可拓展概率逼近中一类矩阵优化问题的有效算法

研究来源于复杂系统离散逼近中的一类可拓展概率逼近模型,欧氏空间中该问题模型可重塑为一类由线性流形和斜流形组成的乘积流形约束矩阵优化问题.结合乘积流形的几何性质,基于Zhang-Hager技术拓展,本文设计一类适用于问题模型的黎曼非线性共轭梯度法,并给出算法全局收敛性分析.数值实验验证所提算法对于问题模型求解是高效可行的,且与其它黎曼梯度类算法及黎曼优化工具箱中已有的黎曼梯度类算法和二阶算法相比在迭代效率上有一定优势.     

稳健矩阵回归模型和方法研究

随着大数据时代的到来,我们面临的数据越来越复杂,其中待估系数为矩阵的模型亟待构造和求解.无论在统计还是优化领域,许多专家学者都致力于矩阵模型的统计性质分析及寻找其最优解的算法设计.当随机误差期望为0且同方差时,采用基于最小二乘的模型可以很好地解决问题.当随机误差异方差,分布为重尾分布(如双指数分布,t-分布等)或数据含有异常值时,需要考虑稳健的方法来求解问题.常用的稳健方法有最小一乘,分位数,Huber等.目前稳健方法的研究大多集中于线性回归问题,对于矩阵回归问题的研究比较缺乏.本文从最小二乘模型讲起,对矩阵回归问题进行了总结和评述,同时列出了一些文献和简要介绍了我们的近期的部分工作.最后对于稳健矩阵回归,我们提出了一些展望和设想.     

线性约束下Hermite-广义反Hamilton矩阵的最佳逼近问题

本文利用对称向量与反对称向量的特征性质,给出了约束矩阵集合非空的充分必要条件及矩阵的一般表达式.运用空间分解理论和闭凸集上的逼近理论,得到了任一n阶复矩阵在约束矩阵集合中的惟一最佳逼近解.     

矩阵方程AX＝B的中心对称解及其最佳逼近

基于一种新的思路,本文将一类约束矩阵方程问题化为无约束矩阵方程问题,从而轻易地获得了简洁且满意的结果.     

基于信息熵的改进鲸鱼优化算法

针对鲸鱼优化算法在面对复杂优化问题时,存在易陷入局部最优和收敛精度低等缺点,在原始鲸鱼算法的基础上,提出了信息熵的改进鲸鱼优化算法.信息熵本身是一种不确定的度量,利用信息熵在路径选择时调控鲸鱼搜索的范围,克服基本鲸鱼优化算法的不足,使算法的全局收敛速度得到提高.通过选取6个标准测试函数进行仿真实验,对改进鲸鱼优化算法、基本鲸鱼优化算法、粒子群算法进行比较,数据结果表明改进鲸鱼算法在处理高维复杂组合优化问题上的可行性与有效性.     

特殊双变量矩阵方程组异类约束解的MCG算法

本文研究了约束矩阵方程问题中异类约束解的迭代算法.利用修正共轭梯度法,求得了特殊双变量线性矩阵方程组的异类约束解,选取特殊的初始矩阵,得到唯一极小范数异类约束解.理论证明和数值算例验证了该方法的有限步收敛性,推广了修正共轭梯度法在求约束矩阵方程问题中的应用范围.     

基于大规模高维线性回归模型的分布式计算方法研究

大数据背景下挖掘大规模高维数据所隐藏的信息备受关注.本文主要目的是采用分布式优化方法解决加SCAD和Adaptive LASSO惩罚的高维线性回归中的参数估计和变量选择问题.主要方法是通过构造全局损失函数的一个交互有效的正则化替代损失函数,把基于全局损失函数的优化问题转化为基于替代损失函数的优化问题.本文设计的修正的ADMM算法,在计算上,只需要子机器基于局部数据计算梯度,而主机器进行参数估计和变量选择.在主从机器交互复杂度上,基于替代损失函数所得的估计误差收敛于基于全局损失函数所得的估计误差.通过模拟和实证研究进一步验证本文提出的分布式计算方法在实际生活中的可行性和实用性.     

基于偏最小二乘变换与Kriging模型的全局优化方法

针对传统Kriging模型在多变量(高维)输入全局优化中因超参数过多而引发收敛速度慢,精度低,建模效率不高问题,提出了基于偏最小二乘变换技术和Kriging模型的有效全局优化方法.首先,构造偏最小二乘高斯核函数;其次,借助差分进化算法寻找满足期望改进准则最大化条件的新样本点;然后,将不同核函数和期望改进准则组合,构建四种有效全局优化算法并进行比较;最后,数值算例结果表明,基于偏最小二乘变换的Kriging全局优化方法在解决高维全局优化问题方面相比于标准的全局优化算法在收敛精度及收敛速度方面更具优势.     

高维多重双正交小波包

本文给出对应于高维多重尺度函数的双正交多小波包的定义及其构造方法.讨论了高维不可分双正交多小波包的双正交性.     

二维有序样本的有约束系统聚类

二维有序样本进行聚类必须满足两个要求：（１）类内各单元的相似性和类间的差异性；（２）各单元在位置上的有序性和类内的连通性。根据这些要求，将各单元观测指标间的距离矩阵作为聚类的指示矩将各单元之间的区位联系矩阵作为聚类的约束矩阵，在约束矩阵给出的约束条件之下，以类间单元指标的最大距离作为类间相似性指标，在指示矩阵中通过逐步聚并而将全部单元合并归类，即可得出满足要求的样本分类。     

The Dynamical System Approach to Multivariate Data Analysis

In this survey a number of problems arising in multivariate data analysis (MDA) are listed and reformulated as matrix fitting (e.g., least-squares, maximum likelihood, etc.) constrained optimization problems (OPs). The goal is to demonstrate that consideration and solution of these diverse MDA problems can be unified by means of the dynamical system approach. The approach transforms the MDA problems into dynamical systems on a manifold defined by the constraints of the original OP.     

High-Dimensional Covariance Matrix Estimation Based on Network

A new method for estimating high-dimensional covariance matrix based on network structure with heteroscedasticity of response variables is proposed in this paper. This method greatly reduces the computational complexity by transforming the high-dimensional covariance matrix estimation problem into a low-dimensional linear regression problem. Even if the size of sample is finite, the estimation method is still effective. The error of estimation will decrease with the increase of matrix dimension. In addition, this paper presents a method of identifying influential nodes in network via covariance matrix. This method is very suitable for academic cooperation networks by taking into account both the contribution of the node itself and the impact of the node on other nodes.     

A two-stage sequential conditional selection approach to sparse high-dimensional multivariate regression models

In this article, we deal with sparse high-dimensional multivariate regression models. The models distinguish themselves from ordinary multivariate regression models in two aspects: (1) the dimension of the response vector and the number of covariates diverge to infinity; (2) the nonzero entries of the coefficient matrix and the precision matrix are sparse. We develop a two-stage sequential conditional selection (TSCS) approach to the identification and estimation of the nonzeros of the coefficient matrix and the precision matrix. It is established that the TSCS is selection consistent for the identification of the nonzeros of both the coefficient matrix and the precision matrix. Simulation studies are carried out to compare TSCS with the existing state-of-the-art methods, which demonstrates that the TSCS approach outperforms the existing methods. As an illustration, the TSCS approach is also applied to a real dataset.

     

On joint probabilistic constraints with Gaussian coefficient matrix

The paper deals with joint probabilistic constraints defined by a Gaussian coefficient matrix. It is shown how to explicitly reduce the computation of values and gradients of the underlying probability function to that of Gaussian distribution functions. This allows us to employ existing efficient algorithms for calculating this latter class of functions in order to solve probabilistically constrained optimization problems of the indicated type. Results are illustrated by an example from energy production.     

Unified Gradient Approach to Performance Optimization Under a Pole Assignment Constraint

General closed-loop performance optimization problems with pole assignment constraint are considered in this paper under a unified framework. By introducing a free-parameter matrix and a matrix function based on the solution of a Sylvester equation, the constrained optimization problem is transformed into an unconstrained one, thus reducing the problem of closed-loop performance optimization with pole placement constraint to the computation of the gradient of the performance index with respect to the free-parameter matrix. Several classical performance indices are then optimized under the pole placement constraint. The effectiveness of the proposed gradient method is illustrated with an example.     

用M方法检测回归系数矩阵的秩

为了确定多重线性回归模型中回归系数矩阵的秩，　本文提出了一个基于Ｍ估计的模型选择程序，　且在较弱的条件下建立了回归系数矩阵的秩的估计的强相合性。     

On minimax eigenvalue problems via constrained optimization

In this paper, we consider the problem of minimizing the maximum eigenvalues of a matrix. The aim is to show that this optimization problem can be transformed into a standard nonlinearly constrained optimization problem, and hence is solvable by existing software packages. For illustration, two examples are solved by using the proposed method.     

A Global Optimization Method,QBB, for Twice-Differentiable Nonconvex Optimization Problem

A global optimization method, QBB, for twice-differentiable NLPs (Non-Linear Programming) is developed to operate within a branch-and-bound framework and require the construction of a relaxed convex problem on the basis of the quadratic lower bounding functions for the generic nonconvex structures. Within an exhaustive simplicial division of the constrained region, the rigorous quadratic underestimation function is constructed for the generic nonconvex function structure by virtue of the maximal eigenvalue analysis of the interval Hessian matrix. Each valid lower bound of the NLP problem with the division progress is computed by the convex programming of the relaxed optimization problem obtained by preserving the convex or linear terms, replacing the concave term with linear convex envelope, underestimating the special terms and the generic terms by using their customized tight convex lower bounding functions or the valid quadratic lower bounding functions, respectively. The standard convergence properties of the QBB algorithm for nonconvex global optimization problems are guaranteed. The preliminary computation studies are presented in order to evaluate the algorithmic efficiency of the proposed QBB approach.