廣義凸條件下一類多目標(biāo)優(yōu)化問(wèn)題的對(duì)偶

張瑞芳, 王海軍

(太原師范學(xué)院 數(shù)學(xué)系, 太原 030012)

廣義凸條件下一類多目標(biāo)優(yōu)化問(wèn)題的對(duì)偶

張瑞芳, 王海軍

(太原師范學(xué)院 數(shù)學(xué)系, 太原 030012)

凸性是最優(yōu)化理論中最常用的假設(shè)之一。在實(shí)際應(yīng)用中目標(biāo)函數(shù)的性質(zhì)可能不是那么理想,為了減弱凸性要求,人們給出了各種各樣的廣義凸性概念。近年來(lái),廣義凸性成為數(shù)學(xué)優(yōu)化研究的新發(fā)展趨勢(shì),越來(lái)越多的學(xué)者致力于討論在各種廣義凸性條件下多目標(biāo)優(yōu)化問(wèn)題的對(duì)偶結(jié)論及其應(yīng)用。在廣義凸條件之下考察一類多目標(biāo)優(yōu)化問(wèn)題,首先介紹一類廣義凸函數(shù)的概念及相關(guān)性質(zhì)。然后建立了多目標(biāo)優(yōu)化問(wèn)題(即原問(wèn)題)的Wolfe對(duì)偶模型,在廣義凸條件下得到了原問(wèn)題與Wolfe對(duì)偶問(wèn)題之間的弱對(duì)偶, 強(qiáng)對(duì)偶和逆對(duì)偶定理。最后建立了多目標(biāo)優(yōu)化問(wèn)題的混合型對(duì)偶模型,并且得到了原問(wèn)題的混合型對(duì)偶問(wèn)題的弱對(duì)偶, 強(qiáng)對(duì)偶和逆對(duì)偶定理。

廣義凸函數(shù); 多目標(biāo)優(yōu)化; Wolfe對(duì)偶; 混合型對(duì)偶

0 引 言

凸性是最優(yōu)化理論中最常用的假設(shè)之一,眾所周知,凸函數(shù)是一類非常重要的函數(shù),它具有一些良好的性質(zhì):如一個(gè)定義在凸集上的凸函數(shù)的局部極小值也是它的全局極小值;可微的凸函數(shù)在某點(diǎn)梯度向量為零,則函數(shù)在這一點(diǎn)取得最小值。正是由于凸函數(shù)的這些良好性質(zhì)引起了學(xué)者們的注意,20世紀(jì)50年代初到60年代末人們對(duì)凸函數(shù)進(jìn)行了大量深入細(xì)致的研究,并將凸函數(shù)應(yīng)用到許多實(shí)際問(wèn)題當(dāng)中,使得凸分析和凸優(yōu)化理論迅速發(fā)展起來(lái)。

然而,在實(shí)際應(yīng)用中目標(biāo)函數(shù)不一定是凸函數(shù),因此,放寬凸性條件限制,推廣凸函數(shù)的概念成為具有理論意義和現(xiàn)實(shí)應(yīng)用背景的問(wèn)題。近年來(lái), 廣義凸性成為數(shù)學(xué)優(yōu)化研究的新發(fā)展趨勢(shì),人們給出了各種各樣的廣義凸性概念[1-15],值得提及的是1981年由Hanson[1]提出的不變凸性。在過(guò)去的20多年中不變凸性引起了眾多學(xué)者的廣泛注意,并對(duì)此概念做了許多推廣。2012年3月,Cheng和Zhang在文獻(xiàn)[7]中,首先給出d-ρ-(η,θ)-univex函數(shù)的概念,并在d-ρηθ-univex條件下討論如下多目標(biāo)優(yōu)化問(wèn)題(P):

其中:f:X→Rk;g:X→Rm;X為Rn的非空子集。

本文將在d-ρηθ-univex條件下建立問(wèn)題(P)的Wolfe型對(duì)偶問(wèn)題(WD)的弱對(duì)偶、強(qiáng)對(duì)偶、逆對(duì)偶結(jié)論以及問(wèn)題(P)的混合型對(duì)偶問(wèn)題(MD)的弱對(duì)偶、強(qiáng)對(duì)偶和逆對(duì)偶結(jié)論。

1 預(yù)備知識(shí)

本文采用Rn中向量之間序關(guān)系的慣用記號(hào)。設(shè)x=(x1,x2,…,xn),y=(y1,y2,…,yn)∈Rn,則有:xy?xiyi,(i=1,2,…,n),xy?xi≥yi,(i=1,2,…,n),x≯y代表xy的反面。相應(yīng)地有類似的記號(hào)x

設(shè)η:X×X→Rn為一向量值函數(shù),在本文中,用f′(u,η(x,u))表示f在η(x,u)方向的方向?qū)?shù)

在以下部分中b0:X×X→R+;φ0:R→R;η,θ:X×X→Rnn。

定義1[7]稱函數(shù)f:X→R在點(diǎn)u∈X處關(guān)于b0,φ0為d-ρηθ-univex,若存在函數(shù)b0,φ0,η,θ以及實(shí)數(shù)ρ使得對(duì)于任意的x∈X有b0(x,u)φ0(f(x)-f(u))≥f′(u,η(x,u))+ρ‖θ(x,u)‖2。 若ρ0,則f(x)稱為強(qiáng)d-ρηθ-univex;若ρ=0,則f(x)為d-univex;若ρ<0,則f(x)稱為弱d-ρηθ-univex。

定義2[7]稱函數(shù)f:X→R在點(diǎn)u∈X處關(guān)于b0,φ0為弱嚴(yán)格偽d-ρηθ-univex,若存在函數(shù)b0,φ0,η,θ以及實(shí)數(shù)ρ使得對(duì)于任意的x∈X有

b0(x,u)φ0(f(x)-f(u))<0?f′(u,η(x,u))+ρ‖θ(x,u)‖2<0。

2 Wolfe對(duì)偶

首先作如下假設(shè):

b00;φ0(t)<0對(duì)任意的t<0成立。

關(guān)于原問(wèn)題(P)應(yīng)考慮它的Wolfe對(duì)偶(WD):

定理1(弱對(duì)偶) 設(shè)x,(y,ξ,μ)分別是(P)和(WD)的可行點(diǎn),若進(jìn)一步假設(shè)ξTf+μTg關(guān)于b0,φ0為弱嚴(yán)格偽d-ρηθ-univex函數(shù);且ρ≥0,則f(x)≮φ(y,ξ,μ)。

ξTf(x)<ξTf(y)+μTg(y)

ξTf(x)+μTg(x)<ξTf(y)+μTg(y)

由于b00;φ0(t)<0 對(duì)任意的t<0成立,可得

b0(x,u)φ0(ξTf(x)+μTg(x)-ξTf(y)-μTg(y))<0

由已知ξTf+μTg關(guān)于b0,φ0為弱嚴(yán)格偽d-ρηθ-univex函數(shù),于是

ξTf′(y,η(x,y))+μTg′(y,η(x,y))+ρ‖θ(x,y)‖2<0

而ρ≥0,故ξTf′(y,η(x,y))+μTg′(y,η(x,y))<0,與對(duì)偶約束條件式(4)相矛盾,即證明了f(x)≮φ(y,ξ,μ)。證畢。

3 混合型對(duì)偶

在這一部分,考慮原問(wèn)題(P)的混合型對(duì)偶:

ξTf(x)<ξTf(y)+μTg(y)

ξTf(x)+μTg(x)<ξTf(y)+μTg(y)

由于b00;φ0(t)<0 對(duì)任意的t<0成立,可得

b0(x,u)φ0(ξTf(x)+μTg(x)-ξTf(y)-μTg(y))<0

由已知ξTf+μTg關(guān)于b0,φ0為弱嚴(yán)格偽d-ρηθ-univex函數(shù),于是

ξTf′(y,η(x,y))+μTg′(y,η(x,y))+ρ‖θ(x,y)‖2<0

而ρ≥0,故ξTf′(y,η(x,y))+μTg′(y,η(x,y))<0,與對(duì)偶約束條件式(6)相矛盾,即證明f(x)≮φ(y,ξ,μ)。證畢。

4 結(jié) 論

本文在d-ρηθ-univex條件下建立問(wèn)題(P)的Wolfe型對(duì)偶問(wèn)題(WD)的弱對(duì)偶, 強(qiáng)對(duì)偶,逆對(duì)偶結(jié)論以及問(wèn)題(P)的混合型對(duì)偶問(wèn)題(MD)的弱對(duì)偶,強(qiáng)對(duì)偶和逆對(duì)偶結(jié)論。今后還可以進(jìn)一步討論d-ρηθ-univex條件下的分式規(guī)劃問(wèn)題及其對(duì)偶。

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Dualityforakindofmultiobjectiveoptimizationproblemundergeneralizedconvexity

ZHANGRuifang,WANGHaijun

(Department of Mathematics, Taiyuan Normal University, Taiyuan 030012, China)

Convexity is the most commonly used hypothesis in optimization theory. In the practical application, the property of objective function is not so ideal. In order to relax the convexity condition, people provide various of generalized convexity concepts. In recent years, generalized convexity become the new trend of mathematical optimization, more and more scholars devote to discuss multiobjective optimization problem duality results and its applications under generalized convexity. This paper considers a kind of multiobjective optimization problem under generalized convexity condition. Firstly, we introduce the concept of a kind of generalized convexity and related properties. Then we set up Wolfe dual problem of the original multiobjective optimization problem. Weak, strong and converse duality results between the original problem and its Wolfe dual problem are given. Finally we establish mixed type dual problem of the original problem, and obtain weak, strong and converse duality between the original problem and its mixed type dual problem.

generalized convex function; multiobjective optimization; Wolfe duality; mixed type duality

2013-10-01。

國(guó)家自然科學(xué)基金資助項(xiàng)目(11171250)。

張瑞芳(1982-),女,山西交口人,太原師范學(xué)院教師,碩士。

1673-5862(2014)04-0482-04

O221.6

: A

10.3969/ j.issn.1673-5862.2014.04.006