### 数学代写|优化算法作业代写optimisation algorithms代考|Noise and Robustness

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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 – Noise

In the context of optimization, three types of noise can be distinguished. The first form is noise in the training data used as basis for learning (i). In many applications of machine learning or optimization where a model $m$ for a given system is to be learned, data samples including the input of the system and its measured response are used for training. Some typical examples of situations where training data is the basis for the objective function evaluation are

• the usage of global optimization for building classifiers (for example for predicting buying behavior using data gathered in a customer survey for training).
• the usage of simulations for determining the objective values in Genetic Programming (here, the simulated scenarios correspond to training cases), and
• the fitting of mathematical functions to $(x, y)$-data samples (with artificial neural networks or symbolic regression, for instance).

Since no measurement device is $100 \%$ accurate and there are always random errors, noise is present in such optimization problems.

Besides inexactnesses and fluctuations in the input data of the optimization process, perturbations are also likely to occur during the application of its results. This category subsumes the other two types of noise: perturbations that may arise from inaccuracies in (ii) the process of realizing the solutions

and (iii) environmentally induced perturbations during the applications of the products.

This issue can be illustrated using the process of developing the perfect tire for a car as an example. As input for the optimizer, all sorts of material coefficients and geometric constants measured from all known types of wheels and rubber could be available. Since these constants have been measured or calculated from measurements, they include a certain degree of noise and imprecision (i).

The result of the optimization process will be the best tire construction plan discovered during its course and it will likely incorporate different materials and structures. We would hope that the tires created according to the plan will not fall apart if, accidently, an extra $0.0001 \%$ of a specific rubber component is used (ii). During the optimization process, the behavior of many construction plans will be simulated in order to find out about their utility. When actually manufactured, the tires should not behave unexpectedly when used in scenarios different from those simulated (iii) and should instead be applicable in all driving scenarios likely to occur.

The effects of noise in optimization have been studied by various researchers; Miller and Goldberg $[136,137]$, Lee and Wong [125], and Gurin and Rastrigin [92] are some of them. Many global optimization algorithms and theoretical results have been proposed which can deal with noise. Some of them are, for instance, specialized

• Genetic Algorithms $[75,119,188,189,217,218]$,
• Evolution Strategies [11, 21, 96], and
• Particle Swarm Optimization [97, 161] approaches.

## 数学代写|优化算法作业代写optimisation algorithms代考|Need for Robustness

The goal of global optimization is to find the global optima of the objective functions. While this is fully true from a theoretical point of view, it may not suffice in practice. Optimization problems are normally used to find good parameters or designs for components or plans to be put into action by human beings or machines. As we have already pointed out, there will always be noise and perturbations in practical realizations of the results of optimization.
Definition 4 (Robustness). A system in engineering or biology is robust if it is able to function properly in the face of genetic or environmental perturbations [225].

Therefore, a local optimum (or even a non-optimal element) for which slight deviations only lead to gentle performance degenerations is usually favored over a global optimum located in a highly rugged area of the fitness landscape [31]. In other words, local optima in regions of the fitness landscape with

strong causality are sometimes better than global optima with weak causality. Of course, the level of this acceptability is application-dependent. Fig. 8 illustrates the issue of local optima which are robust vs. global optima which are not. More examples from the real world are:

• When optimizing the control parameters of an airplane or a nuclear power plant, the global optimum is certainly not used if a slight perturbation can have hazardous effects on the system [218].
• Wiesmann et al [234, 235] bring up the topic of manufacturing tolerances in multilayer optical coatings. It is no use to find optimal configurations if they only perform optimal when manufactured to a precision which is either impossible or too hard to achieve on a constant basis.
• The optimization of the decision process on which roads should be precautionary salted for areas with marginal winter climate is an example of the need for dynamic robustness. The global optimum of this problem is likely to depend on the daily (or even current) weather forecast and may therefore be constantly changing. Handa et al [98] point out that it is practically infeasible to let road workers follow a constantly changing plan and circumvent this problem by incorporating multiple road temperature settings in the objective function evaluation.
• Tsutsui et al $[218,217]$ found a nice analogy in nature: The phenotypic characteristics of an individual are described by its genetic code. During the interpretation of this code, perturbations like abnormal temperature, nutritional imbalances, injuries, illnesses and so on may occur. If the phenotypic features emerging under these influences have low fitness, the organism cannot survive and procreate. Thus, even a species with good genetic material will die out if its phenotypic features become too sensitive to perturbations. Species robust against them, on the other hand, will survive and evolve.

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

For the special case where the problem space corresponds to the real vectors $\left(\mathbb{X} \subseteq \mathbb{R}^{n}\right.$ ), several approaches for dealing with the problem of robustness have been developed. Inspired by Taguchi methods ${ }^{6}$ [209], possible disturbances are represented by a vector $\delta=\left(\delta_{1}, \delta_{2}, \ldots, \delta_{n}\right)^{T}, \delta_{i} \in \mathbb{R}$ in the method of Greiner $[87,88]$. If the distribution and influence of the $\delta_{i}$ are known, the objective function $f(\mathbf{x}): \mathbf{x} \in \mathbf{X}$ can be rewritten as $\tilde{f}(\mathbf{x}, \delta)[235] .$ In the special case where $\delta$ is normally distributed, this can be simplified to $\tilde{f}\left(\left(x_{1}+\delta_{1}, x_{2}+\delta_{2}, \ldots, x_{n}+\delta_{n}\right)^{T}\right)$. It would then make sense to sample the probability distribution of $\boldsymbol{\delta}$ a number of $t$ times and to use the mean values of $\tilde{f}(\mathbf{x}, \boldsymbol{\delta})$ for each objective function evaluation during the optimization process. In cases where the optimal value $y$ of the objective function $f$ is known, Equation 3 can be minimized. This approach is also used in the work of Wiesmann et al $[234,235]$ and basically turns the optimization algorithm into something like a maximum likelihood estimator.
$$f^{\prime}(\mathbf{x})=\frac{1}{t} \sum_{i=1}^{t}\left(y-\bar{f}\left(\mathbf{x}, \delta_{i}\right)\right)^{2}$$
This method corresponds to using multiple, different training scenarios during the objective function evaluation in situations where $\mathrm{X} \nsubseteq \mathbb{R}^{n}$. By adding random noise and artificial perturbations to the training cases, the chance of obtaining robust solutions which are stable when applied or realized under noisy conditions can be increased.

## 数学代写|优化算法作业代写optimisation algorithms代考|Introduction – Noise

• 使用全局优化来构建分类器（例如，使用在客户调查中收集的数据来预测购买行为以进行培训）。
• 使用模拟来确定遗传编程中的目标值（这里，模拟场景对应于训练案例），以及
• 数学函数的拟合(X,是)- 数据样本（例如，使用人工神经网络或符号回归）。

(iii) 产品应用过程中环境引起的扰动。

• 遗传算法[75,119,188,189,217,218],
• 进化策略 [11, 21, 96] 和
• 粒子群优化 [97, 161] 方法。

## 数学代写|优化算法作业代写optimisation algorithms代考|Need for Robustness

• 在优化飞机或核电站的控制参数时，如果轻微的扰动会对系统产生有害影响，则肯定不会使用全局最优值 [218]。
• Wiesmann 等人 [234, 235] 提出了多层光学涂层制造公差的主题。如果它们仅在制造到不可能或太难在恒定基础上实现的精度时才表现最佳，那么找到最佳配置是没有用的。
• 优化冬季气候边缘地区道路应预防性盐渍的决策过程是需要动态稳健性的一个例子。这个问题的全局最优可能取决于每日（甚至当前）天气预报，因此可能会不断变化。Handa 等人 [98] 指出，让道路工作人员遵循不断变化的计划并通过在目标函数评估中结合多个道路温度设置来规避这个问题实际上是不可行的。
• 筒井等[218,217]在自然界中发现了一个很好的类比：个体的表型特征由其遗传密码描述。在解读这段代码的过程中，可能会出现体温异常、营养失衡、受伤、疾病等扰动。如果在这些影响下出现的表型特征具有低适应度，则有机体无法生存和繁殖。因此，如果其表型特征对扰动过于敏感，即使是具有良好遗传物质的物种也会灭绝。另一方面，抵抗它们的物种将生存和进化。

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

F′(X)=1吨∑一世=1吨(是−F¯(X,d一世))2

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## MATLAB代写

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