## 数学代写|凸优化作业代写Convex Optimization代考|CPD131

2023年2月1日

couryes-lab™ 为您的留学生涯保驾护航 在代写凸优化Convex Optimization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。我们的专家在代写凸优化Convex Optimization代写方面经验极为丰富，各种代写凸优化Convex Optimization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
couryes™为您提供可以保分的包课服务

## 数学代写|凸优化作业代写Convex Optimization代考|Minimum and minimal elements via dual inequalities

We can use dual generalized inequalities to characterize minimum and minimal elements of a (possibly nonconvex) set $S \subseteq \mathbf{R}^m$ with respect to the generalized inequality induced by a proper cone $K$.
Dual characterization of minimum element
We first consider a characterization of the minimum element: $x$ is the minimum element of $S$, with respect to the generalized inequality $\preceq_K$, if and only if for all $\lambda \succ_{K^*} 0, x$ is the unique minimizer of $\lambda^T z$ over $z \in S$. Geometrically, this means that for any $\lambda \succ_{K *} 0$, the hyperplane
$$\left{z \mid \lambda^T(z-x)=0\right}$$
is a strict supporting hyperplane to $S$ at $x$. (By strict supporting hyperplane, we mean that the hyperplane intersects $S$ only at the point $x$.) Note that convexity of the set $S$ is not required. This is illustrated in figure $2.23$.

To show this result, suppose $x$ is the minimum element of $S$, i.e., $x \preceq_K z$ for all $z \in S$, and let $\lambda \succ_{K^} 0$. Let $z \in S, z \neq x$. Since $x$ is the minimum element of $S$, we have $z-x \succeq_K 0$. From $\lambda \succ_{K^} 0$ and $z-x \succeq_K 0, z-x \neq 0$, we conclude $\lambda^T(z-x)>0$. Since $z$ is an arbitrary element of $S$, not equal to $x$, this shows that $x$ is the unique minimizer of $\lambda^T z$ over $z \in S$. Conversely, suppose that for all $\lambda \succ_{K^*} 0, x$ is the unique minimizer of $\lambda^T z$ over $z \in S$, but $x$ is not the minimum element of $S$. Then there exists $z \in S$ with $z \nsucceq_K x$. Since $z-x \nsucceq_K 0$, there exists $\tilde{\lambda} \succeq_{K^} 0$ with $\tilde{\lambda}^T(z-x)<0$. Hence $\lambda^T(z-x)<0$ for $\lambda \succ_{K^} 0$ in the neighborhood of $\tilde{\lambda}$. This contradicts the assumption that $x$ is the unique minimizer of $\lambda^T z$ over $S$.

## 数学代写|凸优化作业代写Convex Optimization代考|Dual characterization of minimal elements

We now turn to a similar characterization of minimal elements. Here there is a gap between the necessary and sufficient conditions. If $\lambda \succ_{K^*} 0$ and $x$ minimizes $\lambda^T z$ over $z \in S$, then $x$ is minimal. This is illustrated in figure $2.24$.

To show this, suppose that $\lambda \succ_{K^*} 0$, and $x$ minimizes $\lambda^T z$ over $S$, but $x$ is not minimal, i.e., there exists a $z \in S, z \neq x$, and $z \preceq_K x$. Then $\lambda^T(x-z)>0$, which contradicts our assumption that $x$ is the minimizer of $\lambda^T z$ over $S$.

The converse is in general false: a point $x$ can be minimal in $S$, but not a minimizer of $\lambda^T z$ over $z \in S$, for any $\lambda$, as shown in figure 2.25. This figure suggests that convexity plays an important role in the converse, which is correct. Provided the set $S$ is convex, we can say that for any minimal element $x$ there exists a nonzero $\lambda \succeq_{K^*} 0$ such that $x$ minimizes $\lambda^T z$ over $z \in S$.

To show this, suppose $x$ is minimal, which means that $((x-K) \backslash{x}) \cap S=\emptyset$. Applying the separating hyperplane theorem to the convex sets $(x-K) \backslash{x}$ and $S$, we conclude that there is a $\lambda \neq 0$ and $\mu$ such that $\lambda^T(x-y) \leq \mu$ for all $y \in K$, and $\lambda^T z \geq \mu$ for all $z \in S$. From the first inequality we conclude $\lambda \succeq_K \cdot 0$. Since $x \in S$ and $x \in x-K$, we have $\lambda^T x=\mu$, so the second inequality implies that $\mu$ is the minimum value of $\lambda^T z$ over $S$. Therefore, $x$ is a minimizer of $\lambda^T z$ over $S$, where $\lambda \neq 0, \lambda \succeq_{K * 0}$.

This converse theorem cannot he strengthened to $\lambda \succ_{K^*} 0$. Fixamples show that a point $x$ can be a minimal point of a convex set $S$, but not a minimizer of $\lambda^T z$ over $z \in S$ for any $\lambda \succ_{K^} 0$. (See figure $2.26$, left.) Nor is it true that any minimizer of $\lambda^T z$ over $z \in S$, with $\lambda \succeq_{K^} 0$, is minimal (see figure $2.26$, right.)

## 数学代写|凸优化作业代写Convex Optimization代考|Minimum and minimal elements via dual inequalities

$\backslash$ left ${z \backslash m i d \backslash l a m b d a \wedge T(z x)=0 \backslash$ \right } }

$z-x \succeq_K 0, z-x \neq 0$, 我们得出结论
$\lambda^T(z-x)>0$. 自从 $z$ 是任意元素 $S$ ，不等于 $x$ ，这表 明 $x$ 是的唯一极小值 $\lambda^T z$ 超过 $z \in S$. 相反，假设对于所 有 $\lambda \succ_{K^*} 0, x$ 是的唯一极小值 $\lambda^T z$ 超过 $z \in S$ ，但 $x$ 不 是的最小元素 $S$. 那么存在 $z \in S$ 和 $z \nsucceq \nsucceq_K x$. 自从 $z-x \nsucceq K 0$ ，那里存在
\代字号 $\left{\right.$ Vlambda} $\backslash$ succeq_ $\left{\mathrm{K}^{\wedge}\right} 0$ 和 $\tilde{\lambda}^T(z-x)<0$. 因 此 $\lambda^T(z-x)<0$ 为了 Ilambda Isucc $\left{\mathrm{K}^{\wedge}\right} 0$ 在附近 $\tilde{\lambda}$. 这与以下假设相矛盾 $x$ 是的唯一极小值 $\lambda^T z_{\text {超过 }} S$.

## 数学代写|凸优化作业代写Convex Optimization代考|Dual characterization of minimal elements

$z \preceq_K x$. 然后 $\lambda^T(x-z)>0$, 这与我们的假设相矛盾 $x$ 是的最小值 $\lambda^T z$ 超过 $S$.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。