非Lipshitz一般集值变分不等式的广义投影算法  被引量:1

Generalized Projection Method for Non-Lipschitz General Set-valued Variational Inequalities

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作  者:李观荣[1] 钟莉萍[1] 

机构地区:[1]湛江师范学院数学与计算科学学院,广东湛江524048

出  处:《重庆师范大学学报(自然科学版)》2014年第4期92-95,共4页Journal of Chongqing Normal University:Natural Science

基  金:湛江师范学院自然科学研究青年项目(QL1102)

摘  要:设K是实Hibert空间H的非空闭凸子集,T:H→2H为集值映象,g:H→H为单值映象且Kg(H)。所谓一般集值变分不等式问题,即是指,求x*∈H,使得g(x*)∈K,w∈T(x*)且〈w,g(y)-g(x*)〉≥0,g(y)∈K。在求解以上一般集值变分不等式中,投影算法是常用的算法,但是传统的投影算法需集值映象T关于Hausdoff距离是Lipschtz的。首先,在不需要集值映象T关于Hausdoff距离是Lipschtz的情况下,建立了求解一般集值变分不等式的广义投影算法:第0步:取数列{ρ}j使得0<ρj<1,∑!j=0ρj=+!,∑!j=0ρj2<+!.取g(x0)∈K,令j:=0。第1步:令vj∈T(xj),如果vj=0,则停止,此时xj为问题的解。如果vj≠0,则找wj使得〈vj,g(y)-g(xj)〉+〈wj,g(y)-g(xj)〉≥0,g(y)∈K。如果wj=0,则停止,此时xj是问题的解;否则,进入第2步。第2步:计算xj+1使得g(xj+1)=PK[g(xj)+ρjwj];令j←j+1,回到第1步。然后,在{w}j有界和集值映象T为g-强伪单调的条件下,证明了由该算法产生的序列{x}j强收敛于一般集值变分不等式的解。最后,对广义投影算法作一些修正,保证算法中的序列{w}j是有界的。LetKbe nonempty closed convex subset of real Hibert spaceH,T:H→2Hbe a set-- valued mapping, g :H→Hbe a singlemapping such thatKg(H)The general set--valued variational inequality problem is given as finding x*∈Hsuch that g(x*)∈K,w∈T(x*)and〈w,g(y)-g(x*)〉≥0,g(y)∈K.The projection algorithm is a popular algorithm for general set--valued var-iational inequalities. But classical projection algorithm requires that the set--valued mappingTis Lipschitz with respect to the Haus- dorff distance. Firstly, we establish the generalized projection algorithm for general set--valued variational inequalities, where the set-- valued mapping T is not necessarily Lipschitz with respect to the Hausdorff distance. The algorithm is given as Step 0 : Choosea sequence {ρ}j 0〈ρj〈1,∑!j=0ρj=+!,∑!j=0ρj2〈+!.go back to Step 1. Secondly, under the assumptions that the sequence {w/ } is bounded and the set--valued mappingTis g--strongly pseudomonotone, we proved that the sequence generated by the generalized projection algorithm strongly converges to a solution of the general set--valued variational inequalities. Finally, we make a modification of the generalized projection algorithm to ensure the boundness of the sequence{w}.

关 键 词:一般集值变分不等式 广义投影算法 非Lipschitz映象 强伪单调映象 

分 类 号:O177.91[理学—数学] O178[理学—基础数学]

 

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