数学代写|优化算法作业代写optimisation algorithms代考| Ruggedness and Weak Causality

statistics-lab™ 为您的留学生涯保驾护航 在代写优化算法optimisation algorithms方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写优化算法optimisation algorithms代写方面经验极为丰富，各种代写优化算法optimisation algorithms相关的作业也就用不着说。

• 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|The Problem: Ruggedness

Optimization algorithms generally depend on some form of gradient in the objective or fitness space. The objective functions should be continuous and exhibit low total variation 4 , so the optimizer can descend the gradient easily. If the objective functions are unsteady or fluctuating, i.e., going up and down, it becomes more complicated for the optimization process to find the right directions to proceed to. The more rugged a function gets, the harder it becomes to optimize it. From a simplified point of view, ruggedness is multimodality plus steep ascends and descends in the fitness landscape. Examples of rugged landscapes are Kauffman’s NK fitness landscape [113, 115], the p-Spin model [6], Bergman and Feldman’s jagged fitness landscape [19], and the sketch in Fig. 1.d.

数学代写|优化算法作业代写optimisation algorithms代考|One Cause: Weak Causality

During an optimization process, new points in the search space are created by the search operations. Generally we can assume that the genotypes which are the input of the search operations correspond to phenotypes which have previously been selected. Usually, the better or the more promising an individual is, the higher are its chances of being selected for further investigation. Reversing this statement suggests that individuals which are passed to the

search operations are likely to have a good fitness. Since the fitness of a solution candidate depends on its properties, it can be assumed that the features of these individuals are not so bad either. It should thus be possible for the optimizer to introduce slight changes to their properties in order to find out whether they can be improved any further ${ }^{5}$. Normally, such modifications should also lead to small changes in the objective values and, hence, in the fitness of the solution candidate.

Definition 1 (Strong Causality). Strong causality (locality) means that small changes in the properties of an object also lead to small changes in its behavior $[170,171,180]$.

This principle (proposed by Rechenberg $[170,171]$ ) should not only hold for the search spaces and operations designed for optimization, but applies to natural genomes as well. The offspring resulting from sexual reproduction of two fish, for instance, has a different genotype than its parents. Yet, it is far more probable that these variations manifest in a unique color pattern of the scales, for example, instead of leading to a totally different creature.

Apart from this straightforward, informal explanation here, causality has been investigated thoroughly in different fields of optimization, such as Evolution Strategy $[170,65]$, structure evolution $[129,130]$, Genetic Programming $[65,107,179,180]$, genotype-phenotype mappings [193], search operators [65], and Evolutionary Algorithms in general [65, 182, 207].

In fitness landscapes with weak (low) causality, small changes in the solution candidates often lead to large changes in the objective values, i.e., ruggedness. It then becomes harder to decide which region of the problem space to explore and the optimizer cannot find reliable gradient information to follow. A small modification of a very bad solution candidate may then lead to a new local optimum and the best solution candidate currently known may be surrounded by points that are inferior to all other tested individuals.
The lower the causality of an optimization problem, the more rugged its fitness landscape is, which leads to a degradation of the performance of the optimizer [120]. This does not necessarily mean that it is impossible to find good solutions, but it may take very long to do so.

数学代写|优化算法作业代写optimisation algorithms代考|Countermeasures

To our knowledge, no viable method which can directly mitigate the effects of rugged fitness landscapes exists. In population-based approaches, using large population sizes and applying methods to increase the diversity can decrease the influence of ruggedness, but only up to a certain degree. Utilizing the Baldwin effect $[13,100,101,233]$ or Lamarckian evolution [54, 233], i.e., incorporating a local search into the optimization process, may further help to smoothen out the fitness landscape $[89]$.

Weak causality is often a home-made problem: it results from the choice of the solution representation and search operations. Thus, in order to apply Evolutionary Algorithms in an efficient manner, it is necessary to find representations which allow for iterative modifications with bounded influence on the objective values.

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。