Math_Strong_Induction
python牛顿-拉弗森方法
python牛顿-拉弗森方法标题：Python牛顿-拉弗森方法：探索数学之美在我们的日常生活中，数学无处不在。
无论是解决实际问题，还是探索未知领域，数学都发挥着至关重要的作用。
今天，我们将一起探索一种神奇的数学方法——牛顿-拉弗森方法。
这种方法在科学计算中发挥着重要作用，尤其在解决非线性方程和优化问题时表现出色。
在这篇文章中，我们将深入了解Python中牛顿-拉弗森方法的工作原理，并使用它来解决一些有趣的问题。
一、牛顿-拉弗森方法的概述牛顿-拉弗森方法是求解非线性方程根的一种数值方法。
这种方法基于牛顿法的思想，通过利用泰勒级数的概念，将非线性方程转化为一系列线性方程组，从而进行求解。
这种方法在处理复杂方程和优化问题时具有很高的精度和效率。
二、Python实现Python是一种强大的编程语言，为科学计算提供了丰富的库和工具。
在Python中，我们可以使用SciPy库中的`newton_krylov`函数来实现牛顿-拉弗森方法。
这个函数接受一个非线性方程、初始猜测值、步长等参数，并返回方程的根或近似根。
以下是一个简单的Python代码示例，演示了如何使用牛顿-拉弗森方法求解非线性方程：```pythonfromscipy.optimizeimportnewton_krylovimportnumpyasnp#定义一个非线性方程f(x)=x^3-x-1deff(x):returnx**3-x-1#初始猜测值x_0=1.5x_0=1.5#牛顿-拉弗森方法求解x_new=newton_krylov(f,x_0,tol=1e-6,maxiter=100)print("近似根:",x_new)```三、应用案例：极坐标系下的抛物型偏微分方程在实际应用中，牛顿-拉弗森方法可用于解决各种复杂的数学问题。
让我们尝试解决一个极坐标系下的抛物型偏微分方程，这个方程在物理、工程等领域有广泛应用。
假设我们有一个二维区域D，在该区域内，一个物体在重力作用下的运动满足以下偏微分方程：ρ(x,y)=x^2+y^2-1。
rastringin函数
rastringin函数拉斯特林函数（Rastrigin function）是一种常用的优化问题测试函数，通常用于评估优化算法的性能。
它是一个多维非凸函数，具有多个局部最小值。
下面我会从不同角度对拉斯特林函数进行详细介绍。
首先，拉斯特林函数的数学表达式如下：f(x) = A n + Σ(x_i^2 A cos(2πx_i))。
其中，x是一个n维向量，x_i是向量中的第i个元素，A是一个常数，通常取为10。
函数的取值范围是[-5.12, 5.12]。
从几何角度来看，拉斯特林函数可以看作是一个具有多个峰值和山谷的地形图。
每个峰值代表一个局部最小值，而全局最小值位于函数的中心。
函数的表面呈现出起伏不平的形状，使得优化算法在搜索最优解时面临较大的挑战。
从优化问题的角度来看，拉斯特林函数是一个典型的非凸优化问题。
由于函数具有多个局部最小值，优化算法需要克服陷入局部最小值的困境，以找到全局最小值。
这使得拉斯特林函数成为评估优化算法鲁棒性和全局搜索能力的重要基准函数。
从算法性能评估的角度来看，拉斯特林函数具有以下特点：1. 多个局部最小值，函数具有多个局部最小值，使得优化算法容易陷入局部最优解。
2. 高度非线性，函数的非线性特性增加了优化问题的复杂性。
3. 平滑性，函数的表面相对平滑，没有明显的锐变或间断点。
在实际应用中，拉斯特林函数广泛用于测试和比较不同优化算法的性能。
通过对算法在拉斯特林函数上的表现进行评估，可以帮助研究人员选择合适的优化算法，并进行算法参数的调优。
总结起来，拉斯特林函数是一个具有多个局部最小值的非凸优化问题测试函数。
它在优化算法性能评估和参数调优中起到重要的作用。
对于优化算法的设计和改进，深入理解和研究拉斯特林函数是非常有意义的。
MathematicalInduction.ppt
CS 202 Epp, chapter 4 Aaron Bloomfield
1
How do you climb infinite stairs?
• Not a rhetorical question!
• First, you get to the base platform of the staircase
• Recall the inductive hypothesis:
k
2i 1 k 2
i 1
• Proof of inductive step:
k 1
2i 1 (k 1)2
i 1 k
2(k 1) 1 2i 1 k 2 2k 1 i 1 2k 1
• Three parts:
– Base case(s): show it is true for one element
• (get to the stair’s base platform)
– Inductive hypothesis: assume it is true for any given element
• Show that the sum of the first n odd
integers is n2
– Example: If n = 5, 1+3+5+7+9 = 25 = 52
– Formally, show:
n
n P(n) whereP(n) 2i 1 n2
i 1
• Base case: Show that P(1) is true
• Then, show that if it’s true for some value n, then it is true for n+1
MIT基础数学讲义(计算机系)lecture3
Lecture 3 11 Sep 97
Lecture Notes 1 Strong Induction
Claim 2.1
8n
8, it is possible to produce n cents of postage from 3 and 5 cents stamps.
Now let's preview the proof. A proof by strong induction will have the same four-part structure as an ordinary induction proof. The base case, P (8), will be easy because we can produce 8 cents of postage from a 3 cent stamp and a 5 cent stamp. In the inductive step we assume that we know how to produce 8 9 : : : n cents of postage, and we have to show how to produce n + 1 cents of postage. One way is rst to create n ; 2 cents of postage using strong induction and then to add a three cent stamp. We have to be careful there is a pitfall here. If n is 8 or 9, then we can not use the trick of creating n + 1 cents of postage from n ; 2 cents and a 3 cent stamp. In these cases, n ; 2 is less than 8. None of the assumptions that strong induction permits help us make less than 8 cents of postage. Fortunately, making n + 1 cents of postage when n is 8 or 9 is very easy we can do this directly. Proof. The proof is by strong induction. Let P (n) be the predicate that it is possible to produce n cents of postage from 3 and 5 cent stamps. In the basis step, P (8) is true because 8 cents of postage can be made from a 3 cent stamp and a 5 cent stamp.
离散数学英文版
– during the class,
Discrete Structures
CSCI-235 Dr. Kaitian Cao
Introduce Myself
• What is my name? • Where am I from? • What are my current research fields?
Prerequisite and Description
Bayes’ theorem to dependent events, and solve problems such as Hashing 5. Relate ideas of mathematical induction(归纳) to recursion and apply it to problems in computer science setting 6. Apply the basic counting principles, permutations and combinations to problems in computer science setting 7. Implementing algorithms in C or C++ programs
Learning Outcomes
1. An ability to apply algorithmic concepts and constructs in problem analysis and design (Specific about algorithmic concepts and constructs)
• Prepare two notebooks for the assignments
多变量无约束优化牛顿法python代码
多变量无约束优化牛顿法python代码多变量无约束优化牛顿法是一种常用的数学优化算法，在机器学习、数据分析、统计学等领域广泛应用。
本文将介绍多变量无约束优化牛顿法的基本原理，并提供Python代码实现。
牛顿法是一种求解非线性方程和最优化问题的迭代方法。
它利用函数的二阶导数信息来确定函数的极值点。
对于多变量无约束优化问题，牛顿法的迭代公式为：x_{k+1} = x_k - H(f(x_k))^{-1}abla f(x_k)其中，x_k 是第k次迭代的值，H(f(x_k)) 是函数f在x_k处的Hessian矩阵，abla f(x_k) 是函数f在x_k处的梯度向量。
以下是使用Python实现多变量无约束优化牛顿法的代码：```pythonimport numpy as npdef newton_opt(f, f_grad, f_hess, x0, tol=1e-6,max_iter=100):'''多变量无约束优化牛顿法:param f: 目标函数:param f_grad: 目标函数的梯度:param f_hess: 目标函数的Hessian矩阵:param x0: 初始点:param tol: 迭代停止条件:param max_iter: 最大迭代次数:return: 迭代结果'''x = x0for i in range(max_iter):g = f_grad(x)H = f_hess(x)dx = np.linalg.solve(H, -g)x += dxif np.linalg.norm(dx) < tol:breakreturn x```其中，f_grad(x) 和 f_hess(x) 分别是目标函数f在点x处的梯度向量和Hessian矩阵。
np.linalg.solve(H, -g) 是用于求解线性方程组的函数，用于计算牛顿法中的方向dx。
数学收敛 英文
数学收敛英文Title: Mathematical Convergence: A Comprehensive Overview Introduction:Mathematical convergence refers to the concept of a sequence or a series approaching a specific value or limit as the number of terms increases indefinitely. It is a fundamental concept in mathematics and finds applications in various fields such as analysis, calculus, and probability theory. This essay aims to provide a comprehensive overviewof mathematical convergence, highlighting its different types, properties, and applications.1. Types of Convergence:1.1 Pointwise Convergence:Pointwise convergence occurs when a sequence of functions approaches a specific value for each point in the domain. Itis defined as follows: for every x in a given set D and for every ε > 0, there exists an N such that |fn(x) - f(x)| < ε for all n ≥ N. The concept is commonly used in calculus and analysis.1.2 Uniform Convergence:Uniform convergence is a stronger form of convergence compared to pointwise convergence. A sequence of functions converges uniformly if, for every ε > 0, there exists an N such that |fn(x) - f(x)| < ε for all x in the domain and n ≥ N. Uniform convergence has properties that allow for the interchange of limiting operations and is essential in the study of continuity and integration.1.3 Absolute Convergence:Absolute convergence is applicable to series and refers to the convergence of the series of absolute values. A series is absolutely convergent if the series of absolute values converges. Absolute convergence guarantees that the seriescan be rearranged without changing its sum and is extensively studied in calculus and real analysis.1.4 Conditional Convergence:In contrast to absolute convergence, a series is conditionally convergent if it converges but the series of absolute values diverges. Conditionally convergent series exhibit different behavior when rearranged, and they arise in various mathematical phenomena, such as alternating series.2. Properties of Convergence:2.1 Monotone Convergence Theorem:The monotone convergence theorem states that if a sequence is bounded and monotonic, it will converge. This property is widely used in analysis, particularly in proving the convergence of important sequences, including those that define irrational numbers.2.2 Comparison Test:The comparison test is employed to determine the convergence or divergence of a series by comparing it with a known series. If the known series converges and the given series is less than it for all terms, then the given series also converges. Conversely, if the known series diverges and the given series is greater than it for all terms, then the given series also diverges.2.3 Ratio Test:The ratio test is a powerful tool to determine the convergence of a series when the terms involve factorials or powers. It states that if the absolute value of the ratio of consecutive terms approaches a limit less than 1 as n tends to infinity, then the series converges.3. Applications of Convergence:3.1 Calculus:Convergence plays a crucial role in calculus, where it is used to define limits, continuity, and differentiability. Convergent sequences and series provide a foundation for the development of differential and integral calculus, allowing the analysis of functions and their behavior.3.2 Numerical Analysis:In numerical analysis, convergence is essential in iterative methods for solving equations or approximating solutions. Methods such as the Newton-Raphson method, fixed-point iteration, and the bisection method rely on the convergence of a sequence of approximations to a solution.3.3 Probability Theory:Convergence concepts are widely utilized in probability theory, particularly in the study of random variables and their distributions. The weak law of large numbers and the central limit theorem are prominent examples where theconvergence of sequences of random variables is essential to derive meaningful results.Conclusion:Mathematical convergence is a fundamental concept that underlies many mathematical branches and applications. Understanding the different types of convergence, their properties, and applications provides a solid foundation for studying advanced mathematics and its various real-world applications. By exploring the principles of mathematical convergence, researchers and practitioners can delve deeper into the intricacies of mathematical analysis and its practical implications.。
一种改进的支持向量数据描述故障诊断方法
第26卷第7期V ol.26No.7控制与决策Control and Decision2011年7月Jul.2011一种改进的支持向量数据描述故障诊断方法文章编号:1001-0920(2011)07-0967-06唐明珠1,王岳斌2,阳春华1(1.中南大学信息科学与工程学院，长沙410083；2.湖南理工学院计算机学院，湖南岳阳414006)摘要:针对故障诊断中故障类样本难于获取以及不均衡类问题,提出了基于粒子群和滑动窗口的支持向量数据描述(M-SVDD)故障诊断方法.该方法利用粒子群优化支持向量数据描述的核参数,同时引入滑动窗口技术,通过大窗口大小来控制故障诊断模型的训练样本数,根据小窗口的预测误差变化动态调整大窗口的大小.采用该方法对铜转炉吹炼过程进行故障诊断的实验结果表明,该方法能有效抑制过拟合现象,具有故障敏感性高、泛化能力强等特点.关键词:支持向量数据描述；粒子群；故障诊断；不均衡类；滑动窗口中图分类号:TP18文献标识码:AModified support vector data description for fault diagnosisTANG Ming-zhu1,WANG Yue-bin2,YANG Chun-hua1(1.School of Information Science and Engineering，Central South University，Changsha410083，China；2.School of Computer，Hu’nan Institute of Science and Technology，Yueyang414006，China．Correspondent：YANG Chun-hua, E-mail：ychh@)Abstract：Aiming at the difficulty of obtaining fault samples and class imbalanced problem,a modified support vector data description for fault diagnosis based on both particle swarm optimization and sliding windows(M-SVDD)is proposed in this paper.The kernel parameters of support vector data description are optimized by the particle swarm optimization. At the same time,the sliding window technique is introduced.The number of training samples for fault diagnosis model is controlled by a dynamic adjusted large window.The size of the large window is adjusted dynamically according to the changes of predicting error of the small window.M-SVDD is applied to the fault diagnosis of copper-converting smelting process.The experimental results show that M-SVDD can prevent effectively the phenomenon of over-fitting and has good fault sensitivity and generalization.Key words：support vector data description；particle swarm optimization；fault diagnosis；class imbalanced；sliding window1引言判断复杂工业过程是否发生故障是安全生产的重要步骤之一,及时发现故障对于提高产品的产量和质量具有重要的意义.对该过程采样时,往往很难获取所有故障类别,甚至有些故障样本无法获得.因此,在不考虑或较少考虑故障类样本的前提下,研究如何利用比较容易获取的复杂工业过程正常类样本构造一个能准确判断复杂工业过程是否处于正常状态的诊断模型更具实用价值.近年来,基于数据驱动的故障诊断方法得到了广泛应用,包括代价敏感支持向量机[1]、代价敏感概率神经网络[2]和K-最邻近(KNN)[3]等方法.代价敏感概率神经网络和代价敏感支持向量机在进行故障诊断时均假设样本集中至少有两类样本,而在实际应用中往往类别不均衡,甚至只能获取某一类样本,如正常类样本.因此,这两种方法利用有限的有色冶金过程故障类样本和正常类样本建立的故障诊断模型应用于实际有色冶金过程时受到限制.文献[3]提出的KNN算法用一类样本便可以进行故障诊断,但在预测时存在过拟合现象.Tax等人在SVM的基础上提出了支持向量数据描述(SVDD)[4],在机械故障诊断、图像检测等方面得收稿日期:2010-03-25；修回日期:2010-06-11.基金项目:国家杰出青年科学基金项目(61025015)；国家自然科学基金项目(60874069)；国家863计划项目(2009AA04Z137)；中南大学优秀博士学位论文扶植项目.作者简介:唐明珠(1983−),男,博士生,从事数据挖掘、机器学习等研究；阳春华(1965−),女,教授,博士生导师,从事复杂工业过程建模与优化控制等研究.968控制与决策第26卷到了成功应用[5-6].但是,由于引入了核函数,其核参数的选择直接影响了SVDD的性能,只有选择合适的核参数才能获得理想的效果.目前,交叉验证是人们常用的核参数选择方法[7],但该方法耗时严重,而且是经验上的最优值.文献[8]提出了一种改善支撑向量域描述性能的核优化算法,但该算法要求数据服从高斯分布,且各分量独立、同方差.因此,上述两种核参数优化方法的应用受到限制.粒子群(PSO)优化算法在解决大规模非线性的连续问题中具有更高的优化效率和更好的优化结果,在工业优化控制中得到了广泛应用[9-10],因而可用PSO来优化SVDD的核参数.另外,SVDD很难应用大规模样本集,文献[11-12]采用滑动窗口技术,通过一个动态调整的大窗口来控制故障诊断模型中所包含的最近产生工况,以反映工况的变化,同时能控制样本集的规模.本文利用粒子群优化支持向量数据描述的核参数,通过滑动窗口技术中一个动态调整的大窗口来控制故障诊断模型中包含的最近产生工况,以反映工况的变化,根据小窗口的预测误差变化动态调整大窗口的大小,提出了基于粒子群和滑动窗口的支持向量数据描述(M-SVDD)故障诊断方法.实验结果表明,该方法具有良好的故障敏感性,能有效防止过拟合现象.2支持向量数据描述支持向量数据描述(SVDD)将需要描述的样本作为一个整体,试图建立一个封闭而紧凑的区域Ω,使被描述的样本全部或尽可能多地包容在Ω内,而非该类样本应没有或尽可能少地包含在Ω内[4].设训练样本集X中包含n个样本x i,i=1,2,⋅⋅⋅,n,x i∈R d,设法找到一个以a为中心,以R为半径的能包含所有训练样本的最小超球体.如果直接进行优化处理,所得到的优化区域是一个超球体.为了使优化区域更紧凑,采用核函数替代计算中的点积运算,通过非线性映射函数φ实现低维空间到高维空间的映射,然后求解在高维空间包含所有样本点的最小超球体.为增强支持向量数据域描述的鲁棒性,引入松弛因子ξi⩾0,i=1,2,⋅⋅⋅,n.寻找一个满足mercer条件的核函数K(x i⋅x j),使得K(x i⋅x j)=⟨φ(x i),φ(x j)⟩.这样,优化问题可描述为min R f(R,a)=R2+Cn∑i=1ξi.s.t.(φ(x i)−a)(φ(x i)−a)T⩽R2+ξi,i=1,2,⋅⋅⋅,n;ξi⩾0,i=1,2,⋅⋅⋅,n.(1)其中参数C用于控制最小超球体体积与数据误差之间的平衡,称为SVDD的控制参数C.该问题的对偶形式为maxαiL=n∑i=1αi K(x i⋅x i)−∑i,jαiαj K(x i⋅x i).s.t.n∑i=1αi=1,i=1,2,⋅⋅⋅,n;0⩽αi⩽C,i=1,2,⋅⋅⋅,n.(2)其中参数αi为拉格朗日乘子.解该优化问题可得到参数αi,称αi>0对应的样本x i为支持向量.由定义可知,R2为超球体中心a与边界上任何支持向量之间的距离,即与拉格朗日乘子0<αi<C的支持向量(样本)之间的距离,但不包含拉格朗日乘子为αi=C的支持向量(样本).因此R2=K(x k⋅x k)−2∑iαi K(x i⋅x k)+∑i,jαiαj K(x i⋅x j),∀x k∈SV<C.(3)根据式(3),用任意一个支持向量可求出R2.对于测试样本z,设D(a,z)=K(z⋅z)−2∑iαi K(z⋅x i)+∑i,jαiαj K(x i⋅x j),i=1,2,⋅⋅⋅,n,j=1,2,⋅⋅⋅,n.(4)若D(a,z)⩽R2,则z被判决为正常类样本;否则,z被判决为故障类样本.由文献[4]可知,在常见的4种核函数中,高斯核函数在SVDD中能取得最好的分类效果.当选择高斯核函数K G(x,y)=exp(−∥x−y∥2σ2),σ>0时,式(2)和(4)分别变为L=n∑i=1αi K G(x i⋅x i)−∑i,jαiαj K G(x i⋅x j)= 1−∑iα2i−∑i=jαiαj K G(x i⋅x j);s.t.0⩽αi⩽C,i=1,2,⋅⋅⋅,n,j=1,2,⋅⋅⋅,n.(5) D(a,z)=K G(z⋅z)−2∑iαi K G(z⋅x i)+∑i,jαiαj K G(x i⋅x j)=1−2∑iαi K G(z⋅x i)+∑iα2i+∑i=jαiαj K G(x i⋅x j),i=1,2,⋅⋅⋅,n,j=1,2,⋅⋅⋅,n.(6)式(6)中只有一项2∑iαi K G(z⋅x i)与测试样本z有关系.第7期唐明珠等:一种改进的支持向量数据描述故障诊断方法9693线性权重粒子群算法PSO 初始化为一群随机粒子(随机解),然后通过迭代找到最优解[10].假设d 维搜索空间中的第i 个粒子的位置和速度分别为S i =(s i,1,s i,2,⋅⋅⋅,s i,d )和V i =(v i,1,v i,2,⋅⋅⋅,v i,d ),在每一次迭代中,粒子通过跟踪2个最优化解来更新自己,第1个是粒子本身所找到的最优化解,即个体极值p best ,P i =(p i,1,p i,2,⋅⋅⋅,p i,d );另一个是整个种群目前找到的最优解,即全局最优解g best .p g 在找到这2个最优值后,粒子通过下面公式更新自己的速度和位置:v (t +1)i,j =wv (t )i,j +c 1r 1(p i,j −s (t )i,j )+c 2r 2(p g,j −s (t )i,j );(7)s (t +1)i,j=s (t )i,j +v (t )i,j ,j =1,2,⋅⋅⋅,d.(8)其中:w 为惯性权重因子,c 1和c 2为正的学习因子,r 1和r 2为0到1之间均匀分布的随机数.在PSO 的可调参数中,惯性权重w 是最重要的参数,t 和t +1分别为当前迭代次数和下一次迭代次数.因为较大的惯性权重有利于跳出局部极小点,便于全局搜索,利于算法收敛,因此针对PSO 算法容易早熟以及算法后期易在全局最优解附近产生振荡现象,可采用线性变化的权重,让惯性权重从最大值w max 线性减小到最小值w min .w 随算法迭代次数的变化公式为w =w max −t (w max −w min )t max.(9)其中:w max 和w min 分别表示w 的最大值和最小值,t 和t max 分别为当前迭代次数和最大迭代次数.4M-SVDD针对SVDD 的核参数σ选择问题构造PSO 的适应度函数.通过优化迭代计算核参数σ,可以提高SVDD 的泛化性能和故障敏感性.对于未知样本,预测误差要尽可能地小.用预测误差作为适应度函数,对每一个粒子进行判断,最终选出的预测误差最小时所对应的粒子即为最优解y ∗=min 1NN∑i =1(y (t )i −y (p )i )2.(10)其中:y (t )i ,y (p )i 分别表示第i 个样本真实的类别值(正常类取值为1,异常类取值为−1)和预测的类别值;N 表示训练集样本数;y ∗为适应度函数的最小值,即最优值.为了将训练样本集规模控制在合理范围内,本文采用了滑动窗口技术.滑动窗口技术是指利用一个动态调整的大窗口控制预测模型中所包含的最近工况样本,以反映工况的改变,同时引入一个动态调整的小窗口来决定增加、减少或保持大窗口.当大窗口和小窗口中的预测误差一致增加时,表明过时的工况信息增加了预测误差,因此应缩短2个窗口的长度.当大窗口和小窗口中的预测误差一致减少时,说明新的工况加强了原有模型的预测能力,因此应扩大2个窗口的长度.当大窗口和小窗口中的预测误差稳定时,表明新的工况和原有模型保持一致,此时应保持2个窗口的长度不变.当大窗口和小窗口中的预测误差不包含在上述任何一种情况时,为避免过早地忘掉旧的训练样本,可增加大窗口的大小,并保持小窗口的长度不变.利用M-SVDD 进行故障诊断可分为2步:首先利用PSO 优化SVDD 的核参数,利用滑动窗口技术中的大窗口控制样本的规模,以获得参数优化后的SVDD 模型;然后将该模型用于测试新的工况样本类别.M-SVDD 算法描述如下:算法1M-SVDD 算法.输入参数:PSO 算法参数,大窗口的大小L win ,小窗口的大小S win ,预测误差y ∗L win ,y ∗S win ,样本集dataset ,样本比例p ;输出参数:M-SVDD 的故障敏感性和测试样本集的预测类别.Step 1:初始化PSO 算法参数:粒子数目M ,粒子位置S 和速度V ,学习因子c 1和c 2,最大权重w max 和最小权重w min ,最大迭代次数t max ,问题维数D ,PSO 的预设精度,大窗口大小L win ,小窗口大小S win ,样本集dataset ,样本比例p ,训练样本集trainingset =dataset ∗p ,测试样本集testset=dataset ∗(1−p ),预测误差temp L win =y ∗L win ,temp S win =y ∗S win ;Step 2:While (测试样本集testset 不为空){Step 3:根据文献[4]的方法,计算控制参数C ;Step 4:调用PSO 算法优化SVDD 的核参数,返回训练样本集trainingset SVDD 最优核参数;Step 5:将测试样本集testset 中的采样时间最早的样本作为测试样本z ;Step 6:用决策函数(6)对测试样本进行测试,判断该样本是否为故障类样本;Step 7:如果预测样本为故障类,则将该样本存入故障类样本库;如果预测样本为正常类,则将该样本存入正常类样本库;Step 8:将测试样本z 添加到大窗口和小窗口,同时删除大窗口和小窗口的采样时间最早的样本;Step 9:计算大窗口和小窗口中的新的预测误差y ∗L win ,y ∗S win ;Step 10:如果y ∗L win <temp L win ,y ∗S win <temp S win ,970控制与决策第26卷则增大L win,S win;Step11:如果y∗L win>temp L win,y∗S win>temp S win,则减小L win,S win;Step12:如果y∗L win=temp L win,y∗S win=temp S win,则保持L win,S win;Step13:如果新预测误差为其他情况,则增加大窗口的大小L win,并保持小窗口的长度S win不变;Step14:保存新的预测误差y∗L win,y∗S win;Step15:更新预测误差temp L win=y∗L win,temp S win=y∗S win;Step16:在测试样本集testset中删除测试样本z;}Step17:根据每一个测试样本的预测误差计算故障敏感性,输出故障敏感性和测试样本的类别.PSO算法优化SVDD核参数算法描述如下:算法2M-SVDD核参数算法.输入参数:PSO算法的粒子数目M,粒子位置S 和速度V,学习因子c1和c2,最大权重w max和最小权重w min,最大迭代次数t max,问题的维数D,PSO的预设精度,控制参数C;输出参数:SVDD的预测误差和最优核参数σ.Step1:初始化PSO算法参数:粒子数目M,粒子位置S和速度V,学习因子c1和c2,最大权重w max和最小权重w min,最大迭代次数t max,问题的维数D, PSO的预设精度,控制参数C;Step2:评价每个微粒的适应度,将当前微粒的位置和适应值存储在各微粒的p best中,将所有p best中适应值最优个体的位置和适应值存储于g best中;Step3:分别用式(7)和(8)更新粒子的速度和位移;Step4:用式(9)更新权重;Step5:对于每个微粒,将其适应值与其经历过的最好位置进行比较,如果较好,则将其作为当前的最好位置,比较当前所有p best和g best的值,更新g best;Step6:如果满足预设的运行精度或最大的迭代次数t max,则训练算法结束,输出SVDD的最优核参数值和预测误差;否则,返回Step3继续搜索.5工业应用铜转炉吹炼过程是一个典型的复杂工业过程.冰铜的吹炼造渣期分为S1期和S2期,从风口向炉内熔体中鼓入空气或富氧空气.通过空气的搅拌,冰铜中的FeS被氧化,生成FeO和SO2;FeO再与添加的熔剂中的SiO2进行造渣反应.造渣反应一结束,便暂停送风,进行排渣操作.在转炉吹炼过程中,发生的反应几乎全部是放热反应,为防止由于产生大量的热量而缩短耐火砖的寿命,需向炉内加入包壳、块状烟灰和残极等冷料,以控制炉内温度.铜锍吹炼过程是间歇式作业.通过深入分析铜转炉吹炼过程机理可知:造渣S1期的影响因素有5个,分别为S1期入炉铜锍的总量,S1期入炉铜锍的品位,S1期熔剂投放总量,S1期冷料投放总量,S1期鼓风量;造渣S2期的影响因素有13个,分别为S1期入炉铜锍的总量,S1期入炉铜锍的品位,S1期熔剂投放的总量,S1期冷料投放总量, S1期鼓风量,S1期出炉总渣量,S1期出炉炉渣的含铜量,S1期出炉炉渣的SiO2含量,S2期新入炉的铜锍的总量,S2期新入炉的铜锍的品位,S2期新增熔剂投放总量,S2期新增冷料投放总量,S2期鼓风量.将S1期的5个影响因素和S2期的13个影响因素分别作为M-SVDD的输入变量.以某铜冶炼厂3号PS铜转炉为对象,从生产现场采集的样本集分布如表1所示.表1中:d s1和d s2分别表示2009年1∼6月某铜冶炼厂3号PS铜转炉吹炼过程S1期样本集和S2期样本集,其中正常类样本数分别为454和641,2个样本集都无故障类样本;d s3和d s4分别表示2009年7∼12月某铜冶炼厂3号PS铜转炉吹炼过程S1期样本集和S2期样本集,其中正常类样本数分别为548和736,人工产生故障类样本数分别为128和134.∣d i∣表示第i个样本集的正常类样本数;p表示训练样本数所占对应样本集中正常类样本数的比例,p∈[0,1].∣d i∣p表示训练集的样本数,∣d i∣(1−p)与故障类样本数之和为测试集样本数.对4个样本集中的所有样本进行归一化处理.表14个样本集训练集样本数与测试集样本数分布测试样本数样本集名正常类样本数训练集样本数正常类故障类d s1454∣d1∣p∣d1∣(1−p)0d s2641∣d2∣p∣d2∣(1−p)0d s3548∣d3∣p∣d3∣(1−p)128d s4736∣d4∣p∣d4∣(1−p)134以p的步长为0.1设定M-SVDD故障诊断模型的训练样本数,然后设定线性递减权重PSO种群参数M=25,c1=c2=2,w max=0.9,w min=0.4,v max=1, t max=50,D=1.由文献[4]优化控制参数C理论可知,样本集d s1,d s2,d s3和d s4对应的控制参数C值分别为0.0282,0.0156,0.0148,0.0115.滑动窗口技术中大窗口大小初值L win=12∣d i∣p,第7期唐明珠等:一种改进的支持向量数据描述故障诊断方法971小窗口大小初值S win =14∣d i ∣p.通过线性递减PSO 算法,可获得M-SVDD 最优核参数如表2所示.表2中:p 表示训练样本集所占正常类样本集的比例,σd i 分别表示不同比例p 样本集d si 的M-SVDD 最优核参数.表24个训练样本集10个p 值下M-SVDD 的最优核参数pσd 1σd 2σd 3σd 40.120.709411.068413.27919.11250.220.43639.859812.11629.13150.320.33819.56479.13159.72710.414.308614.501311.552011.55200.510.1141 5.185012.897711.70570.617.83877.31149.727112.11620.716.764113.913013.811712.89770.8 5.525312.621011.705713.27910.910.894113.21419.112513.8117利用表2获得SVDD 的核参数,分别对测试样本集∣d 1∣(1−p )和∣d 2∣(1−p )进行测试,测试结果如图1和图2所示.1.00.90.80.70.60.10.30.50.70.9pd 1d 2图1M-SVDD 在样本集d 1和d 2上故障敏感性比较0.10.30.50.70.9pM-SVDDKNN 0.90.80.60.50.40.7图2M-SVDD 与KNN 在样本集d 3上故障敏感性比较由图1可知,M-SVDD 分别随着S1期和S2期样本集d 1和d 2的训练集所占整个样本集比例p 的增加,故障敏感度越来越高,原因是训练样本数增加有利于M-SVDD 训练更充分,有更多有用的样本信息.为更好地说明M-SVDD 算法性能,将M-SVDD 与一类分类的KNN 算法[3]在样本集d s 3和d s 4上进行对比实验.对不同p 值下M-SVDD 和KNN 的故障敏感性进行了比较.不同的p 值在d 3和d 4上所对应的M-SVDD 上最优核参数σ如表2所示.利用表2获得的M-SVDD 的核参数σd 3对测试集∣d 3∣(1−p )和d s 3故障类样本集进行测试,实验结果如图2所示.用M-SVDD 核参数σd 4对测试集∣d 4∣(1−p )和d s 4故障类样本进行测试,实验结果如图3所示.由图2和图3可知:M-SVDD 在训练样本集所占整个样本集比例p 比较小的情况下,表现出了较好的故障敏感性;M-SVDD 和KNN 随着p 值的增加,其故障敏感性提高,但由于KNN 在训练集上有过拟合现象,在测试集上表现不佳,而M-SVDD 是对样本集边界进行描述,其在测试集上具有良好的故障敏感性.1.00.80.60.40.10.30.50.70.9pM-SVDD KNN图3M-SVDD 与KNN 在样本集d 4上故障敏感性比较滑动窗口技术中大窗口的大小是影响M-SVDD 故障敏感性的关键参数.故障敏感性是指M-SVDD 正确诊断的正类样本比例.故障敏感性定义为Sensitivity =tp /pos .其中:tp 是指使用M-SVDD 模型将原本属于正类的样本正确地诊断为正类数,pos 为总的正类样本数.大窗口大小用于控制M-SVDD故障诊断模型中的训练样本数pos ,根据小窗口的预测误差变化动态调整大窗口的大小.大窗口越大,M-SVDD 故障诊断模型训练越充分,M-SVDD 的故障敏感度越高;大窗口越小,M-SVDD 故障诊断模型训练越不充分,M-SVDD 故障敏感度越小.从图2和图3可知,对于某一个样本集而言,随着p 值增大,大窗口的大小L win =12∣d i ∣p 和小窗口的大小S win =14∣d i ∣p相应增大,M-SVDD 的故障敏感度不断地迅速增加,但增加到一定程度后,故障敏感度增加幅度减小.故障诊断的响应时间是评价故障诊断系统的关键参数.在故障诊断系统中,故障诊断的响应时间越短,越能满足M-SVDD 实时应用需求.M-SVDD 通过滑动窗口技术中的大窗口大小控制故障诊断模型的训练样本数,这样可以有效地减少该方法的训练时间.M-SVDD 通过训练时获得的支持向量对未知故障样本进行诊断.一般情况下,支持向量数远小于训练样本数[4],这样可以有效地减少对未知故障样本诊断的时间.在实验中,M-SVDD 在4个样本集上的平均训练时间分别为0.18s,0.25s,0.21s,0.29s,对一个未知故障样本的平均诊断时间分别为0.15s,0.26s,0.23s,0.3s.因此,M-SVDD 能够满足故障诊断实时应用的需求.972控制与决策第26卷6结论本文针对故障诊断中故障类样本难以获取以及不均衡类问题,提出了基于粒子群和滑动窗口的支持向量数据描述(M-SVDD)故障诊断方法.用支持向量数据描述模型进行故障诊断时,模型的核参数选择非常关键.利用线性粒子群算法优化支持向量数据描述的核参数,可以提高支持向量数据描述的泛化能力和故障敏感性.采用滑动窗口技术,通过一个动态调整的大窗口来控制故障诊断模型中所包含的最近产生工况,以反映工况的变化和控制训练样本集规模,根据小窗口的预测误差变化动态调整大窗口的大小.以典型有色冶金过程—–铜转炉吹炼过程为例,深入分析了该过程机理,选择该过程中合适的输入条件和过程参数作为M-SVDD的输入参数,减少了噪声变量干扰.实验结果表明,与KNN相比,M-SVDD能有效地提高故障敏感性和泛化能力,防止过拟合现象.M-SVDD仅依靠正常类样本,便可以对铜转炉吹炼过程进行故障诊断.该方法在故障诊断领域具有重要的推广价值和广泛的应用前景.参考文献(References)[1]郑恩辉,李平,宋执环.代价敏感支持向量机[J].控制与决策,2006,21(4):473-476.(Zheng E H,Li P,Song Z H.Cost sensitive support vector nachines[J].Control and Decision,2006,21(4):473-476.)[2]唐明珠,阳春华,桂卫华.代价敏感概率神经网络及其在故障诊断中的应用[J].控制与决策,2010,25(7):1074-1078.(Tang M Z,Yang C H,Gui W H.Cost-sensitive probabilistic neural network and its application in the fault diagnosis[J].Control and Decision,2010,25(7):1074-1078.)[3]Angiulli F.Condensed nearest neighbor data domaindescription[J].IEEE Trans on Pattern Analysis and Machine Intelligence,2007,29(10):1746-1758.[4]Tax D M J,Duin R P W.Support vector data description[J].Machine Learning,2004,54(1):45-66.[5]Zhang Y,Liu X D,Xie F D,et al.Fault classifier ofrotating machinery based on weighted support vector data description[J].Expert Systems with Applications,2009, 36(3):7928-7932.[6]Bovolo F,Camps V G,Bruzzone L.A support vectordomain method for change detection in multitemporal images[J].Pattern Recognition Letters,2010,31(10): 1148-1154.[7]Lee K Y,Kim D W,Lee D,et al.Improving support vectordata description using local density degree[J].Pattern Recognition,2005,38(10):1768-1771.[8]赵峰,张军英,刘敬.一种改善支撑向量域描述性能的核优化算法[J].自动化学报,2008,34(9):1122-1127.(Zhao F,Zhang J Y,Liu J.An optimizing kernel algorithm for improving the performance of support vector domain description[J].Acta Automatic Sinica,2008,34(9):1122-1127.)[9]Del V Y,Venayagamoorthy G K,Mohagheghi S,et al.Particle swarm optimization:Basic concepts,variants and applications in power systems[J].IEEE Trans on Evolutionary Computation,2008,12(2):171-195. [10]Chang Y P,Ko C N.A PSO method with nonlineartime-varying evolution based on neural network for design of optimal harmonicfilters[J].Expert Systems with Applications,2009,36(3):6809-6816.[11]Tanbeer S K,Ahmed C F,Jeong B,et al.Sliding window-based frequent pattern mining over data streams[J].Information Sciences,2009,179(22):3843-3865.[12]Li H F,Lee S Y.Mining frequent itemsets over datastreams using efficient window sliding techniques[J].Expert Systems with Applications,2009,36(2):1466-1477.(上接第966页)[18]Zhang Qingfu,Sun Jianyong,Edward Tsang,et al.Hybrid estimation of distribution algorithm for global optimization[J].Engineering Computations,2004,21(1): 91-107.[19]王湘中,喻寿益.适用于高维优化问题的改进进化策略[J].控制理论与应用,2005,23(1):148-151.(Wang X Z,Yu S Y.Improved evolution strategiesfor high-dimensional optimization[J].Control Theory& Application,2003,16(4):390-396.)[20]江中央.正交遗传算法及其应用研究[D].长沙:中南大学信息科学与工程学院,2008.(Jiang Z Y.Researchon orthogonal genetic algorithm and its applications[D].Changsha:School of Information Science and Engineering,Central South University,2008.)。
怎样让数学学好（Howtolearnmathwell）
怎样让数学学好（How to learn math well）How to learn math well?Mathematics has always been an important tool for human beings to practice, and it is one of the most basic courses in basic education. Each student hopes to master mathematics knowledge, train and improve his calculation ability, logical thinking ability, spatial imagination ability, creative ability, and preliminary application to mathematics. However, for a junior high school student who has just entered junior high school from primary school, how can he learn mathematics well? I think we can start by grasping all kinds of study habits. Entering junior high school from primary school is a major turning point in the learning phase. According to people's physical and psychological development of junior high school students, it is at a critical stage of forming the habit, if not timely to seize this opportunity to form good learning habits, it is easy to catch a lot of bad learning habits, and seriously influence the development of intelligence and ability. Good study habit is an important psychological factor for stimulating thinking, developing ability and developing personality. It is the basis of achieving good teaching effect. So it is the key to learn good habits to cultivate good study habits. The following four aspects of how to develop and shape good learning habits.First, reading habitsThis is the basic skill of self-study ability. According to the scores of famous university study in the United States and the former Soviet Union, those successful scientists have 20%~25% knowledge from school, and the knowledge of 75%~80% is to relyon them after leaving work, through self-study and research to obtain the. According to psychological rules, junior high school students already have the ability to read, but because of the influence of intuitive imitation habits in primary schools, many students mistake mathematics textbooks as exercise sets. Therefore, from the beginning of the first year, we should pay attention to correct their mistakes, study habits, and set up mathematical textbooks, also need to read the correct ideas, and pay attention to how to read mathematics textbooks.1., each class must be prepared to preview the habit, and strive to preview the discovery that they do not understand the problem, in order to be able to listen to the problem. In class, pay attention to how the teacher reads the text, from which he can learn how to analyze and define the key words, words and sentences in the theorem, and the connection with the old knowledge.2., often sum up the knowledge learned, training review habits. At the beginning, can follow the teacher summed up a lesson or a unit, a stage according to the teacher's review outline, with their own problems to delve into the text, the final transition to promote themselves by their own induction, repeated reading, timely review, new.Two, note habitsA good memory is worse than a bad pen". The middle school mathematics classroom is rich in content, capacity is relatively large, to learn mathematics, from junior high school must attach great importance to the cultivation of the habitof taking notes in class, class notes can also constrain energy dispersion, improve the efficiency of class. In general, in addition to record the lecture notes outline, is mainly remembered, the teacher summarized the key ideas, methods and contents of the account. Pay special attention to the notes and questions in the lecture at any time. In the two aspects of "listening" and "recording", listening is the foundation".In order to gradually improve the quality of class notes, students should make appropriate exchanges and learn from each other.Three, hands-on practice, cooperation, exchange habitsTrue knowledge comes from practice". Hands-on practice can concentrate attention, improve interest in learning, and can deepen the impression and understanding of learning objects. In practice, we can link the knowledge of the book with the actual things, and form a correct and profound concept. In practice, the brain and the use of expert activities, and gradually formed and developed their own cognitive structure, can form skills, development ability. In practice, develop a "pre - guess - hands-on experiment - operation results - summary summary" habits."Three men go together, and there must be my teachers."". The results of learning exchanges between the students from each one airs his own views. It can achieve the functions of thinking, moving, opening, starting, stimulating thinking, active atmosphere and arousing enthusiasm.Four, homework habitsMathematics homework is an important link to strengthen mathematical knowledge, stimulate learning interest and train mathematical ability. Some students regard homework as a burden. After class, they only rush to answer questions in class. They often have a single solution. Some of them are sloppy, careless, careless in format and even copied. This misses the opportunity to train, seriously affecting the learning effect. We should correctly understand the purpose of doing homework and train good habit of homework. Good habit of homework should include:1., to develop the habit of reading before work. Before you do your homework, read the text carefully and see the solution, format, steps and methods of the examples. This is the "quicken work".2. to develop their habits. After reading the questions, first find out what kind of questions the topic is, what conditions it has and what characteristics it has.3., to develop the habit of independent work. If there are special circumstances can not be completed on time, you can explain the situation to the teacher: if you encounter problems will not do, you can ask the teacher or classmate, understand, after the completion of the independent. Never copy in order to cope with a task.4., get into the habit of thinking about doing your homework again. Many students do not pay attention to doing homework again, and then thinking, resulting in erroneous practices inthe minds of stereotypes. Some questions wrong, the teacher revised, you are wrong, that is the reason. As you go on, you will make greater mistakes in new knowledge and new assignments. In order to consolidate the results of your work, you must always give feedback to your previous day's homework before you make new assignments. The feedback includes: (1) the types of questions; (2) problem solving ideas and methods; (3) cause errors; error correction; (4) (5) (collection of error problem is their mistake to collect in one place, marked above four items for future review when the error correction).Five, thinking habitsScientific methods of thinking and good habit of thinking are keys to developing intelligence and developing ability. Psychology tells us, the first stage is the students from image thinking to abstract thinking of the important period of change, so at this time must pay attention to good habit of thinking training. According to the characteristics of junior high school mathematics, good habits of thinking include logicality, thoroughness, divergence, convergence and converse.1. logic. This requires students to answer "yes" and must not take it for granted. In the process of reasoning, we can understand the basis of each step, do not understand, do not write, try to understand, and then continue to reasoning calculus.2. thoroughness. This requires the students to consider everything in a comprehensive way. Such as: the known point C in the line AB, line AB=8cm, line BC=3cm, to find the lengthof the line AC. Fully consider the issue will be discussed in the C segment of AB extension line two on segment AB and C: when C on segment AB, AC=AB-BC=8-3=5cm; when the C extension line of the line AB, AC=AB+BC=8+3=11cm. To develop this habit, pay special attention to the situations and causes that the teacher points out in the class as "error prone" or "not complete".3. divergent. This requires students to use many different ways to solve a problem. To cultivate this habit, we should pay special attention to the teacher's thinking method, problem, extension and analysis when we are talking about multiple solutions. In the process of mathematics study, we should try to develop a habit of seeking a problem with multiple solutions and a problem.4. convergence. This is based on divergent thinking, summarized, in order to achieve multiple questions, a solution, draw inferences from others. The two kinds of thinking, divergence and convergence, can be combined with each other.5. reverse. This requires students to reverse the order of certain formulas, laws, and theorems. As calculated:(-0.38) * 4.58-0.62 * 4.58 can be obtained by using the multiplicative distribution law in reverse.Six, questioning habitsConfucius, an ancient great educator in China, always insisted that learning should be understood and that it should be known. Is not only to ask what is "what", but also to ask "why?"".Psychologists tell us,When people accept a new problem, there is a general desire to make a real difference. Students in junior middle school students' active thinking, strong curiosity, psychological requirements should have asked. But due to the old social psychological bondage, not daring to problems "like" and "nonsense", this is the classroom, extracurricular too, make their personality severely stifle, is not conducive to healthy growth. To reverse this situation, requires students to speak boldly and actively discuss and practice in the classroom, after-school qinsi ask, cultivate a good habit to create a love question, students should know the teacher ask the students love Dickens, to develop the knowledge to the bottom of the habit, ready to develop the question the habit of doubt.Developing study habits is a systematic project. It requires the students to have determination, perseverance and patience. "The most valuable knowledge is knowledge about methods," Darwin said". As time passes, the method becomes a natural habit, so the cultivation of good study habits is the key to mastering a treasure trove of knowledge, and the energy released by it is incomparable.。
Mathematica入门(二)
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Mathematica入门 入门
三维图形 Plot3D[ f , {x, xmin, xmax}, {y, ymin, ymax}] 这里二元函数f(x,y)的定义域为矩形区域 的定义域为矩形区域 这里二元函数 Plot3D[ {f , s},{x, xmin, xmax}, {y, ymin, ymax}] 其中s是着色表达式， 其中 是着色表达式，用于给曲面着色 是着色表达式 将函数f(x,y)的定义域分割成若干小矩 注：Mathematica将函数 将函数 的定义域分割成若干小矩 计算各小矩形的顶点处的函数值， 形，计算各小矩形的顶点处的函数值，得到曲面上的若 干点，顺次连接这些点便得到曲面的近似图形。 干点，顺次连接这些点便得到曲面的近似图形。 可选参数 Boxed（是否给图形加立体框）、 （是否给图形加立体框）、BoxRatios[r1, r2, r3] ）、 Mesh（是否加网格）、 （是否加网格）、ViewPoint（设置观察点位置） ）、 （设置观察点位置）
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解常微分方程（ 解常微分方程（组） DSolve[eqn, y[x], x] 求方程eqn的通解，x为自变量 的通解， 为自变量 求方程 的通解
离散数学归纳
I.e, prove that 11! + … + kk! + (k+1)(k+1)! = (k+2)! – 1, assuming P(k)
Inductive hypothesis
11! + … + kk! + (k+1)(k+1)! = (k+1)! - 1 + (k+1)(k+1)! = (1 + (k+1))(k+1)! - 1
4 – Inductive Step: show that (k) P(k) P(k+1), assuming P(k). How?
P(k+1)= 1 + 3 + … + (2k-1) + (2k+1) = k2 + (2k + 1) = (k+1)2
By inductive hypothesis
4 – Inductive Step: show that (k) P(k) P(k+1), assuming P(k). How? By inductive hypothesis
P(k+1)= 1 + 2 +
22
+…+
严格强凸函数
严格强凸函数严格强凸函数是数学中的一个重要概念，它在优化问题中有着广泛的应用。
本文将介绍严格强凸函数的定义、性质、求解方法以及相关定理。
一、定义1.1 强凸函数设$f(x)$是定义在$\mathbb{R}^n$上的实值函数，若对于任意$x,y\in \mathbb{R}^n$和任意$\lambda \in [0,1]$，有：$$f(\lambda x+(1-\lambda)y)\leq \lambda f(x)+(1-\lambda)f(y)-\frac{\mu}{2}\lambda(1-\lambda)\|x-y\|^2$$其中$\mu>0$为常数，则称$f(x)$是强凸函数。
1.2 严格强凸函数若上述不等式中等号仅在$x=y$时成立，则称$f(x)$是严格强凸函数。
二、性质2.1 唯一最小值点对于严格强凸函数$f(x)$，存在唯一的最小值点$x^*$，即：$$f(x^*)=\min_{x\in \mathbb{R}^n}f(x)$$2.2 Lipschitz连续性对于严格强凸函数$f(x)$，存在常数$L>0$使得：$$\|f(x)-f(y)\|\leq L\|x-y\|,\forall x,y\in \mathbb{R}^n$$ 其中$\|\cdot\|$为欧几里得范数。
2.3 连续可微性对于严格强凸函数$f(x)$，存在连续可微函数$g(x)$使得：$$f(x)=g(x)+\frac{\mu}{2}\|x\|^2$$其中$\mu>0$为常数。
三、求解方法3.1 梯度下降法梯度下降法是求解优化问题的一种常用方法。
对于严格强凸函数$f(x)$，其梯度$\nabla f(x)$在全局范围内都是单调递增的，因此可以通过不断迭代来逼近最小值点$x^*$。
具体算法如下：(1) 初始化$x_0$和学习率$\alpha>0$；(2) 重复执行以下步骤直至收敛：$$x_{k+1}=x_k-\alpha \nabla f(x_k)$$其中$k$表示迭代次数。
牛顿法代码
牛顿法代码牛顿法，是一种基于函数导数的迭代算法，用于寻找函数的零点或最小值。
它可以高效地优化目标函数，有很广泛的应用。
牛顿法的基础思想是利用当前点附近的信息来构建一个局部二次逼近模型，然后以该模型的零点为下一个点的位置，反复进行迭代，直到达到所需的精度或满足其他停止准则。
接下来，我们将介绍如何在 Matlab 中实现牛顿法。
代码如下：function r = Newton(func, x0, tol, max_iter)% func: 目标函数% x0: 初始点% tol：收敛精度% max_iter：最大迭代次数% r：函数的零点或最小值syms x % 声明符号变量xf = matlabFunction(sym(func)); % 将传入的目标函数转换为Matlab函数df = matlabFunction(diff(sym(func))); % 求导函数转换为Matlab函数iter = 0;xk = x0;while iter < max_iteriter = iter + 1;f_val = f(xk);df_val = df(xk);if abs(df_val) < 1e-9error('无法收敛，梯度过小')endxk1 = xk - f_val/df_val;if abs(xk1-xk) < tolr = xk1;breakendxk = xk1;endif iter >= max_iter || ~isfinite(r)error('无法收敛，达到最大迭代次数')endend函数的输入参数依次为：目标函数、初始点、收敛精度和最大迭代次数，返回值为最终的零点或最小值。
其中，函数目标函数 func 是一个符号表达式，需要先声明符号变量 x，然后将其转换为 Matlab 函数（ f 和 df 分别表示函数本身及其一阶导数）。
在每次迭代中，我们计算函数在当前点的函数值 f_val 和一阶导数值 df_val，然后更新下一个点 xk1 并判断是否满足停止准则（差值小于收敛精度 tol）。
三阶多项式核函数
三阶多项式核函数是一种非线性核函数，用于将数据从低维空间映射到高维特征空间，以便在高维空间中实现线性可分。
三阶多项式核函数的数学表达式通常为K(x, xi) = (γ+ x·xi + b)^3，其中γ、b 是人工设置的参数，γ为正实数，b 为非负实数。
在这个表达式中：
1.x 和xi：分别代表两个数据点在原始特征空间中的表示。
2.γ（gamma）：是一个正实数，它的作用是调节核函数的幅度，防止计算时出现数值问题。
3.b（bias）：是一个非负实数，它可以使核函数沿y轴移动，增加模型的灵活性。
4.d（degree）：表示多项式的阶数，在这里d=3，即三阶多项式。
多项式核函数的特点是能够处理那些在低维空间中线性不可分的数据。
通过将数据映射到一个更高维度的空间，数据点可以变得线性可分，从而可以使用线性分类器进行分类或回归分析。
在选择核函数时，需要根据具体的数据特性和问题需求来决定使用哪种类型的核函数。
例如，如果数据在原始空间中已经是线性可分的，那么使用线性核函数就足够了。
而如果数据比较复杂，可能需要使用多项式核函数或高斯核函数来实现更复杂的非线性映射。
求解绝对值方程的光滑牛顿算法
求解绝对值方程的光滑牛顿算法张焱娇【摘要】背包问题以及大部分的线性互补问题都可以转化成为绝对值方程组来求解，求解绝对值方程Ax＋B｜a｜＝b是较难的问题．将该问题等价为线性互补问题，利用光滑牛顿法算求解该互补问题．当满足一定的条件时，证明了该算法是适定的，更证明了该算法的全局收敛性．利用Matlab软件对200维，500维，800维，和1000维的情况进行了数值试验．每种情况测试了随机产生的50个可解的例子．精度达到了10-6．800维的用时在10 s左右，1000维的用时在20 s左右．%The well-known NP-hard knapsack feasibility problem and linear complementa-ry problems can be formulated as an equivalent absolute value equations (AVEs).A smooth Newton method is proposed to solve the NP -hard absolute value equations:Ax+B|x|=b. And the algorithm is well defined under the following three conditions .Moreover the same condition can ensure that the algorithm is globally convergent .A Matlab implementation of the smoothing Newton method solved randomly generated 500-dimensional AVEs , 800-di-mensional AVEs , and 1000-dimensional AVEs .50 randomly generated AVEs for each case were solved to accuracy of 10 -6 .The average time for the 800-dimensional AVEs was about 10 s.The average time for the 1000-dimensional AVEs was about 20 s.【期刊名称】《哈尔滨商业大学学报（自然科学版）》【年(卷),期】2014(000)005【总页数】5页(P617-620,625)【关键词】绝对值方程;光滑牛顿法;全局收敛性【作者】张焱娇【作者单位】天津大学，天津300072【正文语种】中文【中图分类】X703绝对值方程一般数学表达式如下：Ax+B|x|=b,A∈Rm×n,B∈ Rm×n,b∈Rm(1)其中:|x|表示的是对x的每个分量取绝对值.本文中研究的矩阵A和B矩阵为方阵：Ax+B|x|=b,A∈Rn×n,B∈ Rn×n,b∈Rn(2)绝对值方程是Rohn在中首次出提出的.目前求解绝对值方程的算法针对B=-I，Ax-|x|=b,A∈Rn×n,b∈ Rn(3)参考文献[1-5]中讨论了绝对值方程(1)(2)和(3)的一些性质.下面简单列举一下这些性质：如果绝对值等式(1)、(2)、(3)是可解的，则绝对值方程或者有惟一解，或者有多重解 [3]，一般的绝对值规划(1)可以转换成一个线性互补问题 [4]. Mangasarian提出算法是：将绝对值等式(3)转化成分段线性的凹极小化问题，利用连续化线性算法来求解.之后在参考文献[6]又中提出了一种广义牛顿法来求解绝对值方程(3)，要求A的奇异值严格大于1，此方法的有线性收敛性质已经得到了证明.之后Oleg Prokopyev在参考文献[7]中将绝对值方程(3)问题转化为0-1混合线性规划来进行求解.本文所做的工作是利用光滑牛顿算法来求解问题(2)和(3)，光滑化的思想 [8-9].本文中证明了在矩阵A的主对角线上的元素的绝对值大于矩阵B的主对角元素的绝对值，矩阵A、B对于可交换条件下，算法是适定的，具有全局收敛性质.根据参考文献[10-11],绝对值方程的可解性问题是一个NP难的问题，所以本文不讨论绝对值方程解的存在性.现将本文所使用的符号如下：对于向量x∈ Rn，若x>0表示的是向量x的每个分量都大于0；‖x‖表示向量x的欧式范数，对于向量y，s∈ Rn对于函数F:Rn→Rm,F′(x)表示函数F在点的Jacobi矩阵.1 算法设计1.1 NCP函数定义设函数φ(a,b):R×R→R，如果满足φ(a,b)=0⟺a≥0,b≥0，且ab=0性质. 1.2 光滑牛顿法定理1 对于绝对值等式Ax+B|x|=b，其中x∈Rn，A∈Rn×n，B∈Rn×n若矩阵(-A+B)和矩阵B可逆，矩阵A和矩阵B对于乘法可交换时，则该问题可等价转换为如下：(-A+B)s-(-A-B)y=2Bby≥0,s≥0,yTs=0证明：见参考文献[5]推论1 当B=-I时，求Ax-|x|=b解的解，若矩(A+I)阵可逆，那么(2)等价转化为求解如下的问题：(I+A)s-(I-A)y=-2by≥0,s≥0,yTs=0(4)F-B函数在R2\{0,0}处是连续可微的，为了保证其在整个R2平面上的可微性质，引入光滑化因子μ使函数在R2上是可微的，其具体形式如下：现在在利用光滑后的F-B函数，来构造向量值函数：Ψ(μ,y,s)=(φ(μ,y1,s1),φ(μ,y2,s2),…,φ(μ,yn,sn))T定义的光滑化函数H:R2n+1→R(2n+1)，如下：H(z):=(μ,(-A+B)s+(A+B)y-2Bb,Ψ(μ,y,s))T(5)其中z=(μ,y,s)，显然‖H(z)‖=0当且仅当μ=0，且(y,s)是线性互补问题(4)的解.只要求出z*使得‖H(z*)‖=0，就可以求出绝对值方程(2)的解.对于任意(μ,y,s)∈R++×Rn×Rn，函数H(z)一定是连续可微的.且函数H(z) Jacobi矩阵为：(6)下面对一些符号予以说明：Ψμ=Ψy=算法1 (求解形如(2)的绝对值等式的光滑牛顿法)步骤0选择适当的δ,σ∈(0,1)，μ0>0和(y0,s0)∈R2n是初始向量.令z0:=(μ0,y0,s0)，选择适当的β>1使得‖H(z0)‖≤βμ0，令e0=(1,0,…,0)T，令k:=0；步骤1 如果‖H(zk)‖=0，停止，zk则是我们所求的最优解；否则，跳至步骤2；步骤2 计算Δzk:=(Δμ0,Δy0,Δs0)∈R×Rn×Rn，其中Δzk满足以下的牛顿方程：根据(5)的具体形式，可以得到：步骤3 令λk是1,δ,δ2中使下式成立的最大值.步骤4 令zk+1:zk+λkΔzk取k:=k+1，调到步骤1.步骤5 求解方程y-s=2Bx1.3 算法的性质在参考文献[8]中，有如下结论成立：命题1 设{zk=(μk,yk,sk)是由算法1产生的迭代序列，则有以下的结论成立：迭代过程中，对任意迭代点都有μk>0；序列{‖H(zk)‖}是单调下降序列；算法迭代的过程中序列{μk}是单调下降的；设β>1是算法中给定的常数，定义集合：N(β):={z∈R+×Rn×Rn:‖H(z)‖≤βμ}则我们在迭代过程中，对于任意的k∈{1,2,…,n}，zk∈N(β)；定理2 若果矩阵(A+B)，(-A+B)和矩阵B可逆，矩阵A和B矩阵对于乘法可交换，并且矩阵A的主对角线上的元素的绝对值严格大于矩阵B的主对角线上元素的绝对值，则算法1中的步骤2的牛顿方程一定是可解的，说明此算法一定是适定的. 证明：先介绍一些证明中涉及的符号：cii:=min{-aii-bii,aii+bii},c=(c11,c22,cnn)C:=diag(c)dii:=max{-aii-bii,aii+bii},d=(d11,d22,knn)D:=diag(d),i∈{1,2,…,n}由假设矩阵A的对角线上的元素的绝对值大于对应B的位置元素的绝对值，并且下式成立：|aii-1|bii<1，i∈{1,2,…,n}由参考文献[13]，推论3.2区间[C,D]是正则的.即任意的，m,n∈[0,1],mC+nD是可逆的.Ψs=diag矩阵，Ψy,Ψs不仅仅是可逆的，而且对角线上的元素一定是在(-1,0)之间的.Ψs(A+B)-Ψy(-A+B)一定是正则的.Ψs(A+B)-Ψy(-A+B)是可逆的⟺Ψs(A+B)-Ψy(A+B)-1(A+B)(-A+B)是可逆的⟺Ψs(A+B)-Ψy(A+B)-1(A+B)(-A+B)是可逆的，因为AB=BA⟺Ψs-Ψy(A+B)-1(-A+B)是可逆的，因为矩阵是(A+B)可逆的.那么算法1的步骤2的牛顿方程也一定有解的.推论2当矩阵B=-I时，绝对值方程(3)有以下的结论成立：如果矩阵(I-A)可逆，并且矩阵的主对角线上的元素同时大于1或者小于-1，则算法中的牛顿方程是可解的.2 算法的性质本节证明在矩阵AB满足适当关系的条件下，算法1有全局收敛性.定理3 矩阵(I-A)，(I+A)是可逆的，并且矩阵A的主对角线上的元素同时严格大于1或者严格小于-1时，则通过算法1产生的{zk:=(μk,yk,sk)}是有界序列.证明由命题1的结论可知序列{μk}是单调下降有下界的序列{zk}:={μk,wk}，为了说明序列的有界性{wk}，只需在说明序列是有界序列即可，其中{wk=(yk,sk)}.利用反证法证明序列{wk}是是有界序列{wk}，假设序列无界，则可知其部分序列{yk},{sk}至少有一个是无界序列.则无界序列，必须满足以下三种情况之一：1)序列{yk}无界，序列{sk}有界；2)序列{yk}有界，序列{sk}无界；3)序列{yk}，{sk}无界；1)序列{yk}无界，序列{sk}有界的因为序列{yk}无界，则存在指标i0∈{1,2,…,n}，使得|(yk)|→∞，当k→∞时，|((I+A)sk+(I-A)yk+2b)i0|→∞,k→∞，此时就一定会出现‖H(zk)‖→∞，当k→∞，这与命题2.1中序列‖H(zk)‖是单调下降有界的序列产生矛盾.所以情况(1)是不可能出现的.2)序列{yk}有界，序列{sk}无界的(同(1))3)序列{yk}，{sk}同时是无界的若实数对{i0,j0}其中i0∈{1,2,…,n}，j0∈{1,2,…,n}且满足i0≠j0则此时可以归结为上述的(a)、(b)两种情况之一，会出现矛盾，无界序列无法产生.下面讨论另外一种情况：即为对于任意t0∈{1,2,…n}，只要|(yk)t0|→∞则必有|(sk)t0|→∞并且其逆命题也成立.下面在进一步讨论，当k→∞时：若(yk)t0→+∞,(sk)t0→+∞时，((I+A)sk+(I-A)yk+2b)t0→+∞，若(yk)t0→+∞,(sk)t0→+∞时，((I+A)sk+(I-A)yk+2b)t0→+∞若(yk)t0→+∞,(sk)t0→-∞时，有显然当k→∞，有|φ(μ,(yk)t0,(sk)t0|→∞，则此时同样可以的得到函数H(zk)的范数‖H(zk)‖→∞，与算法性质矛盾.综上所述，假序列{zk:=μk,yk,sk)}也有界的.定理4 假定矩阵(I-A)，(I+A)是可逆的，矩阵A的对角线上的元素同时大于1或者同时小于-1，则产生的序列{zk}有以下性质：证明方法类似于参考文献[13]3 数值计算结果与结论数值实验所用电脑规格为：CPU是2.93 GHz，内存2.00 GHz.数值实验的具体绝对值方程产生过程如下：产生一个矩阵保证其每一个对角元素都同时大于1或者同时小于-1；随机选取向量x∈Rn，其中的每个分量符合[-10,10]的均匀分布；计算出b=Ax-|x|.本文将对n等于500、800、1000的情况分别进行数值实验，每组实验中，连续随机产生50个绝对值方程.在数值实验的程序中‖Ax-|x|-b‖<10-6作为算法的终止条件；循环的最高次数定为40次；当算法终止是若‖Ax-|x|-b‖<10-6，则认为失败.相关参数值为σ=0.8，σ=0.0001，μ0=0.0001；x0为随机产生的维向量其分量符合[-10,10]之间的均匀分布.表1 数值结果(矩阵的主对角线上的元素严格大于1或者严格小于-1)编号平均用时用时最长用时最短平均迭代次数失败次数n=5001～103.28014.153 42.446 213011～203.255 74.580 32.456 014.4021～302.743 55.629 52.416 312.2131～403.383 44.007 92.689 414.0041～502.754 73.087 22.091 712.71n=8001～109.931 411.617 08.446 412.8011～2010.496 114.125 48.575 913.3021～309.614 610.332 87.759 312.1031～4010.506 420.682 97.565 513.5041～5010.219 712.328 67.859 612.80n=1 0001～1019.870 829.083 115.826 013.4011～2019.983 228.324 815.928 813.2021～3023.521950.625116.034215.1031～4021.752233.411714.285714.2041～5019.908226.572416.180313.004 结语本文提出的光滑化牛顿算法，分别对形如Ax+B|x|=b和Ax-|x|=b的绝对值方程组进行求解，并且证明了在一定的条件下此算法是全局收敛的.在此之前，要求的是在区间[I-A,I+A]是正则的，对矩阵A的奇异值等性质要求较高.在理论证明中，条件“当矩阵(B+A)和矩阵(B-A)可逆，矩阵AB对乘法满足交换律，以及矩阵A的主对角线上的元素的绝对值大于矩阵B主对角线上的绝对值时，证明了算法是适定的，并且当矩B=-I阵时，更证明了该算法的全局收敛性，”在未来的研究中，可以尝试证明矩阵B的一般情形是算法的全局收敛性.参考文献：[1] MANGASARIAN O L. Knapsack feasibility as an absolute value equation solvable by successive linear programming [J]. Optimization Letters,2009,3(2): 161-170.[2] OLEG P. On equivalent reformulations for absolute value equations [J]. Computational Optimization and Applications, 2009, 44(3): 363-372.[3] FUKUSHIMA M. The primal Douglas-Rachford splitting algorithm for a class of monotone mappings with application to the traffic equilibrium problem [J].Mathematical Programming, 1996, 72(1): 1-15.[4] NAGURNEY A, DONG J, ZHANG D. A supply chain network equilibrium model [J].Transportation Research, 2002, Part E 38(5):281-303[5] PANG TRINKLE J C. Complementarity formulations and existence of solutions of multi-rigid-body contact problems with Coulomb friction [J].Mathematical Programming, 1996, 73(2): 199-226.[6] CHEN CH, MANGASARIAN O L. Smoothing methods for convex inequalities and linear complementarity problems [J]. Mathematical Programming, 1995, 71: 51-69.[7] MANGASARIAN O L, MEYER R. Absolute Value Equations[J]. Linear Algebra and Its Applications, 2006, 419: 359-367.[8] CHEN C, MANGASARIAN O L. A class of smoothing functions for nonlinear and mixed complementarity problems [J]. ComputationalOptimization and Applications, 1996, 5: 97-138.[9] QI L Q, SUN D, Smoothing functions and smoothing Newton method for complementarity and variational inequality problems[J]. Jouranal of Optimization Theory and Applications, 2002, 113-121.[10] ROHN J. A theorem of the alternatives for the equation Ax + B|x| = b[J].Linear Multilinear Algebra, 2004, 52(6): 421-426.[11] MANGASARIAN O L .Absolute value programming[J]. Computational Optimization and Applications, 2007, 36: 43-53.[12] MANGASARIAN O L.Absolute value equation solution via concave minimization[J]. Optimization Letters, 2007, 1(1): 3-8.[13] 高竹峰. 求解绝对值方程组的牛顿算法[D]. 天津: 天津大学, 2009.。
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Mathematical Induction and Strong Induction
Scope:
1)Mathematical Induction
2)Strong Induction
1. Mathematical Induction
1.1. Introduction & Definition
Many mathematical statements assert that a property is true of all positive
integers.
Example: U=Z+,
Show that
Mathematical induction is a proof technique that is used to prove a property of a set of positive integers, which is based on the rule of inference:
___________________________
Two Steps:
1.Basis step: Show that
2.Inductive step: Show that
Example: We have a line of people.
We know that the first person will be told the secret and a person
will communicate the secret to the next person in the line. Show
that all people will know the secret.
1.Basis step: Since we know that the first person will be told the
secret, it follows
2.Inductive step: Let be arbitrary.
------------------------------------------------- Hypothesis
person will communicate the secret to the next person+1 in the
line ------------------------------------------------- Given fact
___________________________
Because is arbitrary, by universal
generalization. Therefore,
1.2. Examples
General example:
To prove P(n) is true for all integers n ≥ c. We do the following steps:
1. Basis step: Show that P(c) is true (where c is the smallest element).
2. Inductive step: Show that P(k) → P(k + 1) is true for all k ≥ c, i.e. show that
P(k) → P(k + 1). (Assume P(k) is true and show that P(k +1) is true.) A proof by mathematical induction follows a fixed pattern.
1.3. Exercise
Suppose that we have an infinite ladder and we know two things:
1.We can reach the first step of the ladder.
2.If we can reach a particular step of the ladder, then we can reach the next step.
Show that we can reach every step on this ladder.
2. Strong Induction (also called generalized induction)
2.1. Introduction & Definition
Introductory Example:
Suppose that we have an infinite ladder and we know two things:
1.We can reach the first and second steps of the ladder.
2.If we can reach a particular step of the ladder, then we can reach two steps higher.
Show that we can reach every step on this ladder.
Rule of inference:
_________________________________________________
Two Steps:
1.Basis step: Show
2.Inductive step: Show
or for an
arbitrary k
Solution to the introductory example:
1.Basis step: We know that we can reach first step →
2.Inductive step: for an arbitrary k, we prove two cases:
Case#1:
We know that we can reach the second step
Case#2:
Assumption
Given fact
_________________________________________________
2.2. Example: Fundamental Theorem of Arithmetic。