李 陽(yáng),王一伊

(1.大連民族學(xué)院理學(xué)院,遼寧大連 116605;2.吉林大學(xué)數(shù)學(xué)學(xué)院,吉林長(zhǎng)春 130012)

一個(gè)用于構(gòu)造非線性半定規(guī)劃算法的非線性 Lagrange函數(shù)

李 陽(yáng)1,王一伊2

(1.大連民族學(xué)院理學(xué)院,遼寧大連 116605;2.吉林大學(xué)數(shù)學(xué)學(xué)院,吉林長(zhǎng)春 130012)

非線性Lagrange方法是求解非線性半定規(guī)劃的一個(gè)重要方法。給出了一個(gè)可以用于研究非線性半定規(guī)劃的非線性Lagrange函數(shù),并證明了這個(gè)函數(shù)在 KKT點(diǎn)附近具有凸性,從而保證了非線性 Lagrange算法理論成立的前提條件。

半定規(guī)劃;非線性 Lagrange函數(shù);KKT條件

本文所研究的非線性半定規(guī)劃問題的形式如下:

關(guān)于求解非線性規(guī)劃問題 (NLP),人們經(jīng)常使用非線性 Lagrange方法[2]。原因是即使在 KKT點(diǎn)的附近,經(jīng)典的線性 Lagrange函數(shù)通常很難保證凸性。這個(gè)不好的性質(zhì)給算法設(shè)計(jì)帶來了困難。尤其對(duì)于半定規(guī)劃而言,其二階最優(yōu)性條件比非線性規(guī)劃多了一個(gè)σ-項(xiàng),因此要找到一個(gè)有效的非線性Lagrange函數(shù)對(duì)于求解非線性半定規(guī)劃是有意義的工作。下面將證明,函數(shù) (2)就是我們所要得到的非線性 Lagrange函數(shù)。

1 問題的假設(shè)

這里給出本文的假設(shè)條件。問題 (1)的經(jīng)典Lagrange函數(shù)是

2 非線性 Lagrange函數(shù)的微分性質(zhì)

[1]SUN D.Lowner's operator and spectral functions in Euclidean Jordan algebrasnus[J].Mathematics of Operations Research,2008,33:421-445.

[2]BERTSEK AS D P.Constrained Opt imization and Lagrange MultiplierMethods[M].New York:Academic Press,1982.

[3]BONNANS J F,SHAPI RO A.PerturbationAnalysisofOpt imization Problems[M].New York:Springer,2000.

[4]SUN Jie,ZHANG Liwei,WU Yue.Properties of the augmented Lagrangian in nonlinear semidefinite optimization[J].Journal of Optimization Theory and Applications,2006,129:437-456.

[5]DEBREU G.Definite and semidefinite quadratic forms[J].Econometrica,1952,20:295-300.

A Nonlinear Lagrangian for Constructing Algorithm s for Nonlinear Sem idefin ite Programm ing

L IYang1,WANG Y i-yi2
(1.College of Science,Dalian NationalitiesUniversity,Dalian Liaoning 116605,China;2
.School ofMathematics,Jilin University,Changchun Jilin 130012,China)

The nonlinearLagrangian method is very important for solving nonlinear semidefinite programming.This paper provides a nonlinear Lagrangian for studying nonlinear semidefinite programming and proves that this function is convex around the KKT point,thereby guaranteeing the precondition for the theory of nonlinearLagrangian algorithms.

semidefinite programming;nonlinearLagrangian;KKT conditions

O221.2

A

1009-315X(2010)03-0230-03

2010-03-10

大連民族學(xué)院人才引進(jìn)科研啟動(dòng)基金資助項(xiàng)目 (20096208)。

李陽(yáng) (1979-),女,遼寧沈陽(yáng)人,講師,博士,主要從事最優(yōu)化理論、算法及應(yīng)用研究。

(責(zé)任編輯 鄒永紅)