非光滑廣義凸多目標規劃的對偶

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.

1 基本概念

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).

2 對偶定理

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.

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