Posted on Categories:Convex optimization, 凸优化, 数学代写

# 数学代写|凸优化代写Convex Optimization代考|Cutting Plane Schemes

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|凸优化代写Convex Optimization代考|Cutting Plane Schemes

Let us look now at the following minimization problem with set constraint:
$$\min {f(x) \mid x \in Q}$$
where the function $f$ is convex on $\mathbb{R}^n$, and $Q$ is a bounded closed convex set such that
$$\text { int } Q \neq \emptyset, \quad \operatorname{diam} Q=D<\infty$$
We assume that $Q$ is not simple and that our problem is equipped with a separation oracle. At any test point $\bar{x} \in \mathbb{R}^n$, this oracle returns a vector $g(x)$, which is either:

a subgradient of $f$ at $\bar{x}$, if $x \in Q$,

a separator of $\bar{x}$ from $Q$, if $x \notin Q$.
An important example of such a problem is a constrained minimization problem with functional constraints (3.2.22). We have seen that this problem can be rewritten as a problem with a single functional constraint (see (3.2.23)) defining the feasible set
$$Q=\left{x \in \mathbb{R}^n \mid \bar{f}(x) \leq 0\right}$$
In this case, for $x \notin Q$ the oracle has to provide us with any subgradient $\bar{g} \in \partial \bar{f}(x)$. Clearly, $\bar{g}$ separates $x$ from $Q$ (see Theorem 3.1.18).
Let us present the main property of finite-dimensional localization sets. Consider a sequence $X \equiv\left{x_i\right}_{i=0}^{\infty}$ belonging to the set $Q$. Recall that the localization sets generated by this sequence are defined as follows:
\begin{aligned} S_0(X) & =Q, \ S_{k+1}(X) & =\left{x \in S_k(X) \mid\left\langle g\left(x_k\right), x_k-x\right\rangle \geq 0\right} . \end{aligned}

## 数学代写|凸优化代写Convex Optimization代考|Nonsmooth Models of the Objective Function

In the previous section, we looked at several methods for solving the following problem:
$$\min _{x \in Q} f(x),$$
where $f$ is a Lipschitz continuous convex function and $Q$ is a closed convex set. We have seen that the optimal method for problem (3.3.1) is the Subgradient Method (3.2.14), (3.2.16). Note that this conclusion is valid for the whole class of Lipschitz continuous functions. However, if we are going to minimize a particular function from this class, we can expect that it will not be as bad as in the worst case. We usually can hope that the actual performance of the minimization methods can be much better than the worst-case theoretical bound. Unfortunately, as far as the Subgradient Method is concerned, these expectations are too optimistic. The scheme of the Subgradient Method is very strict and in general it cannot converge faster than in theory. It can also be shown that the Ellipsoid Method (3.2.53) inherits this drawback of subgradient schemes. In practice it works more or less in accordance with its theoretical bound even when it is applied to a very simple function like $|x|^2$.

In this section, we will discuss algorithmic schemes which are more flexible than the Subgradient Method and Ellipsoid Method. These schemes are based on the notion of a nonsmooth model of a convex objective function.
Definition 3.3.1 Let $X=\left{x_k\right}_{k=0}^{\infty}$ be a sequence of points in $Q$. Define
$$\hat{f}k(X ; x)=\max {0 \leq i \leq k}\left[f\left(x_i\right)+\left\langle g\left(x_i\right), x-x_i\right\rangle\right]$$
where $g\left(x_i\right)$ are some subgradients of $f$ at $x_i$. The function $\hat{f}_k(X ; \cdot)$ is called a nonsmooth model of the convex function $f$.

Note that $f_k(X ; \cdot)$ is a piece-wise linear function. In view of inequality (3.1.23), we always have
$$f(x) \geq \hat{f}k(X ; x)$$ for all $x \in \mathbb{R}^n$. However, at all test points $x_i, 0 \leq i \leq k$, we have $$f\left(x_i\right)=\hat{f}_k\left(X ; x_i\right), \quad g\left(x_i\right) \in \partial \hat{f}_k\left(X ; x_i\right)$$ Moreover, the next model is always better than the previous one: $$\hat{f}{k+1}(X ; x) \geq \hat{f}_k(X ; x)$$
for all $x \in \mathbb{R}^n$

## 数学代写|凸优化代写Convex Optimization代考|Cutting Plane Schemes

$$\min {f(x) \mid x \in Q}$$

$$\text { int } Q \neq \emptyset, \quad \operatorname{diam} Q=D<\infty$$

$f$在$\bar{x}$的子梯度，如果$x \in Q$，

$$Q=\left{x \in \mathbb{R}^n \mid \bar{f}(x) \leq 0\right}$$

## 数学代写|凸优化代写Convex Optimization代考|Nonsmooth Models of the Objective Function

$$\min _{x \in Q} f(x),$$

3.3.1设$X=\left{x_k\right}_{k=0}^{\infty}$为$Q$中的一个点序列。定义
$$\hat{f}k(X ; x)=\max {0 \leq i \leq k}\left[f\left(x_i\right)+\left\langle g\left(x_i\right), x-x_i\right\rangle\right]$$

$$f(x) \geq \hat{f}k(X ; x)$$为所有$x \in \mathbb{R}^n$。然而，在所有的测试点$x_i, 0 \leq i \leq k$，我们有$$f\left(x_i\right)=\hat{f}_k\left(X ; x_i\right), \quad g\left(x_i\right) \in \partial \hat{f}_k\left(X ; x_i\right)$$而且，下一个模型总是比前一个更好:$$\hat{f}{k+1}(X ; x) \geq \hat{f}_k(X ; x)$$

## 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。