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• 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|Optimization Problems of Interest

The results in [93] related to multi-objective optimization of paths in the graphs, used as models of BPD, imply the hypothesis that for the frequent in practice instances there exist solutions that are optimal with respect to both considered criteria: the length of path and the number of bends. To test such a hypothesis, the investigation of the following optimization problems is essential
\begin{aligned} &\min \sigma(\mathbf{X}), \ &\min \beta(\mathbf{X}), \ &\min (\sigma(\mathbf{X}), \beta(\mathbf{X}))^T, \end{aligned}
where the feasible region is defined by (10.1)-(10.7) and (10.10) with $\tau=2$ for simple instances, and $\tau=3$ for more complicated instances.

The first and second optimization problems are single-objective, and their minimizers are coincident if the Pareto front of two-objective problem (10.13)

consists of a single solution. Otherwise, the weighted sum scalarization method could be applicable to solve $(10.13)$ by means of minimizing $\phi(\mathbf{X})$,
$$\min \phi(\mathbf{X})=w \cdot \sigma(\mathbf{X})+(1-w) \cdot \beta(\mathbf{X}), 0<w<1$$
where the value of $w$ is important only in the case of the incorrect hypothesis on the single-point Pareto front. In that case, $w=\frac{w_2}{w_1+w_2}$ is a reasonable choice, where $w_1$ is the sum of Manhattan distances between the vertices to be connected, and $w_2$ is the minimum number of bends of the connectors which ignore all the restrictions; $w_1$ and $w_2$ are easily computable. This value of $w$ corresponds to the approximately equal importance of both objectives, i.e., length and bends of paths.

To test the hypothesis stated above, it is enough to compare the minima of (10.11) and (10.12) with the values of $\sigma(\mathbf{X})$ and $\beta(\mathbf{X})$ at the minimum point of (10.14). As an alternative to the weighted sum scalarization a method of lexicographic optimization, e.g., [154] could be applied, however, in such a case single-objective optimization problem should be solved two times.

## 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.

## 数学代写|凸优化作业代写凸面优化代考|感兴趣的优化问题

.

[93]中有关图中路径的多目标优化的结果(用作BPD模型)暗示了这样一个假设:对于经常出现的实践实例，存在针对两个考虑标准(路径长度和弯道数量)的最优解。为了检验这样的假设，以下优化问题的研究是必要的
\begin{aligned} &\min \sigma(\mathbf{X}), \ &\min \beta(\mathbf{X}), \ &\min (\sigma(\mathbf{X}), \beta(\mathbf{X}))^T, \end{aligned}
，其中可行区域由(10.1)-(10.7)和(10.10)定义，其中$\tau=2$为简单实例，$\tau=3$为更复杂的实例

$$\min \phi(\mathbf{X})=w \cdot \sigma(\mathbf{X})+(1-w) \cdot \beta(\mathbf{X}), 0<w<1$$

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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。

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