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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|Methodological Problems

Theoretical and algorithmic achievements of multi-objective optimization have implied also the expansion of respective applications. Among applied multiobjective optimization problems, expensive multimodal black-box problems are rather frequent. However, they constitute still a relatively little researched subfield of multi-objective optimization and deserve more attention from researchers. Since the statistical models based single-objective optimization algorithms well correspond to the challenges of single-objective global optimization of expensive black-box functions, they were generalized to the multi-objective case. As shown in the previous sections, the theoretical generalization is rather straightforward. Some experimental investigation was performed to find out how much the generalization corresponds to the expectations of their suitability for multi-objective optimization.
General methodological concepts of testing and comparison of mathematical programming algorithms and software are well developed; see [131]. The methodology, called Competitive Testing in [82], should normally be applied for the comparison of the well-established algorithms. This methodology is also extended for testing and comparison of multi-objective optimization algorithms; see. e.g.. [42, $60,135,266,267]$. In the case of the well-researched classes of problems (e.g., convex multi-objective optimization), this methodology is universally applicable, only the selection of test functions should be specially selected taking into account the properties of the considered sub-class of problems, e.g., considered in $[66,129]$. The tests, based on special cases of real world applied problems, can be very useful for evaluating the efficiency of the respective algorithms; see, e.g., [154] where multi-objective portfolio problems are used for testing the algorithms aimed to distribute solutions uniformly in the Pareto optimal set.

However, the standard testing methodology is not well suitable for the algorithms considered in this chapter. The first difficulty is caused by the main feature of the targeted problems: they are supposed to be expensive. Therefore, a solution, found by an optimization algorithm applied, normally is rather rough. An optimization algorithm is as much useful as much its application aids a decision maker in making a final decision in the conditions of uncertainty reduced because of the application of the algorithm. The quantitative assessment of such a criterion of an algorithm is difficult.

数学代写|凸优化作业代写Convex Optimization代考|Test Functions

Bi-objective problems with one and two variables were chosen for the experiments to enable visual analysis of the results.

Some experiments were performed using objective functions of a single variable. Experimentation with one-dimensional problems was extensive during the development of the single-objective methods based on statistical models; see, e.g., $[139,208,216]$. These test functions have been used also to demonstrate the performance of the Lipschitz model based algorithms in Section 6.2.5: Rastr (6.46), Fo\&Fle (6.47), and Schaf (6.48). The feasible objective regions with the highlighted Pareto front of the considered test functions are shown in Figures 6.5, 6.6, and 6.7.

Two bi-objective test problems of two variables are chosen for the experimentation. The test problems of two variables are chosen similarly to the choice of one-dimensional problems: the first multi-objective test problem is composed using a typical test problem for a single-objective global optimization, and the second one is chosen from the set of functions frequently used for testing multi-objective algorithms. The first test function Shek (1.6) is composed of two Shekel functions which are frequently used for testing global optimization algorithms, see, e.g., [216]. A rather simple multimodal case is intended to be considered, so the number of minimizers of both objectives is selected equal to two. The objective functions are represented by contour lines in Figure 1.3. The second problem Fo\&Fle, (1.5), is especially difficult from the point of view of global minimization, since the functions $f_1(\mathbf{x})$ and $f_2(\mathbf{x})$ in (1.5) are similar to the most difficult objective function whose response surface is comprised of a flat plateau over a large part of the feasible decision region, and of the unknown number of sharp spikes. The estimates of parameters of the statistical model of (1.5) are biased towards the values that represent the “flat” part of response surface. The discrepancy between the statistical model and the modeled functions can negatively influence the efficiency of the statistical models based algorithms.

The selection of test problems can be summarized as follows: two problems, (6.46) and (1.6), are constructed generalizing typical test problems of global optimization, and two other problems, (6.48) and (1.5), are selected from a set of non-convex multi-objective test problems. The former problems are well represented by the considered above statistical model, and the latter ones are not. Both objective functions of problem (1.5) are especially difficult for global optimization, and their properties do not correspond to the properties predictable using the statistical model.

凸优化代写

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有限元方法代写

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MATLAB代写

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