基于凝聚函数的和声搜索算法求解绝对值方程*

绝对值方程Ax-|x|=b是一个不可微的NP-hard问题。在假设矩阵A的奇异值大于1（这里矩阵A的奇异值定义为矩阵ATA特征值的非负平方根）时,给出了求解绝对值方程一个新的光滑化算法。通过引入一种凝聚函数对绝对值方程进行光滑化处理,得到一个非线性方程组;再引入适当的目标函数,进而把绝对值方程化为无约束优化问题,然后利用和声搜索算法对其进行求解。该算法模拟了音乐创作中乐师们凭借自己的记忆,通过反复调整乐队中各乐器的音调,最终达到一个美妙的和声状态的过程。数值结果表明,该算法收敛快,数值稳定性好,是求解绝对     

基于极大熵牛顿法求解绝对值方程

针对绝对值方程Ax-|x|=b的求解问题,在假设矩阵A的奇异值大于1时,给出了求解绝对值方程的一个新方法。通过引入一种极大熵函数将绝对值方程进行光滑化处理,进而把绝对值方程转换为光滑非线性方程组,然后利用极大熵牛顿法对其进行求解。数值实验结果表明了该方法的正确性和有效性。     

基于差分进化—单纯形混合算法求解绝对值方程*

绝对值方程Ax-｜x｜=b是一个不可微的NP-hard问题。在假设矩阵A的奇异值大于1（这里矩阵A的奇异值定义为矩阵ATA特征值的非负平方根）时, 给出了一种求解绝对值方程的新方法：差分进化—单纯形混合算法。该混合算法充分发挥了差分进化算法的群体搜索性和单纯形算法的局部细致搜索性,同时也克服了差分进化算法后期搜索效率降低和单纯形算法对初始点敏感的缺陷。数值实验表明所设计的混合算法是有效的。     

一种全局和声搜索算法求解绝对值方程

绝对值方程Ax-|x|=b是一个不可微的NP-hard问题。在假设矩阵A的奇异值大于1时, 给出了一个求解绝对值方程的全局和声搜索算法。新的和声搜索算法使用了位置更新和小概率变异策略, 实验结果表明, 该算法具有较强的全局搜索能力, 且收敛快、数值稳定性好、参数少等优点, 是求解绝对值方程的一种有效算法。     

一类不可微方程组的区间算法

针对不可微方程组—绝对值方程Ax+B|x|=b的数值解问题进行研究, 提出了通过构造极大熵函数和新的区间算子对方程进行求解的区间极大熵算法。该算法能同时求出绝对值方程的近似解和估算其近似解的误差限, 并在A的奇异值全部大于|B|的奇异值时, 证明了算法的收敛性且收敛速度至少是线性的。理论分析和数值结果均表明提出的算法是有效的。     

求解绝对值方程的捕鱼算法

提出一种求解绝对值方程的捕鱼算法。算法首先将绝对值问题转化为一个最小化问题,然后使用三种搜索模式对目标函数进行寻优。数值实验结果表明,与粒子群算法和人群搜索算法以及他们的改进算法相比,所提算法不仅获得了稳定的求解结果,而且在最小值、最大值、平均值和方差等指标上均明显优于其他对比算法。     

一种多视图分层重构算法研究

本文介绍了基于奇异值分解的射影重构算法的一般框架,以测量矩阵的秩为4作为约束,以仿射投影逼近透视投影,利用共轭梯度法估计射影深度,通过奇异值分解实现射影重构.利用共轭梯度法确定Kruppa方程中的未知比例因子,然后利用所确定的比例因子线性求解Kruppa方程,进而标定摄像机内参数.在摄像机内参数已知的情况下,求解一个满足欧氏重构条件的非奇异矩阵,然后通过此矩阵将射影重构变换为欧氏重构.实验结果表明所给出的算法是行之有效的.     

基于奇异值的信源数估计方法

信源数估计是空间谱估计中的重要内容,在估计采用不同的判决准则时,往往需要利用信号协方差矩阵的特征值来进行信源数估计。新算法采用数据矩阵的奇异值分解,通过奇异值建立不同判决准则的判决函数。该算法无需进行协方差矩阵估计,也不需要利用奇异值求解特征值,减少了运算量和估计误差。同时,对数据矩阵进行平滑操作,可以解决信号相干性问题。通过数学推导和计算机仿真,证明了算法的正确性。     

改进的RANSAC方法在基础矩阵中的应用

为提高求解精度,提出一种基于改进的随机抽样一致性（RANSAC）算法的基础矩阵求解方法。采用加权策略,将局内点占全部匹配点的比例作为权重函数的自变量;利用本质矩阵和基础矩阵的关系,鉴于本质矩阵两个非零奇异值应该相等这个特性,利用加权因子和本质矩阵的奇异值构造目标函数,这两点改进意味着目标函数中有两个约束条件的限制;利用matlab遗传算法工具箱来求解目标函数的最小值,可以得到准确的基础矩阵。模版图像实验和场景图像实验验证了该算法的有效性。     

基于奇异值分解的压缩感知观测矩阵优化算法

针对压缩感知（CS）中从优化后的Gram矩阵求解观测矩阵时会出现较大相关系数的问题,在利用现有算法得到优化后的Gram矩阵的基础上,通过求解等价变换后的目标函数对观测矩阵行向量的导数得到目标函数取极值时行向量的值,并通过对误差矩阵进行奇异值分解（SVD）在上述行向量的值中选出使得目标函数取最值时行向量的解析式,在此基础上给出了观测矩阵的优化算法：通过借鉴K-SVD算法中逐行优化目标矩阵的思想,对观测矩阵进行逐行迭代优化,并将相邻两轮迭代产生的观测矩阵所对应的相关性之差作为衡量迭代是否结束的条件。仿真结果表明：该算法在观测矩阵与稀疏基的相关性方面优于改进前的算法,从而提高了重构精度。     

Shape optimization of continua using NURBS as basis functions

The present paper introduces a numerical solution to shape optimization problems of domains in which boundary value problems of partial differential equations are defined. In the present paper, the finite element method using NURBS as basis functions in the Galerkin method is applied to solve the boundary value problems and to solve a reshaping problem generated by the H1 gradient method for shape optimization, which has been developed as a general solution to shape optimization problems. Numerical examples of linear elastic continua illustrate that this solution works as well as using the conventional finite element method.     

Simplified neural network for generalized least absolute deviation

This paper proposes a simplified neural network for generalized least absolute deviation by transforming its optimization conditions into a system of double projection equations. The proposed network is proved to be stable in the sense of Lyapunov and converges to an exact optimization solution of the original problem for any starting point. Compared with the existing neural networks for generalized least absolute deviation, the new model has the least neurons and low complexity and is suitable to parallel implementation. The validity and transient behavior of the proposed neural network are demonstrated by numerical examples.

     

Initial condition of costate in linear optimal control using convex analysis

The Pontryagin Maximum Principle is one of the most important results in optimal control, and provides necessary conditions for optimality in the form of a mixed initial/terminal boundary condition on a pair of differential equations for the system state and its conjugate costate. Unfortunately, this mixed boundary value problem is usually difficult to solve, since the Pontryagin Maximum Principle does not give any information on the initial value of the costate. In this paper, we explore an optimal control problem with linear and convex structure and derive the associated dual optimization problem using convex duality, which is often much easier to solve than the original optimal control problem. We present that the solution to the dual optimization problem supplements the necessary conditions of the Pontryagin Maximum Principle, and elaborate the procedure of constructing the optimal control and its corresponding state trajectory in terms of the solution to the dual problem.     

A new technique for curve fitting based on minimum absolute deviations

This paper presents a new technique, the authors have developed for curve fitting based on minimum absolute deviations. The developed technique is a non iterative technique which is easy and simple to apply. The computing time and the computer storage requirements are very small compared to the other available techniques, such as least square curve fitting technique and the standard LAV algorithm which uses linear programming (LP).The problem is formulated in the least absolute value sense (LAV), where the objective is to minimize the absolute deviations. A set of overdetermined equations for the unknown vector's components is obtained. These equations are solved using the left pseudo inverse approach to ascertain which points give the least residuals. Having gained this information, we set to zero a number of residuals equal to the rank of matrix H, these residuals have the smallest values. Several examples which compare results obtained using linear programming (LP) and this new technique are then presented.     

Solving differential equations with Fourier series and Evolution Strategies

A novel mesh-free approach for solving differential equations based on Evolution Strategies (ESs) is presented. Any structure is assumed in the equations making the process general and suitable for linear and nonlinear ordinary and partial differential equations (ODEs and PDEs), as well as systems of ordinary differential equations (SODEs). Candidate solutions are expressed as partial sums of Fourier series. Taking advantage of the decreasing absolute value of the harmonic coefficients with the harmonic order, several ES steps are performed. Harmonic coefficients are taken into account one by one starting with the lower order ones. Experimental results are reported on several problems extracted from the literature to illustrate the potential of the proposed approach. Two cases (an initial value problem and a boundary condition problem) have been solved using numerical methods and a quantitative comparative is performed. In terms of accuracy and storing requirements the proposed approach outperforms the numerical algorithm.     

求解互补问题的极大熵社会认知算法

针对传统算法无法获得互补问题的多个最优解的困难,提出了求解互补问题的社会认知优化算法.通过利用NCP函数,将互补问题的求解转化为一个非光滑方程组问题,然后用凝聚函数对其进行光滑化,进而把互补问题的求解转化为无约束优化问题,利用社会认知算法对其进行求解.该算法是基于社会认知理论,通过一系列的学习代理来模拟人类的社会性以及智能性从而完成对目标的优化.该算法对目标函数的解析性质没有要求且容易实现,数值实验结果表明了该方法是有效的.