一致凸的Banach空间上的渐近非扩张映象的迭代序列的收敛性定理

本文把「３」的主要结果从Ｈｉｌｂｅｒｔ空间推广到一致凸的Ｂａｎａｃｈ空间,证明了一致凸的Ｂａｎａｃｈ空间上的渐近非扩张映象的迭代序列的收敛性。     

Banach空间具有Banach—Saks性质的一个充分条件

本文的目标是利用凸性指标给出Ｂａｎａｃｈ空间具有Ｂａｎａｃｈ－Ｓａｋｅ性质的一个充分条件，从而推广Ｋａｎｋｕｔａｎｉ定理。     

关于度量投影的连续性

本文引入的Ｂａｎａｃｈ空间的（Ｃ－Ｉ）、（Ｃ－Ⅱ），（Ｃ－Ⅲ）等几何性质，证明了如下结果。设Ｍ是Ｂａｎａｃｈ空间的逼近凸子集，如果Ｂａｎａｃｈ空间有性质（Ｃ－Ｉ），（Ｃ－Ⅱ）（Ｃ－Ⅲ），则度量投影ＰＭ连续（范数－范数上半连续，范数－弱上半连续）。这些结果推广了文（４，７，８）相应的定理。最近，Ｄ．Ｋｕｔｚａｒｏｖａ，Ｂｏｒ－Ｌｕｈ Ｌｉｎ等引入了一些新的凸性空间，本文还研究了这些凸性空间中度量投影     

Banach空间中强增生算子的非线性方程的解的迭代构造

本文研究ｐ一致光滑Ｂａｎａｃｈ空间Ｘ中Ｉｓｈｉｋａｗａ迭代法．受Ｄｅｎｇ与Ｔａｎ，Ｘｕ的启发，证明了，当Ｔ是从Ｘ到自身的Ｌｉｐｓｃｈｉｔｚ强增生算子时，Ｉｓｈｉｋａｗａ迭代法强收敛到方程Ｔｘ＝ｆ的唯一解；当Ｔ是从Ｘ的有界闭凸子集到自身的Ｌｉｐｓｃｈｉｔｚ严格伪压缩映象时，Ｉｓｈｉｋａｗａ迭代法强收敛到Ｔ的唯一不动点．通过去掉限制ｌｉｍ_ｎ→∞β_ｎ＝０或ｌｉｍ_ｎ→∞α_ｎ＝ｌｉｍ_ｎ→∞β_ｎ＝０，结果改进与推广了Ｔａｎ，Ｘｕ的定理４．１与定理４．２，也把Ｄｅｎｇ的定理１与定理２推广到了ｐ一致光滑Ｂａｎａｃｈ空间的背景．     

关于Banach空间的强凸性

本文讨论强凸性、Ｌ－ｋＲ，ＬωＲ和（Ｇ）性质之间的关系，指出强凸性介于ＬωＲ和（Ｇ）性质之间，证明光滑的有（Ｇ）性质的Ｂａｎａｃｈ空间是强凸的，此外指出存在一个Ｂａｎａｃｈ空间Ｘ，它是ＬωＲ但对任意自然数ｋ，Ｘ不是Ｌ－ｋＲ．     

Banach空间中广义正交分解定理与广义正交可补子空间

本文首先将 Ｈｉｌｂｅｒｔ空间中的Ｒｉｅｓｚ正交分解定理推广到 Ｂａｎａｃｈ空间,得到 Ｂａｎａｃｈ空间广义正交分解定理．然后,利用此定理讨论由Ｊａｍｅｓ Ｒ．Ｃ．［１］引入的Ｂａｎａｃｈ空间中正交概念及 Ｎａｓｈｅｄ Ｍ．Ｚ．［２］引入的 Ｂａｎａｃｈ空间中（广义）正交可补子空间,得到判别子空间广义正交可补的充分必要条件,并由此给出Ｈｉｌｂｅｒｔ空间的一个新特征．     

Banach空间值鞅不等式与弱大数定律

本文证明了Ｂａｎａｃｈ空间值鞅的一些不等式，讨论了Ｂａｎａｃｈ空间的凸性及光滑性与某些鞅不等的式的联系，给出了Ｈｉｌｂｅｒｔ空间的一个鞅不等式刻划，同时还讨论了一致Ｐ光滑空间中鞅的弱大数定律，本文的结论推广与改进了很多熟知的定理。     

算子值加权模式在Banach空间的实现理论

本文处理能在Ｂａｎａｃｈ空间中实现的线性连续时间系统．这里，输入、输出和状态空间都是无穷维Ｂａｎａｃｈ空间．我们证明：加权模式有Ｂａｎａｃｈ空间实现的充要条件是它强连续且为指数阶．对状态算子是解析半群的无穷小生成元的情况，得到了Ｂａｎａｃｈ空间实现的存在性定理．所有定理都是在时间域中给出的．     

Banach空间中渐近正则的Lipschitz半群的不动点定理

本文首先定义了渐近正则的Ｌｉｐｓｃｈｉｔｚ半群的概念．其次，证明了ｐ一致凸Ｂａｎａｃｈ空间中渐近正则的Ｌｉｐｓｃｈｉｔｚ半群的不动点定理．同时也证明了具有正规结构系数的一致凸Ｅａｎａｃｈ空间中的渐近正则的Ｌｉｐｓｃｈｉｔｚ半群的一个新的不动点定理．     

Banach空间的弱*序列紧性

本文研究了Ｂａｎａｃｈ空间的弱序列紧性．Ｂａｎａｃｈ空间Ｘ称为有（ｗ）性质，如果Ｘ（Ｘ的共轭空间）的每个有界序列有弱收敛子列．我们证明了，如果Ｂａｎａｃｈ空间Ｘ有（ｗ）性质，那么ｌｐ（Ｘ）（１≤ｐ＜＋∞）与ｃ０（Ｘ）也有（ｗ）性质．     

Iterative algorithms for systems of extended regularized nonconvex variational inequalities and fixed point problems

This paper points out some fatal errors in the equivalent formulations used in Noor 2011 [Noor MA. Projection iterative methods for solving some systems of general nonconvex variational inequalities. Applied Analysis. 2011;90:777–786] and consequently in Noor 2009 [Noor MA. System of nonconvex variational inequalities. Journal of Advanced Research Optimization. 2009;1:1–10], Noor 2010 [Noor MA, Noor KI. New system of general nonconvex variational inequalities. Applied Mathematics E-Notes. 2010;10:76–85] and Wen 2010 [Wen DJ. Projection methods for a generalized system of nonconvex variational inequalities with different nonlinear operators. Nonlinear Analysis. 2010;73:2292–2297]. Since these equivalent formulations are the main tools to suggest iterative algorithms and to establish the convergence results, the algorithms and results in the aforementioned articles are not valid. It is shown by given some examples. To overcome with the problems in these papers, we consider a new system of extended regularized nonconvex variational inequalities, and establish the existence and uniqueness result for a solution of the aforesaid system. We suggest and analyse a new projection iterative algorithm to compute the unique solution of the system of extended regularized nonconvex variational inequalities which is also a fixed point of a nearly uniformly Lipschitzian mapping. Furthermore, the convergence analysis of the proposed iterative algorithm under some suitable conditions is studied. As a consequence, we point out that one can derive the correct version of the algorithms and results presented in the above mentioned papers.     

非凸半定规划的广义Fakars引理及最优性条件

1引言在本文中,我们用(?),S~n,S_ ~n分别表示有限维向量空间,n阶对称矩阵空间及n阶半正定矩阵锥.我们考虑如下形式的非凸半定规划问题:     

Commutators of BMO functions and singular integral operators with non-smooth kernels on spaces of Homogeneous type of finite measure

Let X be a space of homogeneous type with finite measure. Let T be a singular integral operator which is bounded on Lp (X), 1 ＜ p ＜∞. We give a sufficient condition on the kernel k(x,y) of Tso that when a function b ∈ BMO (X) ,the commutator [b, T] (f) = T (b f) - bT (f) is aounded on spaces Lp for all p, 1 ＜ p ＜∞.     

An improved Wei–Yao–Liu nonlinear conjugate gradient method for optimization computation

In this paper, we take a little modification to the Wei–Yao–Liu nonlinear conjugate gradient method proposed by Wei et al. [Z. Wei, S. Yao, L. Liu, The convergence properties of some new conjugate gradient methods, Appl. Math. Comput. 183 (2006) 1341–1350] such that the modified method possesses better convergence properties. In fact, we prove that the modified method satisfies sufficient descent condition with greater parameter in the strong Wolfe line search and converges globally for nonconvex minimization. We also extend these results to the Hestenes–Stiefel method and prove that the modified HS method is globally convergent for nonconvex functions with the standard Wolfe conditions. Numerical results are reported by using some test problems in the CUTE library.     

A note on the existence of saddle points of p-th power Lagrangian for constrained nonconvex optimization

Li and Sun [D. Li and X.L. Sun, Existence of a saddle point in nonconvex constrained optimization, J. Global Optim. 21 (2001), pp. 39--50; D. Li and X.L. Sun, Convexification and existence of saddle point in a p-th-power reformulation for nonconvex constrained optimization, Nonlinear Anal. 47 (2001), pp. 5611--5622], present the existence of a global saddle point of the p-th power Lagrangian functions for constrained nonconvex optimization, under second-order sufficiency conditions and additional conditions that the feasible set is compact and the global solution of the primal problem is unique. In this article, it is shown that the same results can be obtained under additional assumptions that do not require the compactness of the feasible set and the uniqueness of global solution of the primal problem.     

Lagrange necessary conditions for Pareto minimizers in Asplund spaces and applications

In this paper, new necessary conditions for Pareto minimal points to sets and Pareto minimizers for constrained multiobjective optimization problems are established without the sequentially normal compactness property and the asymptotical compactness condition imposed on closed and convex ordering cones in Bao and Mordukhovich [10] and Durea and Dutta [5], respectively. Our approach is based on a version of the separation theorem for nonconvex sets and the subdifferentials of vector-valued and set-valued mappings. Furthermore, applications in mathematical finance and approximation theory are discussed.     

論雅各必恒等式

<正> §1.設X是一個拓撲空間,其中任何二點可以用弧聯結。以x_o∈X為參考點,那麼可以定義π_r(X,x_o)(r=1,2,…)。這種羣的元素全體,成為一個集E。在E中有魏德海乘積[2],即α,β為E中一點,那麼[α,β]也是E中一點。關於魏德海乘積的重覆使用問題,就文獻而諭,首先在W.S.Massey[6]     

Armijo线性搜索下Hager-Zhang共轭梯度法的全局收敛性

Hager和Zhang^[4]提出了一种新的非线性共轭梯度法(简称 HZ 方法), 并证明了该方法在 Wolfe搜索和 Goldstein 搜索下求解强凸问题的全局收敛性.但是HZ方法在标准Armijo 搜索下求解非凸问题是否全局收敛尚不清楚.该文提出了一种保守的HZ共轭梯度法,并且证明了这种方法在 Armijo 线性搜索下求解非凸优化问题的全局收敛性.此外,作者给出了一些 数值结果以检验该方法的有效性.     

Modified nonlinear conjugate gradient methods with sufficient descent property for large-scale optimization problems

It is well known that the sufficient descent condition is very important to the global convergence of the nonlinear conjugate gradient method. In this paper, some modified conjugate gradient methods which possess this property are presented. The global convergence of these proposed methods with the weak Wolfe–Powell (WWP) line search rule is established for nonconvex function under suitable conditions. Numerical results are reported. This work is supported by Guangxi University SF grands X061041 and China NSF grands 10761001.     

An algorithm for nonconvex programming problems

Branch and bound approaches for nonconvex programming problems had been given in [1] and [4]. Crucial for both are the use of rectangular partitions, convex envelopes and separable nonconvex portions of the objective function and constraints. We want to propose a similar algorithm which solves a sequence of problems in each of which the objective function is convex or even linear. The main difference between this approach and previous approaches is the use of general compact partitions instead of rectangular ones and a different refining rule such that the algorithm does not rely on the concept of convex envelopes and handles non-separable functions.First we describe a general algorithm and prove a convergence theorem under suitable regularity assumptions. Then we give as example an algorithm for concave minimization problems.