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

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## 数学代写|凸优化作业代写Convex Optimization代考|A Design Problem

An optimal design of a process of chemical engineering is considered. Pressure swing adsorption (PSA) is a cyclic adsorption process for gas separation and purification. PSA systems have the potential of achieving a higher productivity for $\mathrm{CO}_2$ capture than alternative separation processes [180], such as absorption. An efficient and cost-competitive PSA unit is one that achieves high levels of purity and recovery of the product [57]. Therefore, a bi-objective optimization in search for an appropriate design is applicable.

To apply an optimization-based design method, a mathematical model of the system is needed. The development of an appropriate mathematical model is crucial for the success of the optimization aided design. However, here we do not go into the technological details and their mathematical description. Since we focus on the visualization of the potential decisions, only few aspects of the mathematical model in question, which are important for the considered visualization problem, will be mentioned. For the technological aspects we refer to $[18,57,180]$. PSA processes are governed by partial differential algebraic equations. The simulation of a PSA process is computationally challenging, and the task to perform PSA simulation can be very time consuming; a single simulation can take minutes, hours, or even days.
In the case study considered here, the number of design parameters (variables of the respective optimization problem) was 6, and they were re-scaled so that the feasible region was reduced to the unit hyper-cube. The number of objectives was 2 . The values of the objective functions are scaled to the intervals $\left[\begin{array}{ll}0 & 1\end{array}\right]$. A mathematical model based on simplified governing equations was used; see $[18,57,263]$ for details. The simulation time for a single design ranged between $10 \mathrm{~min}$ and an hour, depending on the design parameters [263].

## 数学代写|凸优化作业代写Convex Optimization代考|Visualization of the Optimization Results

Visualization in the considered case study was applied at the final stage of decision making. A multi-objective optimization algorithm (NSGA-II) has been applied by the experts in the process engineering, and complete computation results were recorded. The thorough analysis of the available data was requested to substantiate the choice of the design variables.

The application of the chosen multi-objective algorithm to the problem considered resulted in computing of $N=1584$ two-dimensional vectors of objectives $\mathbf{y}i$ at the points $\mathbf{x}_i=\left(x{i 1}, \ldots, x_{i 6}\right)^T, i=1, \ldots, N$, the components of which belong to the unit interval: $0 \leq x_{i j} \leq 1$ where the index $j,(1 \leq j \leq 6)$, denotes the $j$-th component of the $i$-th vector.

The subset of $\mathbf{y}_i, i-1, \ldots, N$, constituted of non dominated points represents the Pareto front of the considered problem $\mathbf{P}(\mathbf{f})_O$; it consists of $N_P=179$ twoby $\mathbf{X}_P$. A graphical drawing of the Pareto front is the standard presentation of results of a bi-objective optimization problem. By a visual analysis of the drawing a decision maker can choose an appropriate trade-off between the objectives. Since the Pareto front is represented by a finite number of points, the representation precision crucially depends on the number and distribution of points. The drawing of the Pareto front of the considered problem is presented in Figure 9.1a. The points are distributed rather densely and uniformly over the whole Pareto front but there is a discontinuity at the beginning of the upper part of the graph which indicates some neighborhood of minimum of $f_1(\cdot)$. The very precise computation of the boundaries of this discontinuity does not seem important since the most interesting part of the Pareto front is that including the kink where trade-off between the objectives seems favorable.

For a better identification of this part of the Pareto front a graph of $f_1\left(\mathbf{x}_i\right)$ and $f_2\left(\mathbf{x}_i\right), \mathbf{x}_i \in \mathbf{X}_P$ is presented in Figure 9.1b where the horizontal axis is for the indices reordered according to the increase of $f_1\left(\mathbf{x}_i\right), \mathbf{x}_i \in \mathbf{X}_P$.

A kink can be observed in the curve corresponding to $f_2\left(\mathbf{x}_i\right)$ for $i \approx 90$ where the horizontal behavior switches to a downward trend. It is therefore of interest to explore the solutions corresponding to indices in the interval $[80,139]$.

By a visual analysis of the graphs in Figure 9.1, an appropriate Pareto solution can be selected as well as the decision $\mathbf{x} \in \mathbf{X}_P$ which corresponds to the selected Pareto solution. However, such a choice is not always satisfactory since it does not pay respect to such properties of the corresponding decision as, e.g., the location of the selected decision vector in the feasible region $\mathbf{A}$. The analysis of the location of the set of efficient points in $\mathbf{A}$ can be especially valuable in cases of structural properties of the considered set important for the decision making. For example, some subsets of $\mathbf{A}$ might not be forbidden but may be unfavorable, and that property may not be easy to introduce into a mathematical model. The analysis of the properties of the set of efficient points can enable the discovery of latent variables, a relation between which essentially defines the Pareto front.

Numerous statistical methods are developed for the estimation of the impact of particular variables to the considered response. Since the optimization problem is bi-objective, the Pareto front and the set of the non-dominated decisions are single-dimensional manifolds; in other words, the latter is a curve in the sixdimensional unit cube. In principle, the parametric description of such a curve could be derived using a least-squares technique. However, the numerical solution of general nonlinear least-squares problems is difficult [256]. Therefore, an exploratory analysis of the available data seems reasonable here since it may highlight important specific properties of the problem in question.

## 数学代写|凸优化作业代写凸优化代考|优化结果的可视化

$\mathbf{y}_i, i-1, \ldots, N$的子集，由非支配点组成，代表所考虑问题$\mathbf{P}(\mathbf{f})_O$的帕累托面;它由$N_P=179$ twoby $\mathbf{X}_P$组成。帕累托前沿的图形图是双目标优化问题结果的标准表示。通过对绘图的可视化分析，决策者可以在目标之间选择适当的权衡。由于帕累托前沿由有限数量的点表示，表示的精度关键取决于点的数量和分布。图9.1a给出了所考虑问题的帕累托面图。这些点在整个帕累托前沿分布得相当密集和均匀，但在图的上部开始有一个不连续点，这表明了$f_1(\cdot)$的某个极小值的邻域。这种不连续边界的精确计算似乎并不重要，因为帕累托前沿最有趣的部分是包括了目标之间权衡似乎有利的扭结

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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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