无约束优化的超记忆梯度算法及其收敛特征

研究一种新的无约束优化超记忆梯度算法,算法在每步迭代中充分利用前面迭代点的信息产生下降方向,利用Wolfe线性搜索产生步长,在较弱的条件下证明了算法的全局收敛性。新算法在每步迭代中不需计算和存储矩阵,适于求解大规模优化问题。     

非凸无约束优化问题的广义拟牛顿法的全局收敛性

本文对无约束优化问题提出一类新的广义拟牛顿法,并采用一类非精确线搜索证明了算法对一般非凸目标函数极小化问题的全局收敛性.     

无约束优化问题的修正PRP共轭梯度法

本文通过结合牛顿法与PRP共轭梯度法提出一修正PRP方法,新方法中包含了二阶导数信息,在适当的假设下算法全局收敛,数值算例表明了算法的有效性.     

求解无约束优化问题的一个秩一适定方法

在本文中,我们给出一个求解无约束优化问题的秩一适定方法,该方法具有下述较好性质:校正矩阵是对称正定的;在适当条件下,对非凸函数拥有全局收敛性.我们还给出数值检验结果.     

一类带非精确线搜索的修改的Broyden算法

对于文（８）和（１４）中提出的修改的Ｂｒｏｙｄｅｎ算法，本文讨论它在线搜索非精确时的收敛性质，证明这类算法作用于梯度满足Ｌｉｐｓｃｈｉｔｚ条件的目标函数时是整体收敛的，当目标函数一致凸时，算法是Ｑ－超线性收敛和二阶收敛的。     

带有固定步长的非单调自适应信赖域算法

提出了求解无约束优化问题带有固定步长的非单调自适应信赖域算法.信赖域半径的修正采用自适应技术,算法在试探步不被接受时,采用固定步长寻找下一迭代点.并在适当的条件下,证明算法具有全局收敛性和超线性收敛性.初步的数值试验表明算法对高维问题具有较好的效果.     

Extended Global Convergence Framework for Unconstrained Optimization

An extension of the global convergence framework for unconstrained derivative-free optimization methods is presented. The extension makes it possible for the framework to include optimization methods with varying cardinality of the ordered direction set. Grid-based search methods are shown to be a special case of the more general extended global convergence framework. Furthermore,the required properties of the sequence of ordered direction sets listed in the definition of grid-based methods are relaxed and simplified by removing the requirement of structural equivalence.     

无约束最优化的一个修正的类BFGS算法

利用前一步得到的曲率信息代替xk到xk+1段二次模型的曲率给出一个具有和BFGS类似的收敛性质的类BFGS算法,并揭示新算法与自调比拟牛顿法的关系.从试验函数库CUTE中选择标准试验函数,对比标准BFGS算法及其它改进BFGS算法进行数值试验.试验结果表明这个新算法的表现有点象自调比拟牛顿算法.     

How bad are the BFGS and DFP methods when the objective function is quadratic?

We study the use of the BFGS and DFP algorithms with step-lengths of one for minimizing quadratic functions of only two variables. The updating formulae in this case imply nonlinear three term recurrence relations between the eigenvalues of consecutive second derivative approximations, which are analysed in order to explain some gross inefficiencies that can occur. Specifically, the BFGS algorithm may require more than 10 iterations to achieve the first decimal place of accuracy, while the performance of the DFP method is far worse. The results help to explain why the DFP method is often less suitable than the BFGS algorithm for general unconstrained optimization calculations, and they show that quadratic functions provide much information about efficiency when the current vector of variables is too far from the solution for an asymptotic convergence analysis.     

多目标优化问题的模糊交叉算法与收敛性

本文研究了目标权重未事先确知的多目标优化问题，建立可以同时确定目标权重与方案相对优属度的模糊交叉迭代算法，严格证明了该算法的局部收敛性．     

Theoretical Convergence Analysis of a General Division–deletion Algorithm for Solving Global Search Problems

Presented in this paper is the prototype of a very general algorithm referred to as Division – Deletion Algorithm (DDA) for solving the most general global search problem. Various necessary conditions, sufficient conditions, and necessary and sufficient conditions for the convergence of the algorithm are proposed and analyzed. As an example of its application, we demonstrate that the convergence of a standard Hansen’s interval algorithm for unconstrained global optimization simply follows from this general theory.     

无约束最优化的Polak—Ribiere和Hestenes—Stiefel共轭梯度法的全局收敛性

本文在很弱的条件下得到了无约束最优化的Polak-Ribiere和Hestenes-Stiefel共轭梯度法的全局收敛性的新结果,这里PR方法和HS方法中的参数β^PRk和β^HSk可以在某个负的区域内取值,这一负的区域与k有关,这些新的收敛性结果改进了文献中已有的结果。数值检验的结果表明了本文中新的PR方法和HS方法是相当有效的。     

Parallel Algorithm for Unconstrained Optimization Based on Decomposition Techniques

We present a numerical implementation of the parallel gradient distribution (PGD) method for the solution of large-scale unconstrained optimization problems. The proposed parallel algorithm is characterized by a parallel phase which exploits the portions of the gradient of the objective function assigned to each processor; then, a coordination phase follows which, by a synchronous interaction scheme, optimizes over the partial results obtained by the parallel phase. The parallel and coordination phases are implemented using a quasi-Newton limited-memory BFGS approach. The computational experiments, carried out on a network of UNIX workstations by using the parallel software tool PVM, show that parallelization efficiency was problem dependent and ranged between 0.15 and 8.75. For the 150 problems solved by PGD on more than one processor, 85 cases had parallelization efficiency below 1, while 65 cases had a parallelization efficiency above 1.     

On the Method of Shortest Residuals for Unconstrained Optimization

The paper discusses several versions of the method of shortest residuals, a specific variant of the conjugate gradient algorithm, first introduced by Lemaréchal and Wolfe and discussed by Hestenes in a quadratic case. In the paper we analyze the global convergence of the versions considered. Numerical comparison of these versions of the method of shortest residuals and an implementation of a standard Polak–Ribière conjugate gradient algorithm is also provided. It supports the claim that the method of shortest residuals is a viable technique, competitive to other conjugate gradient algorithms.     

整体退火遗传算法的几乎处处强收敛性

本文通过鞍论来分析整体退火遗传算法收敛的特性，得出整体退火遗传算法几乎处处强收敛的条件∑n=1^∞e^-δ,Tn＜ ∞，并且给出其收敛率0（1/N N0 （2-Cn^-^N0-mn^-^N0)e^(△-d)/Tm)。     

两类变参数梯度法及收敛性

本文在ＬＭＩＮＮ方法的基础上,提出了两类变参数梯度法,然后证明了这两类方法在非精确线性搜索的Ｗｏｌｆｅ条件下是下降算法且具有全局收敛性。     

广义拟牛顿算法对一般目标函数的收敛性

本文证明了求解无约束最优化的广义拟牛顿算法在Goldstein非精确线搜索下对一般目标函数的全局收敛性，并在一定条件下证明了算法的局部超线性收敛性。     

A Point Balance Algorithm for the Spherical Code Problem

The Spherical Code (SC) problem has many important applications in such fields as physics, molecular biology, signal transmission, chemistry, engineering and mathematics. This paper presents a bilevel optimization formulation of the SC problem. Based on this formulation, the concept of balanced spherical code is introduced and a new approach, the Point Balance Algorithm (PBA), is presented to search for a 1-balanced spherical code. Since an optimal solution of the SC problem (an extremal spherical code) must be a 1-balanced spherical code, PBA can be applied easily to search for an extremal spherical code. In addition, given a certain criterion, PBA can generate efficiently an approximate optimal spherical code on a sphere in the n-dimensional space ⁿ. Some implementation issues of PBA are discussed and putative global optimal solutions of the Fekete problem in 3, 4 and 5-dimensional space are also reported. Finally, an open question about the geometry of Fekete points on the unit sphere in the 3-dimensional space is posed.     

解无约束最优化问题的块共轭方向法

在给出块共轭概念的基础上，提出了适合并行计算的向量组的块共轭化方法，进而得到解无约束最优化问题的并行块共轭方向法．有大量数值结果表明块共轭方向法具有工作量少．适用函数范围广等特点，是一种比较有效的无约束最优化方法．     

A conjugate gradient method with descent direction for unconstrained optimization

A modified conjugate gradient method is presented for solving unconstrained optimization problems, which possesses the following properties: (i) The sufficient descent property is satisfied without any line search; (ii) The search direction will be in a trust region automatically; (iii) The Zoutendijk condition holds for the Wolfe–Powell line search technique; (iv) This method inherits an important property of the well-known Polak–Ribière–Polyak (PRP) method: the tendency to turn towards the steepest descent direction if a small step is generated away from the solution, preventing a sequence of tiny steps from happening. The global convergence and the linearly convergent rate of the given method are established. Numerical results show that this method is interesting.