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

2022年10月10日

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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

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