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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|The Method of kth-Objective Weighted-Constraint

The method named ” $k$ th-objective weighted-constraint” [26] can be applied for the construction of the discrete representation of the set of Pareto optimal solutions under the mild assumptions that the objective functions are continuous and bounded from below with a known lower bound; the latter assumptions without loss of generality can be reformulated as $f_{i}(\mathbf{x})>0, \mathbf{x} \in \mathbf{A}, i=1, \ldots, m$. As discussed in the previous chapter the set of solutions of
\begin{aligned} &\min {\mathbf{x} \in \mathbf{A}} w{k} f_{k}(\mathbf{x}), \ &w_{i} f_{i}(\mathbf{x}) \leq w_{k} f_{k}(\mathbf{x}), i=1, \ldots, m, i \neq k \ &w_{i}>0, \sum_{i=1}^{m} w_{i}=1 \end{aligned}
is coincident with the set of decisions respective to the weakly Pareto optimal solutions of the original multi-objective optimization problem. We briefly present the description of the version of the algorithm for bi-objective problems [26]. An approximation of the set of Pareto optimal solutions is found solving the series of the single-objective optimization problems with varying $k$ and $\mathbf{w}$ where a utopian vector $\mathbf{u}$ can be chosen in view of known lower bounds for the objective functions values:

Step 1 Initialization: utopian vector $\mathbf{u}=\left(u_{1}, u_{2}\right)^{T}$; the number of partition points $N$ for the interval of weights.

Step 2 Objective functions are minimized separately, and the interval of weights is defined
\begin{aligned} \mathbf{z}{i} &=\arg \min f{i}(\mathbf{x}), f_{i}^{*}=f_{i}\left(\mathbf{z}{i}\right), i=1,2, \ \dot{v}{1} &=f_{1}\left(\mathbf{z}{2}\right) . v{2}=f_{2}\left(\mathbf{z}_{1}\right) . \end{aligned}

## 数学代写|凸优化作业代写Convex Optimization代考|An Example of Adaptive Method

The discrete representation of the set of Pareto optimal solutions is computed by means of the solution of a sequence of single-objective optimization problems which depend on some parameters. To obtain a favorable distribution of the representation vectors, an appropriate sequence of parameters should be selected. However, the a priori selection of the sequence of parameters ensuring a desirable distribution of the solutions is possible very rarely, only in exceptional cases. The adaptive selection of parameters during the solution process can be more advantageous than its a priori definition.

To illustrate the adaptive approach let us consider a bi-objective minimization problem
$$\min {\mathbf{x} \in \mathbf{A}} \mathbf{f}(\mathbf{x}), \mathbf{f}(\mathbf{x})=\left(f{1}(\mathbf{x}), f_{2}(\mathbf{x})\right)^{T},$$
with convex objectives. In this case a weighted sum scalarization can be applied, i.e., a Pareto optimal solution $\mathbf{y}{w}$ is found as follows: $$\begin{array}{r} \mathbf{y}{w}=f\left(\mathbf{x}{w}\right), \quad \mathbf{x}{w}=\arg \min {\mathbf{x} \in \mathbf{A}}\left(w f{1}(\mathbf{x})+(1-w) f_{2}(\mathbf{x})\right), \ 0 \leq w \leq 1 \end{array}$$
A reasonable initial sequence of weights is uniformly distributed in the interval $[0,1]$

## 数学代写|凸优化作业代写Convex Optimization代考|The Method of kth-Objective Weighted-Constraint

$$\min \mathbf{x} \in \mathbf{A} w k f_{k}(\mathbf{x}), \quad w_{i} f_{i}(\mathbf{x}) \leq w_{k} f_{k}(\mathbf{x}), i=1, \ldots, m, i \neq k w_{i}>0, \sum_{i=1}^{m} w_{i}=1$$

Step 2 分别最小化目标函数，定义权重区间
$$\mathbf{z} i=\arg \min f i(\mathbf{x}), f_{i}^{*}=f_{i}(\mathbf{z} i), i=1,2, \dot{v} 1 \quad=f_{1}(\mathbf{z} 2) \cdot v 2=f_{2}\left(\mathbf{z}_{1}\right) .$$

## 数学代写|凸优化作业代写Convex Optimization代考|An Example of Adaptive Method

$$\min \mathbf{x} \in \mathbf{A f}(\mathbf{x}), \mathbf{f}(\mathbf{x})=\left(f 1(\mathbf{x}), f_{2}(\mathbf{x})\right)^{T},$$

$$\mathbf{y} w=f(\mathbf{x} w), \quad \mathbf{x} w=\arg \min \mathbf{x} \in \mathbf{A}\left(w f 1(\mathbf{x})+(1-w) f_{2}(\mathbf{x})\right), 0 \leq w \leq 1$$

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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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师