### 数学代写|优化算法作业代写optimisation algorithms代考| Dynamically Changing Fitness Landscape

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 No Free Lunch Theorem

By now, we know the most important problems that can be encountered when applying an optimization algorithm to a given problem. Furthermore, we have seen that it is arguable what actually an optimum is if multiple criteria are optimized at once. The fact that there is most likely no optimization method that can outperform all others on all problems can, thus, easily be accepted. Instead, there exist a variety of optimization methods specialized

in solving different types of problems. There are also algorithms which deliver good results for many different problem classes, but may be outperformed by highly specialized methods in each of them.

These facts have been formalized by Wolpert and Macready [241, 242] in their No Free Lunch Theorems (NFL) for search and optimization algorithms. Wolpert and Macready [242] focus on single-objective optimization and prove that the sum of the values of any performance measure (such as the objective value of the best solution candidate discovered until a time step $m$ ) over all possible objective functions $f$ is always identical for all optimization algorithms.

From this theorem, we can immediately follow that, in order to outperform the optimization method $a_{1}$ in one optimization problem, the algorithm $a_{2}$ will necessarily perform worse in another. Fig. 16 visualizes this issue. The higher the value of the performance measure illustrated there, the faster will the corresponding problem be solved. The figure shows that general optimization approaches (like Evolutionary Algorithms) can solve a variety of problem classes with reasonable performance. Hill Climbing approaches, for instance, will be much faster than Evolutionary Algorithms if the objective functions are steady and monotonous, that is, in a smaller set of optimization tasks. Greedy search methods will perform fast on all problems with matroid structure. Evolutionary Algorithms will most often still be able to solve these problems, it just takes them longer to do so. The performance of Hill Climbing and greedy approaches degenerates in other classes of optimization tasks as a trade-off for their high utility in their “area of expertise”.

## 数学代写|优化算法作业代写optimisation algorithms代考|Concluding Remarks

The subject of this introductory chapter was the question about what makes optimization problems hard, especially for metaheuristic approaches. We have discussed numerous different phenomena which can affect the optimization process and lead to disappointing results. If an optimization process has converged prematurely, it has been trapped in a non-optimal region of the search space from which it cannot “escape” anymore (Section 2). Ruggedness (Section 3) and deceptiveness (Section 4) in the fitness landscape, often caused by epistatic effects (Section 6), can misguide the search into such a region. Neutrality and redundancy (Section 5) can either slow down optimization because the application of the search operations does not lead to a gain in information or may also contribute positively by creating neutral networks from which the search space can be explored and local optima can be escaped

from. The solutions that are derived, even in the presence of noise, should be robust (Section 7). Also, they should neither be too general (oversimplification, Section 8.2) nor too specifically aligned only to the training data (overfitting, Section 8.1). Furthermore, many practical problems are multiobjective, i.e., involve the optimization of more than one criterion at once (Section 9), or concern objectives which may change over time (Section 10). In the previous section, we discussed the No Free Lunch Theorem and argued that it is not possible to develop the one optimization algorithm, the problem-solving machine which can provide us with near-optimal solutions in short time for every possible optimization task. This must sound very depressing for everybody new to this subject.

Actually, quite the opposite is the case, at least from the point of view of a researcher. The No Free Lunch Theorem means that there will always be new ideas, new approaches which will lead to better optimization algorithms to solve a given problem. Instead of being doomed to obsolescence, it is far more likely that most of the currently known optimization methods have at least one niche, one area where they are excellent. It also means that it is very likely that the “puzzle of optimization algorithms” will never be completed. There will always be a chance that an inspiring moment, an observation in nature, for instance, may lead to the invention of a new optimization algorithm which performs better in some problem areas than all currently known ones.

## 数学代写|优化算法作业代写optimisation algorithms代考|The Rationale Behind Seeking Inspiration

Abstract. There are currently numerous heuristic algorithms for combinatorial optimisation problems which are commonly described as nature-inspired. Parallels can certainly be drawn between these algorithms and various natural processes, but the extent of the natural inspiration is not always clear. This chapter attempts to clarify what it means to say an algorithm is nature-inspired. Additionally, we will discuss the features of nature which make it a valuable resource in the design of successful new algorithms. Not only does nature provide processes which can be used for optimisation, but it is also a popular source of useful metaphors, which assist the designer. Finally, the history of some well-known algorithms will be discussed, with particular attention to the role nature has played in their development.

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## MATLAB代写

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