X J. A globally convergent successive approximation methods for non-smooth equation[J].SIAM Journal on Control and Optimization,1995,33(3)：402-418. doi: 10.1137/S036301299223619X
[8]	Ma C F,Liang G P.A new successive approximation damped Newton method for nonlinear complementarity problems[J].Journal of Mathematical Research and Exposition,2003,23(1)：1-6.
[9]	Luo Z Q, Pang J S, Ralph D.Mathematical Programs With Equilibrium Constraints[M].Cambridge：Cambridge University Press,1996.

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