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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|A Brief Review of Non-convex Single-Objective

Single-objective optimization methods are among the mathematical methods most widely used in various applications. The classical methods starting with linear programming have been proved very valuable tools for solving various economic and engineering problems. However, the growing complexity of the applied problems demanded the development of new ideas, methods, and algorithms. The classical mathematical optimization methods are based on the assumption that an objective function is convex. The convexity assumption, which is very fruitful for theoretical investigation, is hardly provable in many practical applications. Moreover, it is not truth very frequently. Therefore in the 1960 s of the last century there begun active research in global optimization of non-convex problems. Naturally, single-objective problems foremost attracted attention of researchers. In this section we briefly review the approaches to single-objective global optimization, the multi-objective presentation we refer to the representative monographs.

The fundamental difficulty of non-convex problems is the possibility of existence of many local optima. Classical methods are not aimed at finding the best of them, the global optimum, but an arbitrary local optimum corresponding to some initial search conditions. Difficulties in extending the classical optimization theory to nonconvex problems motivated a vast development of heuristic methods. Many of these methods exploited randomization of search and ideas from nature. An early development of a heuristic approach was represented, e.g., in [81, 175, 186]. During the initial period, the development of theoretically substantiated methods was considerably slower than that of various heuristics. Nevertheless, several theoretic approaches have emerged, e.g., based on statistics of extreme values [138]. An example of a heuristic method with subsequently well-developed theory is simulated annealing [1, 221]. During the past years research in heuristic and theoretically substantiated global optimization expanded. Very important impact to intensify the theory of global optimization has made the Journal of Global Optimization, the publication of which started in 1990 . The results of the last century are summarized in $[84,152]$. Further in the present book we consider extensions of theoretically substantiated methods of single-objective global optimization to the multi-objective case. Therefore, in this brief review of single-objective global optimization we do not consider heuristic methods, and refer to $[8,69]$ for thorough presentation of that subject.

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

Lipschitz optimization $[78,86,87,159,168,208]$ is based on the assumption that the real-valued objective function $f(\mathbf{x})$ is Lipschitz continuous, i.e.,
$$|f(\mathbf{x})-f(\mathbf{y})| \leq L|\mathbf{x}-\mathbf{y}|, \forall \mathbf{x}, \mathbf{y} \in \mathbf{A}, \quad 0<L<\infty$$
where $L$ is the Lipschitz constant, $\mathbf{A} \subset \mathbb{R}^{d}$ is compact, and $|\cdot|$ denotes a norm. Sometimes it is also assumed that the first derivative of the objective is also Lipschitz continuous. In Lipschitz optimization the Euclidean norm is used most often, but other norms can also be considered [157, 158].

Lipschitz optimization algorithms can estimate how far the current approximation is from the optimal function value, and hence can use stopping criteria that are more meaningful than a simple iteration limit. The methods can guarantee to find an approximation of the solution to a specified accuracy within finite time. Lipschitz optimization may be used in situations when an analytical description of the objective function is not available.

An important question in Lipschitz optimization is how to obtain the Lipschitz constant of the objective function or at least its estimate. There are several approaches [195]:

1. The Lipschitz constant is assumed given a priori $[10,86,136,170]$. This case is very important from the theoretical viewpoint although in practice it is often difficult to use.
2. Adaptive global estimate over the whole search region is used $[86,109,167,208]$.
3. Local Lipschitz constants are estimated adaptively $[112,119,189,190,194,197$, $208]$
4. A set of possible values for the Lipschitz constant is used $[59,96,160,193,194]$.

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

Lipschitz 优化 $[78,86,87,159,168,208]$ 基于实值目标函数的假设 $f(\mathbf{x})$ 是 Lipschitz 连续的，即
$$|f(\mathbf{x})-f(\mathbf{y})| \leq L|\mathbf{x}-\mathbf{y}|, \forall \mathbf{x}, \mathbf{y} \in \mathbf{A}, \quad 0<L<\infty$$

Lipschitz 优化算法可以估计当前近似值与最优函数值的距离，因此可以使用比简单迭代限制更有意义的停止标准。这些方法可以保证在有限时间内找到解的近似 值到指定的精度。Lipschitz 优化可用于目标函数的分析描述不可用的情况。
Lipschitz 优化中的一个重要问题是如何获得目标函数的 Lipschitz 常数或至少是其估计值。有几种方法 [195]:

1. 假设 Lipschitz 常数是先验的 $[10,86,136,170]$. 从理论角度来看，这种情况非常重要，尽管在实践中通常很难使用。
2. 使用整个搜索区域的自适应全局估计 $[86,109,167,208]$.
3. 自适应估计局部 Lipschitz 常数 $[112,119,189,190,194,197,208]$
4. 使用 Lipschitz 常数的一组可能值 $[59,96,160,193,194]$.

## 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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