# 数学代写|凸优化作业代写Convex Optimization代考|CS168

#### Doug I. Jones

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## 数学代写|凸优化作业代写Convex Optimization代考|BASIC THEORY OF PROXIMAL ALGORITHMS

In this section we consider the minimization of a closed proper convex function $f: \Re^n \mapsto(-\infty, \infty]$ using an approximation approach whereby we modify $f$ by adding a regularization term. In particular, we consider the algorithm
$$x_{k+1} \in \arg \min _{x \in \Re^n}\left{f(x)+\frac{1}{2 c_k}\left|x-x_k\right|^2\right},$$
where $x_0$ is an arbitrary starting point and $c_k$ is a positive scalar parameter; see Fig. 5.1.1. This is the proximal algorithm (also known as the proximal minimization algorithm or the proximal point algorithm).

The degree of regularization is controlled by the parameter $c_k$. For small values of $c_k, x_{k+1}$ tends to stay close to $x_k$, albeit at the expense of slower convergence. The convergence mechanism is illustrated in Fig. 5.1.2. Note that the quadratic term $\left|x-x_k\right|^2$ makes the function that is minimized at each iteration strictly convex with compact level sets. This guarantees, among others, that $x_{k+1}$ is well-defined as the unique minimum in Eq. (5.1) [cf. Prop, 3.1.1 and Prop. 3.2.1 in Appendix B; also the broader discussion of existence of minima in Chapter 3 of [Ber09]].

Evidently, the algorithm is useful only for problems that can benefit from regularization. It turns out, however, that many interesting problems fall in this category, and often in unexpected and diverse ways. In particular, as we will see in this and the next chapter, the creative application of the proximal algorithm and its variations, together with duality ideas, can allow the elimination of constraints and nondifferentiabilities, the stabilization of the linear approximation methods of Chapter 4 , and the effective exploitation of special problem structures.

## 数学代写|凸优化作业代写Convex Optimization代考|Convergence

The proximal algorithm has excellent convergence properties, which we develop in this section. We first derive some preliminary results in the following two propositions.

Proposition 5.1.1: If $x_k$ and $x_{k+1}$ are two successive iterates of the proximal algorithm (5.1), we have
$$\frac{x_k-x_{k+1}}{c_k} \in \partial f\left(x_{k+1}\right) .$$
Proof: Since the function
$$f(x)+\frac{1}{2 c_k}\left|x-x_k\right|^2$$
is minimized at $x_{k+1}$, the origin must belong to its subdifferential at $x_{k+1}$, which is equal to
$$\partial f\left(x_{k+1}\right)+\frac{x_{k+1}-x_k}{c_k},$$
(cf. Prop. 5.4.6 in Appendix B, which applies because its relative interior condition is satisfied since the quadratic term is real-valued). The desired relation (5.2) holds if and only if the origin belongs to the above set. Q.E.D.

# 凸优化代写

## 数学代写|凸优化作业代写Convex Optimization代考|BASIC THEORY OF PROXIMAL ALGORITHMS

$$x_{k+1} \in \arg \min _{x \in \Re^n}\left{f(x)+\frac{1}{2 c_k}\left|x-x_k\right|^2\right},$$

## 数学代写|凸优化作业代写Convex Optimization代考|Convergence

$$\frac{x_k-x_{k+1}}{c_k} \in \partial f\left(x_{k+1}\right) .$$

$$f(x)+\frac{1}{2 c_k}\left|x-x_k\right|^2$$

$$\partial f\left(x_{k+1}\right)+\frac{x_{k+1}-x_k}{c_k},$$
(参见附录B第5.4.6条，由于二次项为实值，故满足其相对内条件)。当且仅当原点属于上述集合时，所需关系(5.2)成立。Q.E.D.

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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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