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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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 Generalization to the Multi-Objective Case

Recently several papers have been published which propose multi-objective optimization algorithms that generalize single-objective optimization algorithms based on statistical models of objective functions [53, 101, 105, 106, 142, 224, 252]. The numerical results included there show the relevance of the proposed algorithms to the problems of multi-objective optimization with black-box expensive objectives. We present here a new idea for constructing relevant algorithms.

A multi-objective minimization problem can be stated almost identically to the single-objective problem considered in the previous subsection:
$$\min _{\mathbf{x} \in \mathbf{A}} \mathbf{f}(\mathbf{x}), \mathbf{f}(\mathbf{x})=\left(f_1(\mathbf{x}), f_2(\mathbf{x}), \ldots, f_m(\mathbf{x})\right)^T, \mathbf{A} \subset \mathbb{R}^d,$$ however, the concept of solution in this case is more complicated. For the definitions of the solution to a multi-objective optimization problem with nonlinear objectives, we refer to Chapter 1 .

In the case of multi-objective optimization, a vector objective function $\mathbf{f}(\mathbf{x})=$ $\left(f_1(\mathbf{x}), f_2(\mathbf{x}), \ldots, f_m(\mathbf{x})\right)^T$ is considered. The same arguments, as in the case of single-objective optimization, corroborate the applicability of statistical models. The assumptions on black-box information and expense of the objective functions together with the standard assumptions of rational decision making imply the acceptability of a family of random vectors $\Xi(\mathbf{x})=\left(\xi_1(\mathbf{x}), \ldots, \xi_m(\mathbf{x})\right)^T, x \in \mathbf{A}$, as a statistical model of $\mathbf{f}(\mathbf{x})$. Similarly, the location and spread parameters of $\xi_i(\mathbf{x})$, denoted by $m_i(\mathbf{x}), s_i(\mathbf{x}), i=1, \ldots, r$, are essential in the characterization of $\xi_i(\mathbf{x})$. For a more specific characterization of $\Xi(\mathbf{x})$, e.g., by a multidimensional distribution of $\Xi(\mathbf{x})$, the available information usually is insufficient. If the information on, e.g., the correlation between $\xi_i(\mathbf{x})$ and $\xi_j(\mathbf{x})$ were available, the covariance matrix could be included into the statistical model. However, here we assume that the objectives are independent, and the spread parameters are represented by a diagonal matrix $\Sigma(\mathbf{x})$ whose diagonal elements are equal to $s_1, \ldots, s_m$. Similarly to the case of single-objective optimization, we assume that the utility of choice of the point for the current computation of the vector value $\mathbf{f}(\mathbf{x})$ has the following structure
$$v_{n+1}(\mathbf{x})=V_{n+1}\left(\mathbf{m}(\mathbf{x}), \Sigma(\mathbf{x}), \mathbf{y}^n\right),$$
where $\mathbf{m}(\mathbf{x})=\left(m_1(\mathbf{x}), \ldots, m_m(\mathbf{x})\right)^T$, and $\mathbf{y}^n$ denotes a vector desired to improve.

## 数学代写|凸优化作业代写Convex Optimization代考|Optimization by Heuristic Methods

As shown in the previous sections, the guaranteed solutions of instances of the considered problem can be obtained by means of an algorithm of binary-linear programming. However, the binary-linear problems of interest involve large number of variables and restriction; see Table 10.1. It is clear, that in the case of larger problems, the solution time by that method can be too long for interactive systems. Therefore, the development of heuristic algorithms for this problem is important. In the discussion above, we have mentioned that the algorithm based on the idea of ant colony optimization [92, 93] was not quite satisfactory. One of the possible challenges for application of this algorithm is related to difficulties in assigning to an edge a proper value of “heuristic attractiveness” with respect to the criterion “number of bends.”

As potential alternatives we consider two following heuristic algorithms. The first algorithm is an extension of the classical shortest path algorithm for the specific multi-objective problem considered here. The second algorithm is a version of metaheuristic, called Harmony Search $\lfloor 67\rfloor$, which is claimed efficient in some recent publications. Harmony Search is a random search algorithm where moves are interpreted in musical terms. The similar in many respects random search algorithm, called Evolutionary Strategy, was proposed about 40 years ago (see, e.g., [187]) and was shown efficient in indeed many applications.

The problem considered is similar to the shortest path problem which is a classical problem of computer science. Here the vertices of the grid, presented in Figure 10.2, are vertices of the considered graph. The lengths of edges, the number of which is equal to $4 \times n \times p$, is assumed equal to 1 . Several paths should be found to connect the vertices which represent the shapes, and the appropriateness of a solution is assessed by the characteristics of the set of found paths. As the start and sink vertices of the paths in question can be only the intermediate vertices of the grid. Although the shortest path problem can be efficiently solved by, e.g., Dijkstra’s algorithm, this and other similar algorithms directly are not appropriate here. Our problem is more complicated, since we are interested not only in the total length of paths; the total number of bends, and the number of paths sharing the same edges, are also important. Nevertheless, the similarity between our problem and the shortest path problem induces an idea to construct a heuristic algorithm including the shortest path algorithm as a constituent.

# 凸优化代写

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## 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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