# 数学代写|凸优化作业代写Convex Optimization代考|Smoothing of Nondifferentiable Problems

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## 数学代写|凸优化作业代写Convex Optimization代考|Smoothing of Nondifferentiable Problems

Generally speaking, differentiable cost functions are preferable to nondifferentiable ones, because algorithms for the former are better developed and are more effective than algorithms for the latter. Thus there is an incentive to eliminate nondifferentiabilities by “smoothing” their corners. It turns out that penalty functions and smoothing are closely related, reflecting the fact that constraints and nondifferentiabilities are also closely related. As an example of this connection, the unconstrained minimax problem
$$\begin{array}{ll} \operatorname{minimize} & \max \left{f_1(x), \ldots, f_m(x)\right} \ \text { subject to } & x \in \Re^n, \end{array}$$
where $f_1, \ldots, f_m$ are differentiable functions can be converted to the differentiable constrained problem
\begin{aligned} & \operatorname{minimize} z \ & \text { subject to } f_j(x) \leq z, \quad j=1, \ldots, m, \end{aligned}
where $z$ is an artificial scalar variable. When a penalty or augmented Lagrangian is applied to the constrained problem (2.63), we will show that a smoothing method is obtained for the minimax problem (2.62).

We will now describe a technique (first given in [Ber75b], and generalized in [Ber77]) to obtain smoothing approximations. Let $f: \Re^n \mapsto$ $(-\infty, \infty]$ be a closed proper convex function with conjugate denoted by $f^{\star}$. For fixed $c>0$ and $\lambda \in \Re^n$, define
$$f_{c, \lambda}(x)=\inf _{u \in \Re^n}\left{f(x-u)+\lambda^{\prime} u+\frac{c}{2}|u|^2\right}, \quad x \in \Re^n .$$

## 数学代写|凸优化作业代写Convex Optimization代考|Smoothing and Augmented Lagrangians

The smoothing technique just described can also be combined with the augmented Lagrangian method. As an example, let $f: \Re^n \mapsto(-\infty, \infty]$ be a closed proper convex function with conjugate denoted by $f^{\star}$. Let $F: \Re^n \mapsto \Re$ be another convex function, and let $X$ be a closed convex set. Consider the problem
$$\begin{array}{ll} \operatorname{minimize} & F(x)+f(x) \ \text { subject to } x \in X, \end{array}$$
and the equivalent problem
$$\begin{array}{ll} \operatorname{minimize} & F(x)+f(x-u) \ \text { subject to } & x \in X, u=0 . \end{array}$$
Applying the augmented Lagrangian method (2.53)-(2.54) to the latter problem leads to minimizations of the form
$$\left(x_{k+1}, u_{k+1}\right) \in \arg \min {x \in X, u \in \Re^n}\left{F(x)+f(x-u)+\lambda_k^{\prime} u+\frac{c_k}{2}|u|^2\right}$$ By first minimizing over $u \in \Re^n$, these minimizations yield $$x{k+1} \in \arg \min {x \in X}\left{F(x)+f{c_k, \lambda_k}(x)\right}$$
where $f_{c_k, \lambda_k}$ is the smoothed function
$$f_{c_k, \lambda_k}(x)=\inf {u \in \Re^n}\left{f(x-u)+\lambda_k^{\prime} u+\frac{c_k}{2}|u|^2\right}$$ [cf. Eq. (2.64)]. The corresponding multiplier update (2.54) is $$\lambda{k+1}=\lambda_k+c_k u_{k+1}$$
where
$$u_{k+1} \in \arg \min {u \in \Re^n}\left{f\left(x{k+1}-u\right)+\lambda_k^{\prime} u+\frac{c_k}{2}|u|^2\right} .$$

# 凸优化代写

## 数学代写|凸优化作业代写Convex Optimization代考|Smoothing of Nondifferentiable Problems

$$\begin{array}{ll} \operatorname{minimize} & \max \left{f_1(x), \ldots, f_m(x)\right} \ \text { subject to } & x \in \Re^n, \end{array}$$

\begin{aligned} & \operatorname{minimize} z \ & \text { subject to } f_j(x) \leq z, \quad j=1, \ldots, m, \end{aligned}

## 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。

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