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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|凸优化作业代写Convex Optimization代考|Summary and discussion

In this chapter, we have revisited some mathematical basics of sets, functions, matrices, and vector spaces that will be very useful to understand the remaining chapters and we also introduced the notations that will be used throughout this book. The mathematical preliminaries reviewed in this chapter are by no means complete. For further details, the readers can refer to [Apo07] and [WZ97] for Section 1.1, and [HJ85] and [MS00] for Section 1.2, and other related textbooks.
Suppose that we are given an optimization problem in the following form:
\begin{aligned} \text { minimize } & f(\boldsymbol{x}) \ \text { subject to } & \boldsymbol{x} \in \mathcal{C} \end{aligned}
where $f(\boldsymbol{x})$ is the objective function to be minimized and $\mathcal{C}$ is the feasible set from which we try to find an optimal solution. Convex optimization itself is a powerful mathematical tool for optimally solving a well-defined convex optimization problem (i.e., $f(\boldsymbol{x})$ is a convex function and $\mathcal{C}$ is a convex set in problem $(1.127)$ ), or for handling a nonconvex optimization problem (that can be approximated as a convex one). However, the problem (1.127) under investigation may often appear to be a nonconvex optimization problem (with various camouflages) or a nonconvex and nondeterministic polynomial-time hard (NP-hard) problem that forces us to find an approximate solution with some performance or computational efficiency merits and characteristics instead. Furthermore, reformulation of the considered optimization problem into a convex optimization problem can be quite challenging. Fortunately, there are many problem reformulation approaches (e.g., function transformation, change of variables, and equivalent representations) to conversion of a nonconvex problem into a convex problem (i.e., unveiling of all the camouflages of the original problem).

The bridge between the pure mathematical convex optimization theory and how to use it in practical applications is the key for a successful researcher or professional who can efficiently exert his (her) efforts on solving a challenging scientific and engineering problem to which he (she) is dedicated. For a given opti-mization problem, we aim to design an algorithm (e.g., transmit beamforming algorithm and resource allocation algorithm in communications and networking, nonnegative blind source separation algorithm for the analysis of biomedical and hyperspectral images) to efficiently and reliably yield a desired solution (that may just be an approximate solution rather than an optimal solution), as shown in Figure 1.6, where the block “Problem Reformulation,” the block “Algorithm Design,” and the block “Performance Evaluation and Analysis” are essential design steps before an algorithm that meets our goal is obtained. These design steps rely on smart use of advisable optimization theory and tools that remain in the cloud, like a military commander who needs not only ammunition and weapons but also an intelligent fighting strategy. It is quite helpful to build a bridge so that one can readily use any suitable mathematical theory (e.g., convex sets and functions, optimality conditions, duality, KKT conditions, Schur complement, S-procedure, etc.) and convex solvers (e.g., CVX and SeDuMi) to accomplish these design steps.

## 数学代写|凸优化作业代写Convex Optimization代考|Affine and convex sets

Mathematically, a line $\mathcal{L}\left(\mathbf{x}{1}, \mathbf{x}{2}\right)$ passing through two points $\mathbf{x}{1}$ and $\mathbf{x}{2}$ in $\mathbb{R}^{n}$ is the set defined as
$$\mathcal{L}\left(\mathbf{x}{1}, \mathbf{x}{2}\right)=\left{\theta \mathbf{x}{1}+(1-\theta) \mathbf{x}{2}, \theta \in \mathbb{R}\right}, \mathbf{x}{1}, \mathbf{x}{2} \in \mathbb{R}^{n} .$$
If $0 \leq \theta \leq 1$, then it is a line segment connecting $\mathbf{x}{1}$ and $\mathbf{x}{2}$. Note that the linear combination $\theta \mathbf{x}{1}+(1-\theta) \mathbf{x}{2}$ of two points $\mathbf{x}{1}$ and $\mathbf{x}{2}$ with the coefficient sum equal to unity as in (2.1) plays an essential role in defining affine sets and convex sets, and hence the one with $\theta \in \mathbb{R}$ is referred to as the affine combination and the one with $\theta \in[0,1]$ is referred to as the convex combination. Affine combination and convex combination can be extended to the case of more than two points in the same fashion.

## 数学代写|凸优化作业代写Convex Optimization代考| Summary and discussion

minimize $f(x)$ subject to $\quad x \in \mathcal{C}$

## 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。

assignmentutor™您的专属作业导师
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