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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|Experiments with the P-Algorithm

The MATLAB implementations of the considered algorithms were used for experimentation. The results of minimization obtained by the uniform random search are presented for the comparison. Minimization was stopped after 100 computations of objective function values.

The sites for the first 50 computations of $\mathbf{f}(\mathbf{x})$ were chosen by the $\mathrm{P}$-algorithm randomly with a uniform distribution over the feasible region. That data were used to estimate the parameters of the statistical model as well as in planning of the next 50 observations according to (7.8). The maximization of the improvement probability was performed by a simple version of multistart. The values of the improvement probability (7.9) were computed at 1000 points, generated randomly with uniform distribution over the feasible region. A local descent was performed from the best point using the codes from the MATLAB Optimization Toolbox.

Let us start from the comments about the results of minimization of the functions of one variable. In the experiments with objective functions of a single variable, the algorithms can be assessed with respect to the solutions found in the true Pareto front, while in the case of several variables normally the approximate solutions are solely available for the assessment of the algorithms.

The feasible objective region of problem (6.46) is presented for the visual analysis in Figure 6.5. The (one hundred) points in the feasible objective region, generated by the method of random uniform search (RUS), are shown in Figure 7.1; the non-dominated solutions found (thicker points) do not represent the Pareto front well. The P-algorithm was applied to that problem with two values of the threshold vector. In Figure 7.2, the trial points in the feasible objective region are shown, and non-dominated points are denoted by thicker points. The left-side figure shows the results obtained with the threshold vector equal to $(-1,-1)^T$ (35 non-dominated points found), and the right-hand side figure shows the results obtained with the threshold vector equal to $(-0.75,-0.75)^T$ (51 non-dominated points found). In the first case, the threshold is the ideal point; that case is similar to the case of a single-objective minimization, where the threshold is considerably below the current record. Presumably, for such a case the globality of the search strategy prevails and implies the uniformity (over the Pareto front) of the distribution of the non-dominated solutions found. For the threshold closer to the Pareto front, some localization of observations can be expected in the sense of increased density of the non-dominated points closer to the threshold vector. Figure $7.2$ illustrates the realization of the hypothesized properties.

## 数学代写|凸优化作业代写Convex Optimization代考|Experiments with the Algorithm

A version of the bi-objective $\pi$-algorithm has been implemented as described in Section 7.4.2. A product of two arctangents was used for $\pi(\cdot)$. Then the $n+1$ step of the $\pi$-algorithm is defined as the following optimization problem
$$\mathbf{x}{n+1}=\arg \max {\mathbf{x} \in \mathbf{A}} \arctan \left(\frac{y_1^n-m_1(\mathbf{x})}{s_1(\mathbf{x})}+\frac{\pi}{2}\right) \cdot \arctan \left(\frac{y_2^n-m_2(\mathbf{x})}{s_2(\mathbf{x})}+\frac{\pi}{2}\right),$$
where the information collected at previous steps is taken into account when computing $m_i(\mathbf{x})=m_i\left(\mathbf{x} \mid \mathbf{x}_j, \mathbf{y}_j, j=1, \ldots, n\right)$ and $s_i(\mathbf{x})=s_i\left(\mathbf{x} \mid \mathbf{x}_j, \mathbf{y}_j, j=1, \ldots, n\right)$. The maximization in (7.26) was performed by a simple version of multistart: from the best of 1000 points, generated randomly with uniform distribution over the feasible region, a local descent was performed using the codes from the MATLAB Optimization Toolbox. By this implementation, we wanted to check whether the function $\arctan (\cdot) \cdot \arctan (\cdot)$ chosen rather arbitrarily could be as good as the Gaussian cumulative distribution function for constructing statistical model-based multi-objective optimization algorithms. The experimentation with this version of the algorithm can be helpful also in selecting the most appropriate statistical model for a further development where two alternatives seem competitive: a Gaussian random field versus a statistical model, based on the assumptions of subjective probability [216].

Some experiments have been done for the comparison of the $\pi$-algorithm with the multi-objective P-algorithm described in Section 7.3. The optimization results by RUS from Section 7.5.3 are included to highlight the properties of the selected test problems. The results obtained by a multi-objective genetic algorithm (the MATLAB implementation in [80]) are also provided for the comparison. Two examples are presented and commented; we think that extensive competitive testing would be premature, as argued in Section 7.5.1.

Since the considered approach is oriented to expensive problems, we are interested in the quality of the result obtained computing a modest number of the values of objectives. Following the concept of experimentation above, a termination condition of all the considered algorithms was defined by the maximum number of computations of the objective function values, equal to 100 . The parameters of the statistical model, needed by the $\pi$-algorithm, have heen estimated using a sample of $\mathbf{f}(\mathbf{x})$ values, chosen similarly to the experiments with the P-algorithm: the sites for the first 50 computations of $\mathbf{f}(\mathbf{x})$ were chosen randomly with a uniform distribution over the feasible region; the obtained data were used not only for to (7.26).

An important parameter of the $\pi$-algorithm is $\mathbf{y}^n$. The vector $\mathbf{y}^n$ should be not dominated by the known values $\mathbf{y}_1, \ldots, \mathbf{y}_n$. A heuristic recommendation is to select $\mathbf{y}^n$ at a possibly symmetric site with respect to the global minima of objectives. We have selected the values of $\mathbf{y}^n$ used in the P-algorithm above: $\mathbf{y}^n=(-0.6,-0.6)$ in the case of problem (1.6), and $\mathbf{y}^n=(0.6,0.6)$ in the case of problem (1.5). Typical results are illustrated in Figure 7.8.

# 凸优化代写

## 数学代写|凸优化作业代写凸优化代考|实验与P-Algorithm

.

$\mathbf{f}(\mathbf{x})$的前50次计算的站点由$\mathrm{P}$ -算法随机选择，并在可行区域内均匀分布。这些数据被用来估计统计模型的参数，以及根据(7.8)规划接下来的50个观察结果。改进概率的最大化是由一个简单版本的multistart执行的。改进概率(7.9)的值在1000点处计算，随机生成，在可行区域内分布均匀。使用MATLAB优化工具箱中的代码从最佳点执行局部下降

## 数学代写|凸优化作业代写凸优化代考|实验与算法

.

$$\mathbf{x}{n+1}=\arg \max {\mathbf{x} \in \mathbf{A}} \arctan \left(\frac{y_1^n-m_1(\mathbf{x})}{s_1(\mathbf{x})}+\frac{\pi}{2}\right) \cdot \arctan \left(\frac{y_2^n-m_2(\mathbf{x})}{s_2(\mathbf{x})}+\frac{\pi}{2}\right),$$
，其中在计算$m_i(\mathbf{x})=m_i\left(\mathbf{x} \mid \mathbf{x}_j, \mathbf{y}_j, j=1, \ldots, n\right)$和$s_i(\mathbf{x})=s_i\left(\mathbf{x} \mid \mathbf{x}_j, \mathbf{y}_j, j=1, \ldots, n\right)$时将考虑前面步骤收集的信息。(7.26)中的最大化是通过multistart的一个简单版本实现的:从1000个点中的最佳点(在可行区域内随机生成且分布均匀)开始，使用MATLAB优化工具箱中的代码进行局部下降。通过这个实现，我们想检验任意选择的函数$\arctan (\cdot) \cdot \arctan (\cdot)$是否可以与高斯累积分布函数一样好地构建基于统计模型的多目标优化算法。该算法版本的实验也有助于为进一步发展选择最合适的统计模型，当两个备选方案看起来具有竞争性时:基于主观概率假设的高斯随机场与统计模型[216]

$\pi$ -algorithm的一个重要参数是$\mathbf{y}^n$。向量$\mathbf{y}^n$不应该被已知值$\mathbf{y}_1, \ldots, \mathbf{y}_n$所支配。一个启发式的建议是在一个可能对称的站点上选择$\mathbf{y}^n$，相对于目标的全局最小值。我们选择了上面p算法中使用的$\mathbf{y}^n$的值:在问题(1.6)的情况下是$\mathbf{y}^n=(-0.6,-0.6)$，在问题(1.5)的情况下是$\mathbf{y}^n=(0.6,0.6)$。典型结果如图7.8所示

## 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。

assignmentutor™您的专属作业导师
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