趙 宇,黃金瑩,劉春妍
(佳木斯大學(xué)理學(xué)院,黑龍江 佳木斯 154007)
嚴格F-G廣義凸函數(shù)
趙 宇,黃金瑩,劉春妍
(佳木斯大學(xué)理學(xué)院,黑龍江 佳木斯 154007)
給出一類新的廣義凸函數(shù)——嚴格F-G廣義凸函數(shù),并給出條件P1、P2及其性質(zhì),研究嚴格F-G廣義凸函數(shù)的若干性質(zhì),給出嚴格F-G廣義凸函數(shù)的兩個充分條件,指出在一定條件下,滿足中間點嚴格F-G廣義凸性的F-G廣義凸函數(shù)是嚴格F-G廣義凸函數(shù),滿足中間點嚴格F-G廣義凸性的半嚴格F-G廣義凸函數(shù)是嚴格F-G廣義凸函數(shù).最后給出嚴格F-G廣義凸函數(shù)在極小化問題中的一個實際應(yīng)用.
嚴格F-G廣義凸函數(shù);F-G廣義凸函數(shù);條件P1;條件P2
凸性及廣義凸性是不等式研究的主要內(nèi)容,同時也在最優(yōu)化理論方面起著重要作用.近20年間,在文獻中出現(xiàn)了各種各樣的廣義凸函數(shù),其中比較常見的推廣方式是以討論極值問題最優(yōu)性條件為目的通過將凸性條件弱化來構(gòu)造的廣義凸函數(shù).1981年,Hanson[1]和Craven[2]提出了不變凸、擬不變凸和偽不變凸函數(shù)等概念,并利用其條件建立了分式規(guī)劃對偶理論.1985年,Jeyakumar[3]提出預(yù)不變凸函數(shù)的概念,1991年,Pini[4]定義了更廣義的預(yù)不變擬凸函數(shù).值得一提的是,1995年S.R.Mohan與S.K.Neogy[5]提出了條件C,條件C所蘊含的等式關(guān)系搭建了不變凸函數(shù)類之間的關(guān)系,推導(dǎo)出嚴格、半嚴格預(yù)不變(擬)凸函數(shù)的若干判別準則,極大促進了不變凸函數(shù)類的性質(zhì)研究.近幾年,國內(nèi)的專家學(xué)者針對上述各類廣義凸函數(shù)開展了特性研究,唐萬梅等[6-7]研究了強預(yù)不變凸函數(shù)和強預(yù)擬不變凸函數(shù)的充要條件,劉彩平[8]研究了半嚴格不變凸函數(shù),豐富了無限維空間中的廣義凸理論,黃金瑩等[9-10]對預(yù)不變凸函數(shù)的性質(zhì)做了進一步研究,提出F-G廣義凸函數(shù)概念.
由于它們在一定程度上又保留了凸函數(shù)的良好性質(zhì),所以各類廣義凸函數(shù)之間必然有著類似的性質(zhì),在研究方法和技巧上就不可避免地出現(xiàn)了類比、雷同、重復(fù)的現(xiàn)象,且在處理具體問題時又受到具體形式的影響.該文建立嚴格F-G廣義凸函數(shù)概念,開展一般性研究,將目前相關(guān)結(jié)果加以整合、梳理和推廣.
定義1 稱集合K?Rn是關(guān)于F的廣義凸集,若存在向量值函F:K×K×[0,1]→Rn,使得?λ∈[0,1],?x,y∈K,有F(x,y,λ)∈K.
將上述中K?Rn換成D?R,則會有:稱集合D?R是關(guān)于G的廣義凸集,若存在數(shù)量函數(shù)G:D×D×[0,1]→R,使得 ?λ∈ [0,1],?s,t∈D,有G(s,t,λ)∈D.
定義2 設(shè)K?Rn是關(guān)于F的廣義凸集,D?R是關(guān)于G的廣義凸集,稱數(shù)量函數(shù)f:K→D在K上是F-G 廣義凸函數(shù),若 ?λ∈ [0,1],?x,y∈K,有f[F(x,y,λ)]≤G[f(x),f(y),λ].
定義3 設(shè)K?Rn是關(guān)于F的廣義凸集,D?R是關(guān)于G的廣義凸集,稱數(shù)量函數(shù)f:K→D在K上是嚴格F-G 廣義凸函數(shù),若 ?λ∈ (0,1),?x,y∈K,且x≠y,有f[F(x,y,λ)]<G[f(x),f(y),λ].
特別地,在上述定義中,當(dāng)特取D?R為凸集,并且G(s,t,λ)=λs+(1-λ)t,?s,t∈D,則稱數(shù)量函數(shù)f:K →D 在K 上是嚴格F 凸函數(shù).當(dāng)特取G(s,t,λ)=max{s,t},?s,t∈D,則稱數(shù)量函數(shù)f:K →D在K上是嚴格F擬凸函數(shù).
注1[1]取K?Rn為關(guān)于向量函數(shù)η:Rn×Rn→Rn的不變凸集,D?R為凸集.
令F(x,y,λ)=y(tǒng)+λη(x,y),?x,y∈K,λ∈ [0,1],G(s,t,λ)=λs+(1-λ)t,?s,t∈D,則當(dāng)數(shù)量函數(shù)f:K→D為K 上關(guān)于η的嚴格預(yù)不變凸函數(shù)時,f在K上是嚴格F凸函數(shù).
注2[2]取K?Rn為關(guān)于向量函數(shù)η:Rn×Rn→Rn的不變凸集,D?R.
令F(x,y,λ)=y(tǒng)+λη(x,y),?x,y∈K,λ∈ [0,1],G(s,t,λ)= max{s,t},?s,t∈D,則當(dāng)數(shù)量函數(shù)f:K→D為K上關(guān)于η的嚴格預(yù)不變擬凸函數(shù)時,f在K上是嚴格F擬凸函數(shù).
定義4 設(shè)K?Rn是關(guān)于F的廣義凸集,D?R是關(guān)于G的廣義凸集,稱數(shù)量函數(shù)f:K→D在K上是半嚴格F-G 廣義凸函數(shù),若 ?λ∈ (0,1),?x,y∈K,且f(x)≠f(y),有
定義5 設(shè)K?Rn是關(guān)于F的廣義凸集,稱F在K 上滿足條件P1、P2,若?α,β∈[0,1],且α<β,?x,y∈K,有
設(shè)D?R是關(guān)于G的廣義凸集,稱G在D上滿足條件P1、P2,若 ?α,β∈ [0,1],且α<β,?s,t∈D,有
對于條件P1、P2,有如下幾個性質(zhì):
定理1 若F在K 上滿足條件P1、P2,則 ?λ∈ (0,1),?u1,u2∈ [0,1],u1≠u2,?x,y∈K,有
證明 ?λ∈ (0,1),?u1,u2∈ [0,1],u1≠u2,?x,y∈K,
i)當(dāng)u1<u2時,由條件P1、P2得
定理2 若K ?Rn是關(guān)于F 的廣義凸集,且 ?λ∈ [0,1],?x,y∈K,有f[F(x,y,λ)]=f[F(y,x,1-λ)],則條件P1等價于條件P2.
故條件P2蘊含條件P1.
故條件P1蘊含條件P2.
定理3 若向量函數(shù)η:Rn×Rn→Rn在K 上滿足條件C,則F(x,y,λ)=y(tǒng)+λη(x,y)在K上滿足條件P1、P2.
證明 因η在K 上滿足條件C,即 ?λ∈ [0,1],?x,y∈K,有
故F在K 上滿足條件P1、P2.
定理4 設(shè)f:K →D是嚴格F-G 廣義凸函數(shù).若 ?λ∈ [0,1],?s,t∈D,?α>0,有αs,αt∈D,且
推論1 設(shè)函數(shù)f:K→D是嚴格F凸函數(shù),F(xiàn)滿足條件P1、P2,?λ∈(0,1),?x,y∈K,且x≠y,有F(x,y,λ)≠y,則 ?x,y∈K,且x≠y,Φ(α)=f[F(x,y,α)]是[0,1]上的嚴格凸函數(shù).
推論2 設(shè)函數(shù)f:K→D是嚴格F擬凸函數(shù),F(xiàn)滿足條件P1、P2,?λ∈(0,1),?x,y∈K,且x≠y,有F(x,y,λ)≠y,則 ?x,y∈K,且x≠y,Φ(α)=f[F(x,y,α)]是[0,1]上的嚴格擬凸函數(shù).
下面給出嚴格F-G廣義凸函數(shù)的2個充分條件.
定理9 設(shè)K?Rn是關(guān)于F的廣義凸集,D?R是關(guān)于G的廣義凸集,且有如下條件:
i)F、G在K 上滿足條件條件P1、P2;
ii)?λ∈ (0,1),G(s,t,λ)關(guān)于t在D 上嚴格增加;
iii)函數(shù)f:K→D在K 上是F-G廣義凸函數(shù),且滿足?α∈ (0,1),?x,y∈ K,且x≠y,有f[F(x,y,α)]<G[f(x),f(y),α],則函數(shù)f在K上是嚴格F-G廣義凸函數(shù).
證明 令w =F(x,y,α),由條件iii)知,?x,y∈K,x≠y,有
于是,對 ?λ∈ (0,1),
綜上,函數(shù)f在K上是嚴格F-G廣義凸函數(shù).
定理10 設(shè)K?Rn是關(guān)于F的廣義凸集,D?R是關(guān)于G的廣義凸集,且有如下條件:
i)F、G在K 上滿足條件條件P1、P2;
ii)G(s,t,λ)關(guān)于s,t在D 上非減,且對 ?λ∈ (0,1),?s,t∈D,有 min{s,t}≤G(s,t,λ)≤ max{s,t};
iii)函數(shù)f:K→D在K 上是半嚴格F-G廣義凸函數(shù),且滿足
?α∈ (0,1),?x,y∈ K,且x≠y,有f[F(x,y,α)]<G[f(x),f(y),α],
則函數(shù)f在K上是嚴格F-G廣義凸函數(shù).
證明 因為函數(shù)f:K→D在K上是半嚴格F-G廣義凸函數(shù),所以只須證明下式成立:
對 ?λ∈ (0,1),f(x)=f(y)且x ≠y,有f[F(x,y,λ)]<G[f(x),f(y),λ].
因為對 ?λ∈ (0,1),?s,t∈ D,有 min{s,t}≤G(s,t,λ)≤ max{s,t},所以當(dāng)f(x)=f(y)時,G[f(x),f(y),λ]=f(x)=f(y).
令w =F(x,y,α),由條件iii)知,?x,y∈K,x≠y,f(x)=f(y),有f(w)=f[F(x,y,α)]<G[f(x),f(y),α]=f(x)=f(y),于是,對 ?λ∈ (0,1),
接下來作幾個推論,將現(xiàn)有文獻相關(guān)結(jié)論作為其特例.
推論3 設(shè)集合K是關(guān)于η的不變凸集,η滿足條件C,如果f:K→R是預(yù)不變凸函數(shù),且?α∈(0,1),?x,y∈K,x≠y,有f(y+αη(x,y))<αf(x)+(1-α)f(y),則f是嚴格預(yù)不變凸函數(shù).
根據(jù)定理3及定理9可以證明.
推論4 設(shè)集合K是關(guān)于η的不變凸集,η滿足條件C,如果f:K→R是半嚴格預(yù)不變凸函數(shù),且?α∈ (0,1),?x,y∈K,x≠y,有f(y+αη(x,y))<αf(x)+(1-α)f(y),則f是嚴格預(yù)不變凸函數(shù).
根據(jù)定理3及定理10可以證明.
推論5 設(shè)集合K是關(guān)于η的不變凸集,η滿足條件C,如果f:K→R是預(yù)不變擬凸函數(shù),且?α∈(0,1),?x,y∈K,x≠y,有f(y+αη(x,y))< max{f(x),f(y)},則f是嚴格預(yù)不變擬凸函數(shù).
根據(jù)定理3及定理9可以證明.
推論6 設(shè)集合K是關(guān)于η的不變凸集,η滿足條件C,如果f:K→R是半嚴格預(yù)不變擬凸函數(shù),且?α∈ (0,1),?x,y∈K,x≠y,有f(y+αη(x,y))< max{f(x),f(y)},則f是嚴格預(yù)不變擬凸函數(shù).
根據(jù)定理3及定理10可以證明.
考慮極小化問題(P)
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Abstract:This paper gives a new class of generalized convex functions-strictly F-Ggeneralized convex functions,and the conditions P1,P2as well as their properties.It studies on the properties of strictly F-Ggeneralized convex functions,gives two sufficient conditions of strictly F-Ggeneralized convex functions,and points out that under certain conditions,F(xiàn)Ggeneralized convex functions which satisfies intermediate-point strictly F-Ggeneralized convexity is strictly F-Ggeneralized convex functions,and semi-strictly F-G generalized convex functions which satisfies intermediate-point strictly F-G generalized convexity is strictly F-Ggeneralized convex functions.And the paper provides a realistic application of strictly FGgeneralized convex functions in minimization problem.
Key words:strictly F-Ggeneralized convex functions;F-Ggeneralized convex functions;condition P1;condition P2
Strictly F-G Generalized Convex Functions
ZHAO Yu,HUANG Jin-ying,LIU Chun-yan
(College of Science,Jiamusi University,Jiamusi 154007,China)
O174.13 MSC2010:90C25;26B25
A
1674-232X(2011)01-0020-07
10.3969/j.issn.1674-232X.2011.01.005
2010-05-10
黑龍江省教育廳科學(xué)技術(shù)研究項目(11551499).
趙 宇(1980—),女,黑龍江佳木斯人,講師,碩士,主要從事凸分析與凸規(guī)劃研究.E-mail:zhaoyu19800801@sina.com