## 数学代写|数值分析代写numerical analysis代考|Inequality Constrained Optimization

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数值分析代写numerical analysis代考|Inequality Constrained Optimization

Inequality constrained optimization is more complex, both in theory and practice. The theorem giving necessary conditions for inequality constrained optimization was only discovered in the middle of the twentieth century, while Lagrange used Lagrange multipliers in his Mécanique Analytique 151. The necessary conditions for inequality constrained optimization are called Kuhn-Tucker or Karush-Kuhn-Tucker conditions. The first journal publication with these conditions was a paper by Kuhn and Tucker in 149, although the essence of these conditions was contained in an unpublished Master’s thesis of Karush 139.

The work of Kuhn and Tucker was intended to build on the work of G. Dantzig and others [69] on linear programming:
$\min x c^T x \quad$ (8.6.7) subject to $$A \boldsymbol{x} \geq \boldsymbol{b}$$ where ” $\boldsymbol{a} \geq \boldsymbol{b}$ ” is understood to mean ” $a_i \geq b_i$ for all $i$ “. It was Dantzig who created the simplex algorithm in 1946 [69], being the first general-purpose and efficient algorithm for solving linear programs (8.6.7, 8.6.8). The simplex method can be considered an example of an active set method as it tracks which of the inequalities $(A x)_i \geq b_i$ is actually an equality as it updates the candidate optimizer $\boldsymbol{x}$. Since then there has been a great deal of work on alternative methods, most notably interior point methods that typically minimize a sequence of penalized problems such as $$c^T \boldsymbol{x}-\alpha \sum{i=1}^m \ln \left((A x)_i-b_i\right)$$
where $\alpha>0$ is a parameter that is reduced to zero in the limit. The first published interior point method was due to Karmarkar 138. Another approach is the ellipsoidal method of Khachiyan 120, which at each step $k$ minimizes $\boldsymbol{c}^T \boldsymbol{x}$ over $\boldsymbol{x}$ lying inside an ellipsoid centered at $\boldsymbol{x}_k$ that is guaranteed to be inside the feasible set ${\boldsymbol{x} \mid A x \geq \boldsymbol{b}}$. Khachiyan’s ellipsoidal method built on previous ideas of N.Z. Shor but was the first guaranteed polynomial time algorithm for linear programming. Karmarkar’s algorithm also guaranteed polynomial time, but was much faster in practice than Khachiyan’s method and the first algorithm to have a better time than the simplex method on average.

## 数学代写|数值分析代写numerical analysis代考|Proving the Karush–Kuhn–Tucker Conditions

To prove the Karush-Kuhn-Tucker conditions we need a constraint qualification to ensure that
$$T_{\Omega}(\boldsymbol{x})=\left{\boldsymbol{d} \mid \nabla g_i(\boldsymbol{x})^T \boldsymbol{d}=0 \text { for all } i \in \mathcal{E},\right.$$
(8.6.9) $\nabla g_i(\boldsymbol{x})^T \boldsymbol{d} \geq 0$ for all $i \in \mathcal{I}$ where $\left.g_i(\boldsymbol{x})=0\right}=C_{\Omega}(\boldsymbol{x})$.
A constraint $g_i(\boldsymbol{x}) \geq 0$ is called active at $\boldsymbol{x}$ if $g_i(\boldsymbol{x})=0$ and inactive at $\boldsymbol{x}$ if $g_i(\boldsymbol{x})>0$. Inactive inequality constraints at $x$ do not affect the shape of the feasible set $\Omega$ near to $\boldsymbol{x}$. We designate the set of active constraints by
$$\mathcal{A}(\boldsymbol{x})=\left{i \mid i \in \mathcal{E} \cup \mathcal{I} \text { and } g_i(\boldsymbol{x})=0\right} .$$
The equivalence (8.6.9) holds under a number of constraint qualifications, the most used of which is the Linear Independence Constraint Qualification (LICQ) for inequality constrained optimization:
(8.6.11) $\quad\left{\nabla g_i(\boldsymbol{x}) \mid i \in \mathcal{A}(\boldsymbol{x})\right}$ is a linearly independent set.
Weaker constraint qualifications that guarantee (8.6.9) include the MangasarianFromowitz constraint qualification (MFCQ):
$\left{\nabla g_i(\boldsymbol{x}) \mid i \in \mathcal{E}\right}$ is a linearly independent set, and there is $\boldsymbol{d}$ where $\nabla g_i(\boldsymbol{x})^T \boldsymbol{d}=0$ for all $i \in \mathcal{E}$, and
$$\nabla g_i(\boldsymbol{x})^T \boldsymbol{d}>0 \text { for all } i \in \mathcal{I} \cap \mathcal{A}(\boldsymbol{x}) .$$
With a suitable constraint qualification, we can prove the existence of Lagrange multipliers satisfying the Karush-Kuhn-Tucker conditions.

# 数值分析代考

## 数学代写|数值分析代写numerical analysis代考|Inequality Constrained Optimization

Kuhn 和 Tucker 的工作旨在建立在 G. Dantzig 和其他人 [69] 关于线性规划的工作之上: $\min x c^T x \quad(8.6 .7)$ 受制于
$$A \boldsymbol{x} \geq \boldsymbol{b}$$

$$c^T \boldsymbol{x}-\alpha \sum i=1^m \ln \left((A x)_i-b_i\right)$$

## 数学代写|数值分析代写numerical analysis代考|Proving the Karush–Kuhn–Tucker Conditions

为了证明 Karush-Kuhn-Tucker 条件，我们需要一个约束条件来确保
$$T_{\Omega}(\boldsymbol{x})=\left{\boldsymbol{d} \mid \nabla g_i(\boldsymbol{x})^T \boldsymbol{d}=0 \text { for all } i \在 \mathcal{E} 中，\对。$$
(8.6.9) $\nabla g_i(\boldsymbol{x})^T \boldsymbol{d} \geq 0$ 对于所有 $i \in \mathcal{I}$ 其中 $\left.g_i(\boldsymbol{x })=0\right}=C_{\Omega}(\boldsymbol{x})$。

$$\mathcal{A}(\boldsymbol{x})=\left{i \mid i \in \mathcal{E} \cup \mathcal{I} \text { and } g_i(\boldsymbol{x})=0\正确的} 。$$

(8.6.11) $\quad\left{\nabla g_i(\boldsymbol{x}) \mid i \in \mathcal{A}(\boldsymbol{x})\right}$ 是线性独立集。

$\left{\nabla g_i(\boldsymbol{x}) \mid i \in \mathcal{E}\right}$ 是线性独立集，有 $\boldsymbol{d}$ 其中 $\nabla g_i( \boldsymbol{x})^T \boldsymbol{d}=0$ 对于所有 $i \in \mathcal{E}$，并且
$$\nabla g_i(\boldsymbol{x})^T \boldsymbol{d}>0 \text { for all } i \in \mathcal{I} \cap \mathcal{A}(\boldsymbol{x}) 。$$

## 有限元方法代写

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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。

## 数学代写|数值分析代写numerical analysis代考|Constrained Optimization

statistics-lab™ 为您的留学生涯保驾护航 在代写数值分析numerical analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写数值分析numerical analysis代写方面经验极为丰富，各种代写数值分析numerical analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数值分析代写numerical analysis代考|Constrained Optimization

Constrained optimization can be represented most abstractly in terms of a feasible set, often denoted $\Omega \subseteq \mathbb{R}^n$ :
(8.6.1) $\min _x f(x) \quad$ subject to $x \in \Omega$
Solutions exist if $f$ is continuous and either $\Omega$ is a compact (closed and bounded) subset of $\mathbb{R}^n$, or if $\Omega$ is closed and $f$ is coercive. Usually $\Omega$ is represented by equations and inequalities:
(8.6.2) $\Omega=\left{\boldsymbol{x} \in \mathbb{R}^n \mid g_i(\boldsymbol{x})=0\right.$ for $i \in \mathcal{E}$, and $g_i(\boldsymbol{x}) \geq 0$ for $\left.i \in \mathcal{I}\right}$.
If $\mathcal{I}$ is empty but $\mathcal{E}$ is not empty, then we say (8.6.1) is an equality constrained optimization problem. If $\mathcal{I}$ is non-empty, we say (8.6.1) is an inequality constrained optimization problem.

For a general constrained optimization problem, first-order conditions can be given in terms of the tangent cone
$(8.6 .3)$
$$T_{\Omega}(\boldsymbol{x})=\left{\lim _{k \rightarrow \infty} \frac{\boldsymbol{x}_k-\boldsymbol{x}}{t_k} \mid \boldsymbol{x}_k \in \Omega, \boldsymbol{x}_k \rightarrow \boldsymbol{x} \text { as } k \rightarrow \infty \text {, and } t_k \downarrow 0 \text { as } k \rightarrow \infty\right}$$

Lemma 8.19 If $\boldsymbol{x}=x^$ minimizes $f(x)$ over $\boldsymbol{x} \in \Omega$ and $f$ is differentiable at $\boldsymbol{x}^$, then
(8.6.4) $\nabla f\left(x^\right)^T d \geq 0 \quad$ for all $d \in T_{\Omega}\left(x^\right)$.
Proof Suppose $x=x^* \in \Omega$ minimizes $f(x)$ over $x \in \Omega$ and $f$ is differentiable. Then for any $\boldsymbol{d} \in T_{\Omega}\left(\boldsymbol{x}^\right)$, there is a sequence $\boldsymbol{x}k \rightarrow \boldsymbol{x}^$ as $k \rightarrow \infty$ with $\boldsymbol{x}_k \in \Omega$ where $\boldsymbol{d}_k:=\left(\boldsymbol{x}_k-\boldsymbol{x}^\right) / t_k \rightarrow \boldsymbol{d}$ as $k \rightarrow \infty$. Since $f\left(\boldsymbol{x}^\right) \leq f\left(\boldsymbol{x}_k\right)=f\left(\boldsymbol{x}^+t_k \boldsymbol{d}_k\right)$, $$0 \leq \lim {k \rightarrow \infty} \frac{f\left(x^+t_k \boldsymbol{d}k\right)-f\left(x^\right)}{t_k}=\nabla f\left(x^\right)^T \lim {k \rightarrow \infty} \boldsymbol{d}k=\nabla f\left(x^\right)^T \boldsymbol{d} .$$ This holds for any $d \in T{\Omega}\left(x^\right)$ showing (8.6.4), as we wanted.
Constraint qualifications relate the tangent cone $T_{\Omega}(\boldsymbol{x})$ to the linearizations of the constraint functions:
\begin{aligned} & C_{\Omega}(\boldsymbol{x})=\left{\boldsymbol{d} \in \mathbb{R}^n \mid \nabla g_i(\boldsymbol{x})^T \boldsymbol{d}=0 \text { for all } i \in \mathcal{E},\right. \ & \left.\quad \nabla g_i(\boldsymbol{x})^T \boldsymbol{d} \geq 0 \text { for all } i \in \mathcal{I} \text { where } g_i(\boldsymbol{x})=0\right} . \end{aligned}
For equality constrained optimization $(\mathcal{I}=\emptyset)$, the LICQ (8.1.2) implies that $T_{\Omega}(\boldsymbol{x})=C_{\Omega}(\boldsymbol{x})$ as noted in Section 8.1.3.

## 数学代写|数值分析代写numerical analysis代考|Equality Constrained Optimization

The theory of Section 8.1.3 for Lagrange multipliers and equality constrained optimization (8.1.5) can be immediately turned into a numerical method. To solve
\begin{aligned} & \mathbf{0}=\nabla f(\boldsymbol{x})-\sum_{i \in \mathcal{E}} \lambda_i \nabla g(\boldsymbol{x}) \ & 0=g_i(\boldsymbol{x}), \quad i \in \mathcal{E} \end{aligned}
for $(\boldsymbol{x}, \boldsymbol{\lambda})$ with $\boldsymbol{\lambda}=\left[\lambda_i \mid i \in \mathcal{E}\right]$ we can apply, for example, Newton’s method. For unconstrained optimization, we can then perform a line search to ensure that the step improves the solution estimate. The issue in constrained optimization is that $f(\boldsymbol{x})$ alone is no longer suitable for measuring improvements. Constrained optimization problems have two objectives: staying on the feasible set, and minimizing $f(\boldsymbol{x})$. It may be necessary to increase $f(\boldsymbol{x})$ in order to return to the feasible set. Solving the Newton equations for $(8.6 .5,8.6 .6)$ gives a direction $\boldsymbol{d}$. Because of the curvature of the feasible set $\Omega$ for general functions $g_i$, moving in the direction $\boldsymbol{d}$ even if $\boldsymbol{x}$ is feasible may take the point $\boldsymbol{x}+s \boldsymbol{d}$ off the feasible set. This can be offset by having a second order correction step to move back toward the feasible set. This second order correction uses a least squares version of Newton’s method to solve $g(\boldsymbol{x})=\mathbf{0}$.

Since this is an under-determined system for $|\mathcal{E}|<n$, we find the solution $\delta \boldsymbol{x}$ for $\nabla g_i(\boldsymbol{x})^T \delta \boldsymbol{x}=-g_i(\boldsymbol{x}), i \in \mathcal{E}$, that minimizes $|\delta \boldsymbol{x}|_2$, which can be done using the QR factorization of $\left[\nabla g_i(\boldsymbol{x}) \mid i \in \mathcal{E}\right]$.

For line search algorithms, we can use a merit function to determine the quality of the result of the step. Often, merit functions of the form $\boldsymbol{x} \mapsto f(\boldsymbol{x})+\alpha \sum_{i \in \mathcal{E}}\left|g_i(\boldsymbol{x})\right|$ are used where $\alpha>\max _{i \in \mathcal{E}}\left|\lambda_i\right|$. A basic method for solving equality constrained optimization problems is shown in Algorithm 82.

If the second-order correction is skipped, then the Newton method may fail to give rapid convergence, as was noted by N. Maratos in his PhD thesis [170].

# 数值分析代考

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

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