# 数学代写|最优化理论作业代写optimization theory代考|CSC591

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## 数学代写|最优化理论作业代写optimization theory代考|Additional Notes on Newton’s Method

Newton’s Method for finding a zero of a mapping $g \in C^r\left(\mathbb{R}^n, \mathbb{R}^n\right), r \geq 1$, mainly solves the linearized equation in each iteration step. In fact, a Taylor expansion of first order around the iterate $x^k$ gives:
$$g(x)=\underbrace{g\left(x^k\right)+D g\left(x^k\right)\left(x-x^k\right)}_{\text {Linearization }}+o\left(\left|x-x^k\right|\right) .$$
Suppose that $D g\left(x^k\right)$ is nonsingular. Then the zero of the linearization is precisely the point $x^k-D g\left(x^k\right)^{-1} \cdot g\left(x^k\right)$; see Figure 9.3 for the case $n=1$.

Now, suppose that $f \in C^r\left(\mathbb{R}^n, \mathbb{R}\right), r \geq 2$. Let $\bar{x} \in \mathbb{R}^n$ be a local minimum for $f$ with $D^2 f(\bar{x})$ positive definite. Newton’s Method for the determination of $\bar{x}$ (as a zero of the mapping $x \mapsto D^{\top} f(x)$ ) minimizes in each step the quadratic approximation of $f$. In fact, a Taylor expansion of second order around the iterate $x^k$ gives:
\begin{aligned} f(x)= & \underbrace{f\left(x^k\right)+D f\left(x^k\right)\left(x-x^k\right)+\frac{1}{2}\left(x-x^k\right)^{\top} D^2 f\left(x^k\right)\left(x-x^k\right)}_{\text {quadratic approximation }}+ \ & +o\left(\left|x-x^k\right|^2\right) . \end{aligned}
For $x^k$ close to $\bar{x}$, the Hessian $D^2 f\left(x^k\right)$ is also positive definite; then, the minimum of the quadratic approximation is taken in the point $x^k-D^2 f\left(x^k\right)^{-1}$. $D^{\top} f\left(x^k\right)$ (exercise).

From a geometric point of view this is an ellipsoid method: consider in $x^k$ the ellipsoid tangent to the level surface $\left{x \mid f(x)=f\left(x^k\right)\right}$ having the same curvature as that level surface in $x^k$. The new iterate $x^{k+1}$ is precisely the center of this ellipsoid (see Figure 9.4).

## 数学代写|最优化理论作业代写optimization theory代考|Lagrange–Newton Method

For optimization problems with constraints one can also apply Newton’s Method in order to find a local minimum. To this aim, the optimization problem is reformulated into a problem of finding a zero of an associated mapping. Then, as in the unconstrained case, one recognizes that a Newton step is equivalent with solving a quadratic optimization problem; the latter then can be carried over to problems with inequality constraints.

As usual, let $f, h_i, g_j \in C^r\left(\mathbb{R}^n, \mathbb{R}\right), i \in I, j \in J, r \geq 2$, be given. Here, $f$ is the objective function and
$$M:=M[h, g]=\left{x \in \mathbb{R}^n \mid h_i(x)=0, i \in I, g_j(x) \geq 0, j \in J\right}$$
is the feasible set. We assume that LICQ is satisfied at each point of $M$.
First we discuss the case without inequality constraints, i.e. $J=\emptyset$. Let $\bar{x} \in M[h]$ be a critical point for $f_{\mid M[h]}$ with Lagrange multiplier vector $\bar{\lambda}$. Then, $(\bar{x}, \bar{\lambda})$ is a zero of the associated mapping $\mathcal{T}$ :
$$\mathcal{T}:\left(\begin{array}{l} x \ \lambda \end{array}\right) \longmapsto\left(\begin{array}{c} D^{\top} f(x)-\sum_{i \in I} \lambda_i D^{\top} h_i(x) \ -h_i(x), \quad i \in I \end{array}\right) .$$
The Jacobian matrix $D \mathcal{T}(\bar{x}, \bar{\lambda})$ has the following structure (compare with $(3.2 .7)$ and $(3.2 .10))$ :
$$D \mathcal{T}(\bar{x}, \bar{\lambda})=\left(\begin{array}{c|c} A & B \ \hline B^{\top} & 0 \end{array}\right)$$
where $A=D^2 L(\bar{x}), L$ the associated Lagrange function, and where $B$ consists of the vectors $-D^{\top} h_i(\bar{x}), i \in I$.

# 最优化代写

## 数学代写|最优化理论作业代写optimization theory代考|Additional Notes on Newton’s Method

$$g(x)=\underbrace{g\left(x^k\right)+D g\left(x^k\right)\left(x-x^k\right)}_{\text {Linearization }}+o\left(\left|x-x^k\right|\right) .$$

\begin{aligned} f(x)= & \underbrace{f\left(x^k\right)+D f\left(x^k\right)\left(x-x^k\right)+\frac{1}{2}\left(x-x^k\right)^{\top} D^2 f\left(x^k\right)\left(x-x^k\right)}_{\text {quadratic approximation }}+ \ & +o\left(\left|x-x^k\right|^2\right) . \end{aligned}

## 数学代写|最优化理论作业代写optimization theory代考|Lagrange–Newton Method

$$M:=M[h, g]=\left{x \in \mathbb{R}^n \mid h_i(x)=0, i \in I, g_j(x) \geq 0, j \in J\right}$$

$$\mathcal{T}:\left(\begin{array}{l} x \ \lambda \end{array}\right) \longmapsto\left(\begin{array}{c} D^{\top} f(x)-\sum_{i \in I} \lambda_i D^{\top} h_i(x) \ -h_i(x), \quad i \in I \end{array}\right) .$$

$$D \mathcal{T}(\bar{x}, \bar{\lambda})=\left(\begin{array}{c|c} A & B \ \hline B^{\top} & 0 \end{array}\right)$$

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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