### 数学代写|优化算法作业代写optimisation algorithms代考|Premature Convergence

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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
• 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|优化算法作业代写optimisation algorithms代考|Premature Convergenceuction

An optimization algorithm has converged if it cannot reach new solution candidates anymore or if it keeps on producing solution candidates from a “small” 2 subset of the problem space. Global optimization algorithms will usually converge at some point in time. One of the problems in global optimization is that it is often not possible to determine whether the best solution currently known is situated on a local or a global optimum and thus, if convergence is acceptable. In other words, it is usually not clear whether the optimization process can be stopped, whether it should concentrate on refining the current optimum, or whether it should examine other parts of the search space instead. This can, of course, only become cumbersome if there are multiple (local) optima, i.e., the problem is multimodal as depicted in Fig. 1.c.

A mathematical function is multimodal if it has multiple maxima or minima $[195,246]$. A set of objective functions (or a vector function) $\mathbf{f}$ is multimodal if it has multiple (local or global) optima – depending on the definition of “optimum” in the context of the corresponding optimization problem.

## 数学代写|优化算法作业代写optimisation algorithms代考|The Problem

An optimization process has prematurely converged to a local optimum if it is no longer able to explore other parts of the search space than the area currently being examined and there exists another region that contains a superior solution $[192,219]$. Fig. 3 illustrates examples of premature convergence.

The existence of multiple global optima itself is not problematic and the discovery of only a subset of them can still be considered as successful in many cases (see Section 9). The occurrence of numerous local optima, however, is more complicated.

The phenomenon of domino convergence has been brought to attention by Rudnick [184] who studied it in the context of his BinInt problem [184, 213]. In principle, domino convergence occurs when the solution candidates have features which contribute significantly to different degrees of the total fitness. If these features are encoded in separate genes (or building blocks) in the genotypes, they are likely to be treated with different priorities, at least in randomized or heuristic optimization methods.

Building blocks with a very strong positive influence on the objective values, for instance, will quickly be adopted by the optimization process (i.e., “converge”). During this time, the alleles of genes with a smaller contribution are ignored. They do not come into play until the optimal alleles of the more “important” blocks have been accumulated. Rudnick [184] called this sequential convergence phenomenon domino convergence due to its resemblance to a row of falling domino stones [213].

In the worst case, the contributions of the less salient genes may almost look like noise and they are not optimized at all. Such a situation is also an instance of premature convergence, since the global optimum which would involve optimal configurations of all blocks will not be discovered. In this situation, restarting the optimization process will not help because it will always turn out the same way. Example problems which are often likely to exhibit domino convergence are the Royal Road [139] and the aforementioned BinInt problem [184].

## 数学代写|优化算法作业代写optimisation algorithms代考|One Cause: Loss of Diversity

In biology, diversity is the variety and abundance of organisms at a given place and time $[159,133]$. Much of the beauty and efficiency of natural ecosystems is based on a dazzling array of species interacting in manifold ways. Diversification is also a good investment strategy utilized by investors in the economy in order to increase their profit.

In population-based global optimization algorithms as well, maintaining a set of diverse solution candidates is very important. Losing diversity means approaching a state where all the solution candidates under investigation are similar to each other. Another term for this state is convergence. Discussions about how diversity can be measured have been provided by Routledge $[183]$, Cousins $[49]$, Magurran $[133]$, Morrison and De Jong [148], and Paenke et al $[159]$.

Preserving diversity is directly linked with maintaining a good balance between exploitation and exploration $[159]$ and has been studied by researchers from many domains, such as

• Genetic Algorithms $[156,176,177]$,
• Evolutionary Algorithms $[28,29,123,149,200,206]$,
• Genetic Programming $[30,38,39,40,53,93,94]$,
• Tabu Search $[81,82]$, and
• Particle Swarm Optimization [238].
The operations which create new solutions from existing ones have a very large impact on the speed of convergence and the diversity of the populations $[69,203]$. The step size in Evolution Strategy is a good example of this issue: setting it properly is very important and leads to the “exploration versus exploitation” problem [102] which can be observed in other areas of global optimization as well. ${ }^{3}$

## 数学代写|优化算法作业代写optimisation algorithms代考|The Problem

Rudnick [184] 在他的 BinInt 问题 [184, 213] 的背景下研究了多米诺骨牌收敛现象。原则上，当候选解决方案具有对不同程度的总适应度有显着贡献的特征时，就会发生多米诺骨牌收敛。如果这些特征在基因型中的不同基因（或构建块）中编码，则它们可能会以不同的优先级进行处理，至少在随机或启发式优化方法中是这样。

## 数学代写|优化算法作业代写optimisation algorithms代考|One Cause: Loss of Diversity

• 遗传算法[156,176,177],
• 进化算法[28,29,123,149,200,206],
• 遗传编程[30,38,39,40,53,93,94],
• 禁忌搜索[81,82]， 和
• 粒子群优化[238]。
从现有解决方案创建新解决方案的操作对收敛速度和人口多样性有很大影响[69,203]. 进化策略中的步长是这个问题的一个很好的例子：正确设置它非常重要，并导致“探索与利用”问题[102]，这也可以在全局优化的其他领域观察到。

## 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。