任詠紅，徐志敏，張曉有

（遼寧師范大學(xué) 數(shù)學(xué)學(xué)院，遼寧 大連 116029）

一個(gè)基于NCP函數(shù)的非線性Lagrange函數(shù)

任詠紅，徐志敏，張曉有

（遼寧師范大學(xué) 數(shù)學(xué)學(xué)院，遼寧 大連 116029）

基于修正的Fischer-Burmeister NCP函數(shù)，提出了一個(gè)求解具有不等式約束的非線性優(yōu)化問(wèn)題的非線性Lagrange函數(shù)，討論了該函數(shù)在K-T點(diǎn)處的性質(zhì).收斂定理表明，在適當(dāng)?shù)臈l件下，當(dāng)懲罰參數(shù)小于某一閾值時(shí)，基于該非線性Lagrange函數(shù)的算法產(chǎn)生的點(diǎn)列具有局部收斂性.

非線性優(yōu)化；非線性Lagrange函數(shù)；NCP函數(shù)；收斂性

1 引言

考慮具有不等式約束的非線性優(yōu)化問(wèn)題其中 x∈IRn,fi(x)∶IRn→IR1,i=0,…,m 是實(shí)值函數(shù).

近年來(lái)，求解問(wèn)題（1）的非線性Lagrange方法倍受國(guó)內(nèi)外學(xué)者的關(guān)注.由于非線性Lagrange函數(shù)可用于發(fā)展非線性規(guī)劃問(wèn)題的對(duì)偶算法，該算法對(duì)原始變量的可行性沒(méi)有限制，因此，非線性La?grange函數(shù)的構(gòu)造方法成為研究熱點(diǎn)之一，迄今為止，已出現(xiàn)許多有效的非線性Lagrange函數(shù)，具有代表性的工作參見(jiàn)文獻(xiàn)[1-3].

值得注意的是，極小NCP函數(shù)通過(guò)積分運(yùn)算

2 一個(gè)基于NCP函數(shù)的非線性Lagrange函數(shù)

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[2]Polyak R A.Modified barrier function:theory and meth?ods.Mathematical Programming[J].1992,54(2):177-222.

[3]Polyak R A.Log-Sigmoid multipliers method in con?strained optimization[J].Annals of operations Research,2001,101:427-460.

[4]Ren Y H,Zhang L W,Xiao X T.A nonlinear Lagrangian based on Fischer-Burmeister NCP function[J].Applied Mathematics and Computation,2007(188):1344-1363.

[5]Kanzow C,Kleinmichel H.A new class of semismooth Newton method for nonlinear complementarity problems[J].Comput Optim Appl,1998,11:227-251.

A Nonlinear Lagrangian Based on NCP Function

REN Yonghong，XU Zhimin，ZHANG Xiaoyou
（School of Mathematics，Liaoning Normal University，Dalian116029，China）

This paper proposes a nonlinear Lagrangian based on a modified Fischer-Burmeister NCP function for solv?ing nonlinear optimization problem with inequality constraints.Properties of proposed nonlinear Lagrangian at K-T point are discussed.The convergence theorem shows that the sequence of points generated by nonlinear Lagrange algorithm is locally convergent when the penalty parameter is less than a threshold under a set of suitable conditions.

Nonlinear Optimization；Nonlinear Lagrangian；NCP Function；Convergence

O 41

A

1674-4942（2011）04-0365-05

2011-09-18

遼寧省博士科研啟動(dòng)基金項(xiàng)目（20091046）

畢和平