Different Conjugate Dual Problems in Vector Optimization and Their Relations

In this paper, three kinds of conjugate dual problems are constructed by virtue of different perturbations to a constrained vector optimization problem. Weak duality, strong duality, and some inclusion relations for the image sets of the three dual problems are established. This research was partially supported by the National Natural Science Foundation of China (Grant Number 60574073) and the Natural Science Foundation Project of CQ CSTC (Grant Number 2007BB6117). The authors thank two anonymous referees for valuable comments and suggestions, which helped improving the paper.     

ON THE GLOBAL CONVERGENCE OF CONJUGATE GRADIENT METHODS WITH INEXACT LINESEARCH

In this paper we consider the global convergence of any conjugate gradient method of the form d1=-g1,dk+1=-gk+1+βkdk(k≥1)with any βk satisfying sume conditions,and with the strong wolfe line search conditions.Under the convex assumption on the objective function,we preve the descenf property and the global convergence of this method.     

On the accelerating property of an algorithm for function minimization without calculating derivatives

This paper studies the speed of convergence of a general algorithm for function minimization without calculating derivatives. This algorithm contains Powell's 1964 algorithm as well as Zangwill's second modification of this procedure. The main results are Theorems 3.1 and 4.1 which show that, if the algorithm behaves well, then asymptotically almost conjugate directions are built; therefore, the algorithm has an every-iteration superlinear speed of convergence. The paper hinges on ideas of McCormick and Ritter and Powell.The authors wish to thank the Namur Department of Mathematics, especially its optimization group, for many discussions and encouragements. The authors also thank the reviewer for many helpful suggestions.     

Synthesis of 4,6-dimethyl-tetrahydro- and hexahydro-dibenzothiophene

2-Bromo-3-methylcyclohexanone was synthesized by conjugate addition of trimethylaluminium to 2-bromo-2-cyclohexen-1-one with copper bromide as catalyst, coupled with 2-methylthiophenol and annulated with the aid of polyphosphoric acid to 4,6-dimethyl-1,2,3,4-tetrahydrodibenzothiophene. The latter was hydrogenated to 4,6-dimethyl-1,2,3,4,4a,9b-hexahydrodibenzothiophene, another intermediate in the hydrodesulfurization of 4,6-dimethyldibenzothiophene, by zinc and trifluoroacetic acid, and dehydrogenated to 4,6-dimethyldibenzothiophene.     

AN AFFINE SCALING INTERIOR ALGORITHM VIA CONJUGATE GRADIENT PATH FOR SOLVING BOUND-CONSTRAINED NONLINEAR SYSTEMS

In this paper we propose an affine scaling interior algorithm via conjugate gradient path for solving nonlinear equality systems subject to bounds on variables. By employing the affine scaling conjugate gradient path search strategy, we obtain an iterative direction by solving the linearize model. By using the line search technique, we will find an acceptable trial step length along this direction which is strictly feasible and makes the objective func- tion nonmonotonically decreasing. The global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions. Furthermore, the numerical results of the proposed algorithm indicate to be effective.     

复变函数中由调和函数求解析函数的方法

本文给出了由解析函数的实部或虚部求该解析函数的几个定理,再以例题说明其应用。这些方法及应用,解决了复变函数中这一问题的理论与计算,而且所给方法能广泛进行推广。     

A new geometric condition for Fenchel's duality in infinite dimensional spaces

In 1951, Fenchel discovered a special duality, which relates the minimization of a sum of two convex functions with the maximization of the sum of concave functions, using conjugates. Fenchel's duality is central to the study of constrained optimization. It requires an existence of an interior point of a convex set which often has empty interior in optimization applications. The well known relaxations of this requirement in the literature are again weaker forms of the interior point condition. Avoiding an interior point condition in duality has so far been a difficult problem. However, a non-interior point type condition is essential for the application of Fenchel's duality to optimization. In this paper we solve this problem by presenting a simple geometric condition in terms of the sum of the epigraphs of conjugate functions. We also establish a necessary and sufficient condition for the ε-subdifferential sum formula in terms of the sum of the epigraphs of conjugate functions. Our results offer further insight into Fenchel's duality. Dedicated to Terry Rockafellar on his 70th birthday     

一个充分下降的有效共轭梯度法

对于大规模无约束优化问题,本文提出了一个充分下降的共轭梯度法公式,并建立相应的算法.该算法在不依赖于任何线搜索条件下,每步迭代都能产生一个充分下降方向.若采用标准Wolfe非精确线搜索求步长,则在常规假设条件下可获得算法良好的全局收敛性最后,对算法进行大规模数值试验,并采用Dolan和More的性能图对试验效果进行刻画,结果表明该算法是有效的.     

一种方向梯度直方图的降维方法

为描述对象的局部外观和形状,方向梯度直方图首先将图像划分成小区域(被称为cell),然后在其上累加像素梯度方向的一维直方图.在被称为block的较大区域(由数个相邻的cell组成)上连接cell的直方图,经归一化处理形成特征向量.为减弱由block引起的区域量化走样,在计算检测窗口的特征时,采取部分重叠block的措施,从而大大增加了特征维度以及目标检测时的计算量.通过扩大参与相邻cell之间像素梯度插值的面积,并设置适当的高斯平滑核尺度,可消除block重叠,从而将64×128尺寸的窗口的方向梯度直方图特征维度由3 780降低为1 152.INRIA的行人数据集实验表明,该方法也可减弱区域量化走样,且其性能与原方向梯度直方图几乎相当,而检测速度却显著提高.     

Necessary conditions for functional differential inclusions

We consider an optimal control problem in which the dynamic equation and cost function depend on the recent past of the trajectory. The regularity assumed in the basic data is Lipschitz continuity with respect to the sup norm. It is shown that, for a given optimal solution, an adjoint arc of bounded variation exists that satisfies an associated Hamiltonian inclusion. From this result, known smooth versions of the Pontryagin maximum principle for hereditary problems can be easily derived. Problems with Euclidean endpoint constraints are also considered.