一类广义Lipschitz非线性算子的带误差的Ishikawa迭代程序

借助于周海云和陈东青［４］新近引入的广义Ｌｉｐｓｃｈｉｔｚ概念,研究了实Ｂａｎａｃｈ空间中广义Ｌｉｐｓｃｈｉｔｚ　 －强伪压缩算子的不动点和广义Ｌｉｐｓｃｈｉｔｚ －强增算子方程解的迭代逼近问题,所得结果改进和扩展了近期许多相关的结果,并部分地回答了周海云［３］提出的一个问题．     

N指标d维广义Wiener过程的极函数

研究了犖指标犱维广义Ｗｉｅｎｅｒ过程极函数的特征,得到了满足Ｌｉｐｓｃｈｉｔｚ条件的连续函数类与广义Ｗｉｅｎｅｒ过程极函数类之间的关系．此结果包含并推广了ＢｒｏｗｎｉａｎＳｈｅｅｔ的结果．     

关于“弱Lipschitz函数,它的广义次梯度及其在优化中的应用”一文的注记

本文证明张玉忠同志在“弱Ｌｉｐｓｃｈｉｔｚ函数、它的广义次梯度及其在优化中的应用”一文中定义的广义次梯度ｆ（π），当ｎ≥２时即为Ｒ￣ｎ，因此这种广义次梯度是没有多少应用价值的。     

一类具有广义Lipschitz条件的非线性映象的迭代过程

本文研究了广义Ｌｉｐｓｃｈｉｔｚ强增生映象的Ｉｓｈｉｋａｗａ型迭代和Ｍａｎｎ型迭代过程的收敛性．所得结果统一和扩展了近期相关结果     

局部凸空间上的泛函的非光滑分析及多目标规划

该文对定义于局部凸线性拓扑空间Ｘ上的泛函引入广义方向导数、广义梯度及满足Ｌｉｐｓ－ｃｈｉｔｚ条件等概念,证明了它们的几个重要性质,并举例说明这里满足Ｌｉｐｓｃｈｉｔｚ条件的概念是Ｂａ－ｎａｃｈ空间情形的严格推广．最后,作为上述结论及方法的应用,讨论定义于Ｘ的多目标数学规划,得出若干关于弱有效解的最优性条件．     

非光滑Lipschitz规划的Mond—Weir对偶

本文建立了非光滑Ｌｉｐｓｃｈｉｔｚ规划的两种Ｍｏｎｄ－Ｗｅｉｒ对偶形式，在引入一定的非光滑广义凸性下证明了相应的对偶定理。     

广义伪凸函数的一个性质

广义伪凸函数的一个性质杨新民（重庆师范学院数学系，６３００４７）关键词广义伪凸，严格广义伪凸，判别准则．分类号ＡＭＳ（１９９１）９０Ｃ２５，２６Ｂ２５／ＣＣＬＯ２２１．２设Ｂ是Ｒ￣ｎ中单位开球，ｆ：Ｒ￣ｎ→Ｒ在点ｘ处称为Ｌｉｐｓｃｈｉｔｚ函数，即存在...     

非光滑Lipschitz规划的Mond－Weir对偶

本文建立了非光滑Ｌｉｐｓｃｈｉｔｚ规划的两种Ｍｏｎｄ－Ｗｅｉｒ对偶形式，在引入一定的非光滑广义凸性下证明了相应的对偶定理．     

Lipschitz常数缩减的散乱数据插值

在计算机辅助设计几何中，变差缩减是一个非常重要的概念，本文分析了函数的变差和Ｌｉｐｓｃｈｉｔｚ常数的关系，指出可以用Ｌｉｐｓｃｈｉｔｚ常数来控制变差，由于变差的概念只限于一维的情形，而Ｌｉｐｓｃｈｉｔｚ常数适用于任意维，这样在高维时就可用Ｌｉｐｓｃｈｉｔｚ常数缩减的概念来代替变差缩减的概念，文中构造性地证明了Ｌｉｐｓｃｈｉｔｚ常数缩减的散乱数据插值函数的存在性，并且对这类函数的性质及光滑性条件进行了讨论．     

非线性Lipschitz算子的Lipschitz对偶算子及其应用

在文山中我们对非线性Ｌｉｐｓｃｈｉｔｚ算子定义了其Ｌｉｐｓｃｈｉｔｚ对偶算子，并证明了任意非线性Ｌｉｐｓｃｈｉｔｚ算子的Ｌｉｐｓｃｈｉｔｚ对偶算子是一个定义在Ｌｉｐｓｃｈｉｔｚ对偶空间上的有界线性算子．本文还进一步证明：设Ｃ为 Ｂａｎａｃｈ空间 Ｘ的闭子集，Ｃ＊Ｌ为Ｃ的 Ｌｉｐｓｃｈｉｔｚ对偶空间，Ｕ为 Ｃ＊Ｌ上的有界线性算子，则当且仅当 Ｕ为 ｗ＊－ｗ＊连续的同态变换时，存在Ｌｉｐｓｃｈｉｔｚ连续算子Ｔ，使Ｕ为Ｔ的Ｌｉｐｓｃｈｉｔｚ对偶算子．这一结论的理论意义在于：它表明一个非线性Ｌｉｐｓｃｈｉｔｚ算子的可逆性问题可转化为有界线性算子的可逆性问题．作为应用，通过引入一个新概念──ＰＸ－对偶算子，在一般框架下给出了非线性算子半群的生成定理．     

Lipschitz B-Vex Functions and Nonsmooth Programming

In this paper, the equivalence between the class of B-vex functions and that of quasiconvex functions is proved. Necessary and sufficient conditions, under which a locally Lipschitz function is B-vex, are established in terms of the Clarke subdifferential. Regularity of locally Lipschitz B-vex functions is discussed. Furthermore, under appropriate conditions, a necessary optimality condition of the Slater type and a sufficient optimality condition are obtained for a nonsmooth programming problem involving B-vex functions.     

Optimality and duality for nonsmooth multiobjective programming problems with V-r-invexity

In the paper, we consider a class of nonsmooth multiobjective programming problems in which involved functions are locally Lipschitz. A new concept of invexity for locally Lipschitz vector-valued functions is introduced, called V-r-invexity. The generalized Karush–Kuhn–Tuker necessary and sufficient optimality conditions are established and duality theorems are derived for nonsmooth multiobjective programming problems involving V-r-invex functions (with respect to the same function η).     

一类Lipschitz B-(p,γ)-不变凸函数与非光滑规划

设本文给出了一类新的Lipschitz B-(p,γ)-不变凸函数,它是B-不变凸函数和(p,γ)-不变凸函数的推广.在这类Lipschitz B-(p,γ)一不变凸性下,建立了非光滑规划的必要和充分最优性条件,讨论了Mond-Weir型对偶和Wolfe型对偶,证明了弱对偶、强对偶和逆对偶定理.所得结果推广了涉及凸函数、B-不变凸函数和(p,γ)-不变凸函数的规划问题的相应结果.     

Optimality Conditions for C 1,1 Constrained Multiobjective Problems

The present paper is concerned with the study of the optimality conditions for constrained multiobjective programming problems in which the data have locally Lipschitz Jacobian maps. Second-order necessary and sufficient conditions for efficient solutions are established in terms of second-order subdifferentials of vector functions.     

Necessary optimality conditions for nonsmooth semi-infinite programming problems

In this paper, for a nonsmooth semi-infinite programming problem where the objective and constraint functions are locally Lipschitz, analogues of the Guignard, Kuhn-Tucker, and Cottle constraint qualifications are given. Pshenichnyi-Levin-Valadire property is introduced, and Karush-Kuhn-Tucker type necessary optimality conditions are derived.     

Exact penalty functions and stability in locally Lipschitz programming

In this paper we extend the theory of exact penalty functions for nonlinear programs whose objective functions and equality and inequality constraints are locally Lipschitz; arbitrary simple constraints are also allowed. Assuming a weak stability condition, we show that for all sufficiently large penalty parameter values an isolated local minimum of the nonlinear program is also an isolated local minimum of the exact penalty function. A tight lower bound on the parameter value is provided when certain first order sufficiency conditions are satisfied. We apply these results to unify and extend some results for convex programming. Since several effective algorithms for solving nonlinear programs with differentiable functions rely on exact penalty functions, our results provide a framework for extending these algorithms to problems with locally Lipschitz functions.     

PARALLEL MULTIPLICATIVE ITERATIVE METHODS FOR CONVEX PROGRAMMING

1IntroductionInthispaper,wewillconsiderthebehavinrOfparallelmultiplicativeiterativemethodsfortheconstrainedopt~ationproblemwhereIisaconvexcontinuouslydtherentiablefunctiononR;withcompactlevelsetsandlOCallyLiP8chitzcontinuousgradient.In[1]Eggerinontintroducedapracticalapprodriatemethodforsolving(1,1),theresultingalgorithmhastheformwhereVI(x*)isthegradientofl(:)atac,andwbisarelaxation/steplengthparameter,andinwhichMisanarbitrarylfordconstant,andListheLipschitzconstantofVI(x),thatistosaylth…     

一类G-(F,ρ)凸多目标分式规划的最优性条件

本文讨论了一类多目标分式规划问题，其中所包含的函数是局部Lipschitz的和Clarke次可微的．首先，在G-(F，ρ)凸的条件下，证明了择一定理．然后，证明了该多目标分式规划问题在Geoffrion意义下的真有效解的充分条件和必要条件．     

不可微多目标优化非控解的最优性条件与对偶

本文对由一类局部Ｌｉｐｓｃｈｉｔｚ的ρ－ｉｎｖｅｘ函数所构成的不可微多目标优化问题进行了讨论；给出了最优性条件。并且对Ｗｏｌｆｅ、Ｗｅｉｒ－Ｍｏｎｄ和Ｃｒａｖｅｎ型对偶问题进行了研究，得到了相应的对偶定理。     

Lower bounds of controlling parameters of exact penalty functions in locally Lipschitz programming

In this note, lower bounds of penalty parameters of general exact penalty functions in locally Lipschitz programming are directly derived from Rosenberg's results.