### 数学代写|优化算法作业代写optimisation algorithms代考|Why Is Optimization Difficult

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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|Introduction

Optimization, in general, is concerned with finding the best solutions for a given problem. Its applicability in many different disciplines makes it hard to give an exact definition. Mathematicians, for instance, are interested in finding the maxima or minima of a real function from within an allowable set of variables. In computing and engineering, the goal is to maximize the performance of a system or application with minimal runtime and resources.

In the business industry, people aim to optimize the efficiency of a production process or the quality and desirability of their current products.

All these examples show that optimization is indeed part of our everyday life. We often try to maximize our gain by minimizing the cost we need to bear. However, are we really able to achieve an “optimal” condition? Frankly, whatever problems we are dealing with, it is rare that the optimization process will produce a solution that is truly optimal. It may be optimal for one audience or for a particular application, but definitely not in all cases.

As such, various techniques have emerged for tackling different kinds of optimization problems. In the broadest sense, these techniques can be classified into exact and stochastic algorithms. Exact algorithms, such as branch and bound, $A^{*}$ search, or dynamic programming can be highly effective for small-size problems. When the problems are large and complex, especially if they are either NP-complete or NP-hard, i.e., have no known polynomialtime solutions, the use of stochastic algorithms becomes mandatory. These stochastic algorithms do not guarantee an optimal solution, but they are able to find quasi-optimal solutions within a reasonable amount of time.

In recent years, metaheuristics, a family of stochastic techniques, has become an active research area. They can be defined as higher level frameworks aimed at efficiently and effectively exploring a search space [25]. The initial work in this area was started about half a century ago (see $[175,78,24]$, and [37]). Subsequently, a lot of diverse methods have been proposed, and today, this family comprises many well-known techniques such as Evolutionary Algorithms, Tabu Search, Simulated Annealing, Ant Colony Optimization, Particle Swarm Optimization, etc.

There are different ways of classifying and describing metaheuristic algorithms. The widely accepted classification would be the view of nature-inspired vs. non nature-inspired, i.e., whether or not the algorithm somehow emulates a process found in nature. Evolutionary Algorithms, the most widely used metaheuristics, belong to the nature-inspired class. Other techniques with increasing popularity in this class include Ant Colony Optimization, Particle Swarm Optimization. Artificial Immune Systems, and so on. Scatter search, Tabu Search, and Iterated Local Search are examples of non nature-inspired metaheuristics. Unified models of metaheuristic optimization procedures have been proposed by Vaessens et al $[220,221]$, Rayward-Smith $[169]$, Osman $[158]$, and Taillard et al [210].

## 数学代写|优化算法作业代写optimisation algorithms代考|Basic Terminology

In the following text, we will utilize a terminology commonly used in the Evolutionary Algorithms community and sketched in Fig. 2 based on the example of a simple Genetic Algorithm. The possible solutions $x$ of an optimization problem are elements of the problem space $X$. Their utility as solutions is evaluated by a set $\mathbf{f}$ of objective functions $f$ which, without loss of generality, are assumed to be subject to minimization. The set of search operations utilized by the optimizers to explore this space does not directly work on them. Instead, they are applied to the elements (the genotypes) of the search space $\mathbb{G}$ (the genome). They are mapped to the solution candidates by a genotype-phenotype mapping gpm : $\mathbb{X}$. The term individual is used for both, solution candidates and genotypes.

## 数学代写|优化算法作业代写optimisation algorithms代考|The Term “Difficult”

Before we go more into detail about what makes these landscapes difficult, we should establish the term in the context of optimization. The degree of difficulty of solving a certain problem with a dedicated algorithm is closely related to its computational complexity, i.e., the amount of resources such as time and memory required to do so. The computational complexity depends on the number of input elements needed for applying the algorithm. This dependency is often expressed in the form of approximate boundaries with the Big-D-family notations introduced by Bachmann [10] and made popular by Landau [122]. Problems can be further divided into complexity classes. One of the most difficult complexity classes owning to its resource requirements is NP, the set of all decision problems which are solvable in polynomial time by non-deterministic Turing machines $[79]$. Although many attempts have been

made, no algorithm has been found which is able to solve an NP-complete [79] problem in polynomial time on a deterministic computer. One approach to obtaining near-optimal solutions for problems in NP in reasonable time is to apply metaheuristic, randomized optimization procedures.

As already stated, optimization algorithms are guided by objective functions. A function is difficult from a mathematical perspective in this context if it is not continuous, not differentiable, or if it has multiple maxima and minima. This understanding of difficulty comes very close to the intuitive sketches in Fig. $1 .$

In many real world applications of metaheuristic optimization, the characteristics of the objective functions are not known in advance. The problems are usually NP or have unknown complexity. It is therefore only rarely possible to derive boundaries for the performance or the runtime of optimizers in advance, let alone exact estimates with mathematical precision.

Most often, experience, rules of thumb, and empirical results based on the models obtained from related research areas such as biology are the only guides available. In this chapter we discuss many such models and rules, providing a better understanding of when the application of a metaheuristic is feasible and when not, as well as with indicators on how to avoid defining problems in a way that makes them difficult.

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## MATLAB代写

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