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

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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|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

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## MATLAB代写

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