王彩玲,王澤升
(吉林大學 數學學院,長春 130012)
非光滑多目標優化理論應用廣泛,而廣義凸性在對偶理論中具有重要作用,已廣泛用于解決優化問題.文獻[1-6]給出了不同條件下廣義凸函數的最優性條件與對偶理論.本文通過定義復合Q-ρ不變凸和S-δ不變凸函數,對其構成的不可微復合凸多目標規劃問題給出了新的對偶理論,推廣了文獻[3-7] 的結果.設Q?p和S?m分別是具有內部的閉凸錐.
考慮復合多目標規劃問題(VP):
其中:g(G(x))=(g1(G1(x)),…,gm(Gm(x)))T;x∈X(X是Banach空間);f和g分別是n上實值局部Lipschitz向量函數;Fi和Gj:X→n分別是局部Lipschitz函數和Gateaux可微函數,它們的Gateaux導數分別記為(·)(i=1,2,…,p;j=1,2,…,m).記
Ω={x∈X|-g(G(x))∈S},
Q*和S*分別為Q和S的對偶錐,如Q*={λ∈p|λTv≥0,?v∈Q}.
定義1如果存在ρ∈p,使得對?及?有
定義2如果存在δ∈m,使得對?及?有
其中U=W∩(-W).
考慮(VP)的Mond-Weir型對偶問題(VD):
定理1(弱對偶性) 設x和(y,Λ)分別是(VP)和(VD)的可行解,如果f(F)在y處關于函數η和θ:X×X→X是廣義復合Q-ρ不變凸的,g(G)在y處關于函數η和θ是廣義復合S-δ不變凸的,且ρ+Λδ∈Q,則
f(F(x))-f(F(y))∈W.
證明:由已知條件可得
由于對任意(VD)的可行解(y,Λ),由定理1可得
證明:由已知條件及定理1可知,對?x∈Ω,有
?W.
[1] Mishra S K,WANG Shou-yang,LAI Kin-keung.Nondifferentiable Multiobjective Programming under Generalizedd-Univexity [J].European Journal of Operation Research,2005,160(1): 218-226.
[2] Mishra S K,WANG Shuo-yang,LAI Kin-keung.Generalized Convexity and Vector Optimization [M].Nonconvex Optimization and Its Applications.Vol.90.Berlin: Springer-Verlag,2009.
[3] YANG Xin-min,Teo K L,YANG Xiao-qi.Duality for a Class of Nondifferentiable Multiobjective Programming Problems [J].J Math Anal Appl,2000,252(2): 999-1005.
[4] Mishra S K,Giorgi G,WANG Shou-yang.Duality in Vector Optimization in Banach Spaces with Generalized Convexity [J].Journal of Global Optimization,2004,29(4): 415-424.
[5] Niculescu C.Optimality and Duality in Multiobjective Fractional Programming Involvingρ-Semilocally Type Ⅰ-Preinvex and Related Functions [J].J Math Anal Appl,2007,335(1): 7-19.
[6] Shapiro A.On a Class of Nonsmooth Composite Functions [J].Mathematics of Operational Research,2003,28(4): 677-692.
[7] Jeyakumar V,Yang X Q.Convex Composite Multiobjective Nonsmooth Programming [J].Math Programming,1993,59(1/2/3): 325-343.
[8] WANG Cai-ling.Saddle-Point Theory of Nonsmooth Invex Multiobjective Programming [J].Journal of Jilin University: Science Edition,2011,49(4): 693-695.(王彩玲.非光滑凸多目標規劃的鞍點定理 [J].吉林大學學報: 理學版,2011,49(4): 693-695.)