数学代写|优化算法代写optimization algorithms代考| Complexity of Real Computation Processes

statistics-lab™ 为您的留学生涯保驾护航 在代写优化算法optimization 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optimization algorithms代写方面经验极为丰富，各种代写优化算法optimization 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

数学代写|优化算法代写optimization algorithms代考|On the Computer Constructing Technology

Scheme of constructing (choice) of $T$-effective computational algorithm depends on many factors (class problems, input data, dimension and characteristics of the problems, computational resources that are available to the user, constrains (2.1), (2.2), and (2.3)); therefore, in the class problem $F$, it is advisable to distinguish multitude (subclasses) of problems that have common features in the context of computing [14]:

• One-off problems with a small amount of computing and moderate constraints on process time
• Problems (or series of problems) that are needed to be solved in real time
• Problems with a very large amount of computations that are needed to be solved in a practically reasonable amount of time (that cannot be achieved on traditional computing machines)

The performance of the conditions (2.1), (2.2), and (2.3) depending upon the statement of the problem can be achieved by choosing one of the following combinations of computing resources: $X,\left(X, I_{n}\right),(X, Y),\left(X, Y, I_{n}\right)$. In the first two situations, the possibilities of the computer are fixed. In the first situation, the information $I_{n}$ is also fixed; conditions (2.1), (2.2), and (2.3) are satisfied by the choice of the algorithm and its parameters; in the second one, it is still possible to select the set $I_{n}$ for this type of information operator. In the third situation, the information is fixed, and the parameters of the computer can be chosen besides the algorithm. In the fourth situation, all computing resources are used.

The first group of problems can be solved by the choice $X$ or $\left(X, I_{n}\right)$ of a regular sequential computer. Herewith, it is possible to devote three levels of detalization of the computing model. At the first level, there are algorithms that are focused on class $F$ solving problems using the information $I_{n}$. Herewith, there is support: approximation, stability, convergence of the approximate solution, the possibility to achieve a certain accuracy for the given input information, and the volume of computations as a function of the problem size (volume of input data). At this level, there is a possibility to discover the impossibility of computation of the $\varepsilon$-solution using specific input data, and there might be a possibility to clarify the class of problems and the requirements for the input information to provide a certain accuracy of the approximate solution, and it is possible (in this regard) to choose a new algorithm.
The second level (detalization) is related to the use of elements of the multitude $Y$ (machine word length, rounding rule) to compute the error estimate of rounding. Herewith, a multitude $A(\varepsilon)$ can be defined as conclusions in the case of the advisability of certain algorithms using from the multitude $A(\varepsilon)$ to save process time.
At the third level, where computational algorithm is a program for computing the $\varepsilon$-solution on a certain computer, time $T(\varepsilon)$ and memory $M(\varepsilon)[114]$ are estimated.
The variants $(X, Y)$ and $\left(X, Y, I_{n}\right)$ are specific to the second group of problems, for example, for digital signals processing and digital images processing using specialized computers. To achieve high rapid rates, the computer architecture is coherent with the computational algorithm $[131,277]$.

It is possible to use the third or fourth variants of the organization of computing to solve the problems of the third group. Herewith, the one purpose high-end computers [220] and computers of all purposes can be used [199].

数学代写|优化算法代写optimization algorithms代考|Specificity of Using Characteristic Estimates

In constructing real computational processes of computations, $\varepsilon$-solution is often used by some estimates of global error, its component and process time. Herewith, they distinguish estimates in the following way: a priori and a posteriori, majorizing and asymptotic, and determinate and stochastic. The possibility and advisability of these estimates using and the methods of their construction depend on the type, structure, and accuracy of a priori data, the problem, and the CA from that why the estimate is computed, and it also depends on the computational resources [114,238].

Majorizing a priori estimate guarantees the upper bound of the estimated deriv atives, and they are performed through known derivatives. Their computation does not require some significant computational expenses, but the value of estimates are often overrated; therefore, the conclusions based on them as for the possibility of computing of the solution under the conditions $(2.1)$ and $(2.2)$ may be false.

Asymptotic estimates approximate the estimated derivative. The variability of the parameter can be achieved by the desirable estimate proximity to the estimated derivative, but the computation of such estimates is related to significant computational expenses, and these estimates are usually a posteriori.

In the algorithmic support of solving problems under the conditions (2.1) and (2.2), given the properties of the estimates, it must be expected the possibility of computing of the various types of estimates of characteristics $E\left(E_{\mathrm{u}}, E_{\mu}, E_{\tau}\right)$ [238]. By the relaxed constraints (2.1) and (2.2), less precise and less complex (computational) estimates may be sufficient. By the tighten constrains (2.1) and (2.2), asymptotic (a posteriori) estimates are used. For example, the condition (2.2) may apply strict requirements to the accuracy of estimates of computational process parameters that are computed on the basis of errors estimate of the solution.

数学代写|优化算法代写optimization algorithms代考|Classes of Computational Problems, Informational

In the given technology of constructing problems solution per time that does not exceed the given $T$, available information plays a great significance. The more a priori information of different principles is known on the problem and algorithm uses it, the more accuracy effective and time it can be solved.

Note that the effectiveness of the algorithms is determined by the estimate of their characteristics so that the estimates should be of high quality (constants that are included in majorizing estimates of errors, accurate, estimates, unimprovable, etc.). And yet even high-quality estimates are constructed on a class of problems. And the wider this class is, the less suitable this estimate may be for a particular problem. Therefore, it is important to have a classification of problems that considers the additional a priori information. This will make a possibility to “select” such a class for a solved problem that is most likely to be used to obtain the required solution of a certain quality.

Consequently, the improvement of the quality of solving problems depends on the “narrowing” of the class of problems to which the solved problem belongs and the building of algorithms of such solving problems and the most accurate estimates of their characteristics.

However, it is not always possible to obtain $\varepsilon$-solution of some problems (although the total input information may be enough for this) using the given technology, or it cannot be checked that the solution was achieved. In these cases, it is important to have algorithms that are accuracy optimal (all available information on the problem is used as much as possible to improve accuracy) and a posteriori error estimates (that are more accurate next to a priori ones).

On the back of the accuracy optimal algorithm of this solving problem and a posteriori estimate of the error, it is often possible to obtain a solution that satisfies the user or draw a conclusion that it was not possible to obtain such a solution. We consider key principles of the problems classification and algorithms through the examples of some specific classes of problems of computational and applied mathematics.

数学代写|优化算法代写optimization algorithms代考|On the Computer Constructing Technology

• 计算量小、处理时间适度限制的一次性问题
• 需要实时解决的问题（或一系列问题）
• 需要在实际合理的时间内解决的大量计算问题（传统计算机无法实现）

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。

数学代写|优化算法代写optimization algorithms代考|Analytic Computational Complexity

statistics-lab™ 为您的留学生涯保驾护航 在代写优化算法optimization 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optimization algorithms代写方面经验极为丰富，各种代写优化算法optimization 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

数学代写|优化算法代写optimization algorithms代考|Input Information, Algorithms, and Complexity

Consider the idealized computation model: Information $I_{n}(f)$ is given accurately, and the model $c$ is fixed. Here are some characteristics that are related to the lower and upper estimates of the error on the example of the passive pure minimax strategy $[102,253]$ (see Chap. 1 for more details):

• $\rho_{\mu}(F, a)=\sup {f \in F} \rho\left(E{\mu}\left(I_{n}(f)\right), a\right)$ is an error of algorithm $a \in A$ on the class of the problems $F$ using information $I_{n}(f)$ (global error [270]).
• $\rho_{\mu}(F, A)=\inf {a \in A} \rho{\mu}(F, a)$ is a lower boundary of error of algorithms of class $A$ in the class of problems $F$ using information $I_{n}(f)$ (radius of information [270])
If there is an algorithm $a_{0} \in A$ for which $\rho_{\mu}\left(F, a_{0}\right)=\rho_{\mu}(F, A)$, then it is called accuracy optimal in class $F$ using information $I_{n}(f)$.

The narrowing of the $F$ class that is provided by the incompleteness of information (in relation to $f \in F) F_{n}(f)=\left{\varphi: I_{n}(\varphi)=I_{n}(f), \varphi, f \in F\right}$ allows to introduce the characteristics that are equivalent to the mentioned one above: $\rho_{\mu}\left(F_{n}(f), a\right)$ and $\rho_{\mu}\left(F_{n}(f), A\right)$, which are also called the local error and the local radius of information [270], respectively.

Let $U(f)$ be the multitude of solving problems from $F_{n}(f)$, and $\gamma(f)$ is the center of this multitude (the Chebyshev center). The algorithm $a^{\gamma} \in A$ is called a central one if $a^{y}\left(I_{n}(f)\right)=\gamma(f)$. These algorithms are accuracy optimal. Their important quality is that they minimize the local error of the algorithm:
$$\inf {a \in A} \sup {\varphi \in F_{n}(f)} \rho_{\mu}\left(I_{n}(f), a\right)=\rho_{\mu}\left(I_{n}(f), a^{\gamma}\right)=\operatorname{rad} U(f) .$$
Note that $A(\varepsilon) \neq \varnothing$ only when $\rho_{\mu}\left(F_{n}(f), A\right)<\varepsilon$.
Consider that class $A$ contains stable congruent algorithms and
$$\rho_{\mu}\left(F_{n}(f), a\right) \rightarrow 0 \text { при } n \rightarrow \infty$$
The algorithm $a$ is called asymptotically accuracy optimal (accuracy order optimal) if the condition is performed:
$$\rho_{\mu}\left(F_{n}(f), A\right) / \rho_{\mu}\left(F_{n}(f), a\right) \rightarrow 1(\leq \text { const }), \quad n \rightarrow \infty$$
Let $A(\varepsilon) \neq \varnothing$. The value $T(F, a, \varepsilon)=\sup {f \in F} T\left(I{n}(f), a, \varepsilon\right)$ is called $\varepsilon$-complexity of the algorithm $a \in A(\varepsilon)$ on the class of problems $F$ using information $I_{n}(f)$; $T(F, A, \varepsilon)=\inf {a \in A(\varepsilon)} T(F, a, \varepsilon)$ is $\varepsilon$-complexity of class $F$ problems using algorithms $A(\varepsilon)$ and information $I{n}(f)$. If there is an algorithm $a^{0} \in A(\varepsilon)$ for which

$T\left(F, a^{0}, \varepsilon\right)=T(F, A, \varepsilon)$, then it is called complexity optimal algorithm in this computational model. As in the case of error, it is possible to enter characteristics $T\left(F_{n}(f), a, \varepsilon\right)$ and $T\left(F_{n}(f), A, \varepsilon\right)$ on the class $F_{n}(f)$ and definitions of the optimal, asymptotically optimal, and complexity order optimal of the algorithm.

The mentioned characteristics are about the so-called “worst”-case model. The estimates are focused on the “worst” (the most complex) problem of the class. Of course, these estimates are guaranteed, and they are achieved on any problem of class $\Pi$, but this problem can be atypical for a given class. Therefore, there are some possible situations where the $\varepsilon$-solution can be obtained at lower expenses $(T)$. Consequently, in order to minimize the computational complexity of the $\varepsilon$-solution construction, the current question of the problems of classes narrowing, the ways where input data is presented, and the use of a priori information on a problem is relevant.

数学代写|优化算法代写optimization algorithms代考|Computer Architecture and the Complexity

There is an opinion (see, for example, [218]) that the optimization of the mathematical support of applied problems and the progress of computing techniques make equal contributions to the increasing possibilities of complex solving problems, in a point of fact in decreasing the computational complexity.

Consider the effect of rounding of the numbers on the computational complexity. Hypothesis [22] on that it is enough to compute the estimate of function $f$ for obtaining the solution with accuracy $O(\varepsilon)$ and perform intermediate computations in implemented CA with $O\left(\ln \varepsilon^{-1}\right)$ binary digit bits found confirmation in solving many problems $[42,106,114]$.

Thus, in the building of the $\varepsilon$-solution, the program uses numeric arrays with a total volume of $N$ numbers, and then memory $O\left(N \ln \varepsilon^{-1}\right)$ is required to store them.
Next, there is a possibility to consider the example of separate classes of problems and how the rounding error affects the possibility of $\varepsilon$-solution computation and the complexity of CP.
Let
$$\begin{gathered} \rho\left(E_{\mathrm{H}}\left(I_{n}(f), a, c\right) \leq \varepsilon_{1}<\varepsilon\right. \ \rho\left(E_{\mu \tau}\left(I_{n}(f), a, c\right)>\varepsilon_{2}, \quad \varepsilon_{2}=\varepsilon-\varepsilon_{1}\right. \end{gathered}$$
where $E_{\mu \tau}=E_{\mu}+E_{\tau}, a \in A$, and the relations are performed (in a point of fact in the numerical integration of $\mathrm{ODE}$, the computation of integrals and other classes of problems) $[106,114]$;

$$E_{\mu}=O\left(n^{-p}\right), \quad E_{\tau}=O\left(n 2^{-\tau}\right),$$
where $p$ is the order of the numerical method accuracy, and $\tau$ is the length of the mantissa in binary number notation in the floating-point mode. Herewith,
$$\varepsilon_{\mu \tau}^{0}(\tau)=\min {n} \rho\left(E{\mu z}\left(I_{n}(f), a, c\right)=O\left(n_{0}^{-p}(\tau)\right),\right.$$
where $n_{0}(\tau)=O\left(2^{\tau /(p+1)}\right)$, and $E_{\mu}\left(n_{0}(\tau)\right)=O\left(n_{0}^{-p}(\tau)\right), E_{\tau}\left(n_{0}(\tau)\right)=O\left(n_{0}^{-p}(\tau)\right)$.
Method error predominates when $n \ll n_{0}$. It can be decreased using optimal sets $I_{n}$, increasing $n$ (considering (2.10)), using accuracy optimal and close to them CA, moving to another class of input data $I_{n}$ (to increase the order of accuracy) and relevant CA.

The rounding error predominates when $n \gg n_{0}$. The decreasing $\rho\left(E_{\mu \mathrm{r}}\right)$ can be achieved by immediate increasing $n$ (considering $(2.10)$ ) or using the same capabilities as when $n \gg n_{0}$ (except increasing), as well as by increasing of $\tau$. From the relations (2.9) and (2.10), it follows that the performance of the constrain $\varepsilon_{\mu \tau}^{0} \leq \varepsilon_{2}$ is related to the conditions:
$$n=O\left(\varepsilon_{2}^{-1 / p}\right), \quad \tau=O\left(\log \varepsilon_{2}^{-1}\right), \quad \varepsilon_{2} \rightarrow 0$$
Consider the case (2.6). Let $\varepsilon$-solution be computed on the one processor using a linear computational algorithm per hour:
$$T(\varepsilon)=T_{l}(\varepsilon)+T_{a}(\varepsilon),$$
where $T_{I}(\varepsilon)$ is the process time of computation of the set of functionals $I_{n}(f)$ (information complexity), and $T_{a}(\varepsilon)$ is the process time of CA implementation for given information $I_{n}(f)$ (combinatorial complexity). Thus,
$$\begin{gathered} T_{l}(\varepsilon)=n(\varepsilon) \beta_{f}(\varepsilon) \alpha(\varepsilon), \ T_{a}(\varepsilon)=n(\varepsilon) \beta_{a} \alpha(\varepsilon), \end{gathered}$$
where $\alpha(\varepsilon)$ is the time performance of the “mean” operation in the computation of $\varepsilon$ solution; $\beta_{f}(\varepsilon)$ is an average number of computation operations of the functional $i_{j}$; and $\beta_{a}$ is an average number of operations that are related to the use of one function in implementation of CA.

Note that $\beta_{f}(\varepsilon)$ does not depend on $\varepsilon$ if the functionals in $I_{n}(f)$ can be computed closely to the accurate arithmetical operations. Then,
$$T(\varepsilon)=n(\varepsilon) \beta \alpha(\varepsilon), \quad \beta=\beta_{f}+\beta_{a} .$$

数学代写|优化算法代写optimization algorithms代考|Optical Models of Computations

Optical Models of Computations Dynamic holography [102] is a promising way of implementing a variety of optical converters. For instance, the effect of the energy transfer of a beam of light into another coherent light of beam that goes in the other direction (courtesy of their transverse in a dynamic environment) is an optical analogue of the transistor. It is possible to control the temporal variations in its intensity by changing the intensity of the amplifying beam. In another variant of the optical analogue of the electronic transistor, such control is achieved by changing not the intensity but the phase of the intensity beam.

Another example can be an optical switching device that is similar to a high-speed electronic commutator that is an inherent part of the most important devices of the computing techniques.

The advantage of holography is the possibility of a single transformation of the most complex images and not just the primary plane or spherical waves.

For today, the experiments have been performed on the creation of optical of bistable devices that switch $10^{-12}$ with elements of optical fiber communication lines, the information of which is moved with the help of optical solutions with the duration of reaching $10^{-13}$. With such a switching time, the productivity of a digital optical processor $10^{5}-10^{6}$ has simultaneous channels that would be equal to $10^{18}$ operations per second; in other words, it is on six order higher than the productive potential of electronic schemes. Examples of primitive actions for an optical computer include addition and subtraction of mappings, computation of Fourier transform, mapping identification, and so on.

数学代写|优化算法代写optimization algorithms代考|Input Information, Algorithms, and Complexity

• ρμ(F,一种)=支持F∈Fρ(和μ(一世n(F)),一种)是算法错误一种∈一种关于问题的类别F使用信息一世n(F)（全局错误 [270]）。
• ρμ(F,一种)=信息一种∈一种ρμ(F,一种)是类算法的误差下界一种在问题类别中F使用信息一世n(F)（信息半径[270]）
如果有算法一种0∈一种为此ρμ(F,一种0)=ρμ(F,一种)，则称其为类中精度最优F使用信息一世n(F).

приρμ(Fn(F),一种)→0 在 n→∞

ρμ(Fn(F),一种)/ρμ(Fn(F),一种)→1(≤ 常量 ),n→∞

数学代写|优化算法代写optimization algorithms代考|Computer Architecture and the Complexity

ρ(和H(一世n(F),一种,C)≤e1<e ρ(和μτ(一世n(F),一种,C)>e2,e2=e−e1

eμτ0(τ)=分钟nρ(和μ和(一世n(F),一种,C)=这(n0−p(τ)),

n=这(e2−1/p),τ=这(日志⁡e2−1),e2→0

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。

数学代写|优化算法代写optimization algorithms代考| Algebraic Computing Complexity

statistics-lab™ 为您的留学生涯保驾护航 在代写优化算法optimization 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optimization algorithms代写方面经验极为丰富，各种代写优化算法optimization 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

数学代写|优化算法代写optimization algorithms代考|Formal Computational Models

Turing machines (TM) [3] is a class of the most well-known formal models for the analysis of the problem of limited complexity. The problem is considered to be algorithmically solved if its solution can be built using the corresponding TM. It

should be noted that the class of problems that can be solved with TM is left to be solved moving from TM to another formal model [3,241, 271]. All problems of algebraic complexity are divided into two classes (the class $\mathrm{P}$ is a problem that can be solved with polynomial complexity on deterministic Turing machines (DTM), and the NP class is the class where the problems can be solved with polynomial complexity on nondeterministic Turing machines (NTM) [3]). As the characteristics of the computational complexity, computing time (number of steps that are necessary to use the solving problem of the algorithm) and memory (the amount of operating domain that is used by the algorithm) are used.

Here are some known relations between the time complexity $(T(n), n$ is the amount of input data) and by the amount of memory $(S(n))$ [249] that are obtained for TM.

Let DTIME $(T(n))$ (DSPACE $(S(n)))$ be a class of problem that suppose DTM per hour $T(n)$ (with a memory $(S(n)$ ). The classes of problems are determined likewise NTIME $(\cdot)$, NSPACE $(\cdot)$ for HTM. Then
$$\begin{gathered} \operatorname{DTIME}(T(n)) \subseteq \operatorname{NTIME}(T(n)) \ \operatorname{NTIME}(T(n)) \subseteq \operatorname{DTIME}\left(2^{O(T(n))}\right) \ \operatorname{DSPACE}(S(n)) \subseteq \operatorname{NSPACE}(S(n)) \ \operatorname{NSPACE}(S(n)) \subseteq \operatorname{DSPACE}\left(S^{2}(n)\right) \ \operatorname{NTIME}(T(n)) \subseteq \operatorname{DSPACE}(T(n)) \ \operatorname{DTIME}(T(n)) \subseteq \operatorname{DSPACE}\left(T(n) / \log _{2}(T(n))\right) \ \operatorname{NSPACE}(T(n)) \subseteq \operatorname{DTIME}\left(2^{O(S(n))}\right) \end{gathered}$$
An important example of complex problems is NP-complete problems. The problem $f$ is considered to be NP-completed if it belongs to the NP class and each NP problem can be polynomial complexity that is reduced to $f$. The central point in the theory of NP-completeness is whether or not the classes $\mathrm{P}$ and NP are congruent, in other words if the problem (from the class NP) is provided by practice that is related to problems (of class P) that can be solved. There are reasons to assume that the solution of the most complex problems of the NP class (NP-complete problems) requires (as it can be seen from the estimates) the deterministic exponential time; in other words, the classes P and NP are different. The NP-completeness of many problems is proved $[3,48]$. The difficulty is to prove that each NP problem can be polynomially transformed to this problem.

It should be noted that the definition of the NP class and the proof of the polynomial complexity of many “reset” problems had great practical importance. Together with practical valuation, it destroyed some illusions regarding the practical constructing of solving a problem that has a solution; it has been found that the existence of only one algorithm for solving a certain mass issue is not enough for

practice. On the other hand, the algorithms for which acceptable polynomial upper estimated were proved and found some practical use.

The basic possibility of classification by complexity is provided by the so-called theorems on the hierarchy. The hierarchy theorem for a given complexity (by time or memory) determines which decrease in the upper complexity estimate leads to the narrowing of the class of functions that can be computed with this complexity.

数学代写|优化算法代写optimization algorithms代考|Asymptotic Qualities of “Fast” Algorithms

The purpose of a lower complexity estimate construction is to prove that none of the algorithms in this computational model has less complexity of computation than the given function $\varphi(t)$. Unfortunately, the well-known “high” (nontrivial) lower estimates are perhaps the exception, not the rule.

The scheme of upper estimates of complexity constructing is as following. Based on some methods of solving problem, CA is built in a particular computational model, and it is proved that the computational complexity does not exceed some function from input data in the class. This function is called the upper estimate of the computational complexity of solving problem constructing.

There are several types of CA (which these estimates are implemented on). They are optimal, order optimal, and asymptotically optimal. Optimal CA corresponds to the case when the upper and lower boundaries are congruent. Two other types of CA concem, respectively, the estimates with the “accuracy to the multiplicative constant” and “accuracy to additive constants.” The practical use of algorithms is based on estimates that have an explicit specificity.

Consider these questions briefly. Let $A(0, X) \neq \varnothing A$ consider the computer model of sequential computations. Then
$$T\left(I_{n}(f), X, Y\right)=T_{I}\left(I_{n}(f), Y\right)+T_{a}(X, Y),$$
where $T_{I}(\cdot)=\sum_{1}^{r} \alpha_{i} n_{i}(n), T_{a}(\cdot)=\sum_{1}^{r} \alpha_{i} m_{i}(n, a), \alpha_{i}$ is a price of the $i$-operation from the model $c ; n_{i}(n), m_{i}(n, a)$ is the number of operations of the $i$-type that are necessary for the computation of the set of functionals $I_{n}(f)$ and the solution of the problem $f$ by the algorithm $a \in A$, provided that the set $I_{n}(f)$ is known; and $n$ is a number of functionals in the set. The values $T_{l}, T_{a}$ are called, respectively, informational and combinatorial (computational) complexities (solving computation) [270].

Note that the value $T$ depends essentially on $n$ and the character of the dependence $\left{n_{i}, m_{i}\right}$ from $n$. For example, by solving a system of $n$ linear equations, $A x=b$ by Gaussian elimination (for given $A, b) n_{i}=0, m_{i}=O\left(n^{3}\right), i=1,2$ (there is about the operations of addition and multiplication of two numbers).

In the general case, there is a possibility to assume that $n_{i}=O(n)$ (the functional $I_{n}(f)$ has a limited complexity) and $m_{i}(n)$ can be functions of $n$, for example, polynomial or exponential (or higher) complexity. Then the question arises on the

possibility of a solution computation with less computational complexity (see, for example, the class of NP-complete problems).

Of course, the character of dependence $m_{i}$ from $n$ is not determinative in the practical acceptability of the algorithm for solving a specific problem. It must be also considered that the constants in the functional dependences $m_{i}(n)$ can be that sort of algorithms with a lower order of complexity increasing, and advantage will be only for infinite values $n$. For example, offered algorithms of solving systems of linear algebraic equations for which $m_{i}=O\left(n^{\beta}\right), \beta<3$, have advantages over the complexity of Gaussian elimination for infinite values $n$. In addition, it is needed to pay attention to the possible loss of numerical stability of the algorithm. The fast Fourier transform (FFT) algorithm is used to multiply two numbers, and it has the complexity $O(n \log n)$ where $n$ is the number of binary digit bits for the number notation. The practical advantage of a high speed next to the traditional way of multiplication $\left(O\left(n^{2}\right)\right)$ is achieved for $n>100$.

数学代写|优化算法代写optimization algorithms代考|Accuracy and Complexity of Computations

The theory of analytic complexity is closely related to the theory of errors in the approximate solving problem. The value of the processing time is often determined by the requirements to the accuracy of the approximate solution; the relation of the components of the global error; the dependence of the error on the type, structure, volume of input data and their accuracy, bit grid of computer, and rounding rules; the type of error estimates; and the method of estimates constructing from below and from above. Therefore, there is a good reason to consider advisably these two characteristics: the error of the approximate solution and the process time [297, 301$]$.
Considering that it is difficult to build high lower and lower upper estimates in the given model of computation (when $E$ is a global error), some idealized models are considered that to consider only individual components of the global error (more often the errors of the method) and the influence of the individual components of computational models on error and complexity. For such incomplete models, it is possible to conclude the impossibility of constructing $\varepsilon$-solution based on this information.

The dependence of the approximate solution accuracy and the complexity of the $\varepsilon$-solution computation from the various components of the computational model will be considered next.

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。

数学代写|优化算法代写optimization algorithms代考|Theories of Computational Complexity

statistics-lab™ 为您的留学生涯保驾护航 在代写优化算法optimization 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optimization algorithms代写方面经验极为丰富，各种代写优化算法optimization 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

数学代写|优化算法代写optimization algorithms代考|Theories of Computational Complexity

Despite the achievements in the application software of modern computers, today there are many problems for which it is impossible to obtain a solution with given accuracy at limited computing resources. This is all about the problems of mathematical modeling, crystallography, radio astronomy, control of fleeting processes, cryptanalysis, and problems of high dimension.

As a rule, the solution of the applied problems is reduced to the solving typical classes of problems of computational and applied mathematics. Thus, it is important to create methods for building high-speed efficient algorithms for calculating $\varepsilon$-solutions of problems that use minimal computer memory for software. This will improve applied mathematical software and provide an opportunity to solve problems with less computing resources and reduce losses from the uncertainty of conclusions based on approximate solutions.

The main attention in the chapter is given to the creation of the elements of the complexity theory. With the use of it, this would be possible to construct effective complexity algorithms for computation of $\varepsilon$-solutions problems of numerical mathematics with limited computing resources.

Important results in the theory of computing optimization on the computing machinery were obtained by M. S. Bakhvalov, P. S. Bondarenko, V. V. Voievodin, H. Vozhniakovsky, V. V. Ivanov, M. P. Komeichuk, I. M. Molchanov, S. M. Nikolski, A. Sard, I. V. Sergienko, S. L. Sobolev, J. Traub, and others. These results allow estimating $\varepsilon$.

Computational complexity is less investigated than other characteristics. The complexity of the problem in time essentially depends on the computing model (computer architecture). A question of problem classes narrowing, the ways of input data presentation, and the complete use of a priori information on the problem are relevant for computational complexity minimizing of algorithm complexity of $\varepsilon$ solution constructing.

Today, many works are devoted to the study of the possibility of increasing the high speed of computing algorithms by paralleling the computations using traditional (with the focus on sequential computation) numerical methods. The general disadvantage of most of these studies lies in their obtainment of ideal computational models that lead to incomplete use of a priori information about the problem.

This chapter is devoted to the presentation of the general provisions of the complexity theory, statement of problems, algebraic and analytic complexity, and complexity of real computational processes. Key attention is given to the asymptotic qualities of “fast” algorithms, computer architecture, and the complexity and specificity of the characteristic estimate use. There are examples of the elements use of the complexity theory to the $\varepsilon$-solution construction of some practical important problems of computational and applied mathematics [285].

数学代写|优化算法代写optimization algorithms代考|General Provisions. Statement of the Problem

Let $F\left(I_{0}\right), A(X)$, and $C(Y)$ be the classes of problems of computational (or applied) mathematics, algorithms, and models of computing tools (computers), and $I_{0}$, $X, Y$ are a multitude of parameters that are dependent on the essentially suitable classes.

It is assumed that for the $\varepsilon$-solution constructing of the problem $f \in F$ (approximate solution, any error that does not exceed $\varepsilon \geq 0$ ), we use the algorithm $a \in A$ that was implemented on the computer $c \in C$ that is oriented on the use of information $I_{0}$ on class $F$ and information $I_{n}(f)$ on the specific class problem. The information (information operator) $I_{n}(f)$ can be given, for example, as a set of functionals $I_{n}(f)=\left(i_{1}(f), i_{2}(f), \ldots, i_{n}(f)\right)^{T}$ from the elements of the problem $f$.

Therefore, computation model is used for $\varepsilon$-solution construction that is described using $I_{0}, I_{n}(f), X, Y$.

The quality of the computational process (CP) of input data reduction, the result of which is $\varepsilon$-solution that is characterized by the computational complexity-the amount of a random computational resource that is necessary to the $\varepsilon$-solution constructing that is also called cost or expenses. The most widely used computing complexity characteristics is a processing time $T=T\left(I_{n}(f), X, Y, \varepsilon\right)$ and computer memory $M=M\left(I_{n}(f), X, Y, \varepsilon\right)$ that are required for $\varepsilon$-solution computing. Dependence of characteristics $T, M$ from $I_{0}$ is not specified since this information does not change.

They say that the problem has a restricted (algebraic) complexity (in this computational model) if there is an algorithm $a \in A$, by which it can be accurately solved $(\varepsilon=0)$ with limited computational complexity.

The problem has unrestricted (analytic) complexity if it cannot be solved precisely $(\varepsilon=0)$ in this computational model with restricted computational complexity.
A specific problem can have an algebraic or analytic complexity depending on input data and set of the computing model operations.

数学代写|优化算法代写optimization algorithms代考|solving problem computation of a system of linear algebraic

For example, solving problem computation of a system of linear algebraic equations by Gaussian elimination has an algebraic complexity providing that input data is given accurately and arithmetic operations are performed accurately either. If this condition is not performed, then the problem has an analytic complexity.

In real sets of operations, the great majority of problems of computational and applied mathematics are the problems of unlimited computational complexity; in other words, they are solved approximately ( $\varepsilon>0$ ). The exception is combinatorial and some algebraic problems [3].

The theory of analytic computational complexity is engaged in the optimization of the processes of approximate solving problems. The problems of algebraic complexity are used as an auxiliary in the theory of analytic complexity. On the other hand, the problems of algebraic complexity can have very high complexity and can be solved approximately [10].

The general situation of an approximate $\varepsilon$-solution of a problem constructing with constrained computing resources can be described by the following conditions $[14,106,114,237]$ :
$$\begin{gathered} E(I, X, Y) \leq \varepsilon, \ T(I, X, Y, \varepsilon) \leq T_{0}(\varepsilon), \ M(I, X, Y, \varepsilon) \leq M_{0}(\varepsilon), \end{gathered}$$
where $\varepsilon, T_{0}, M_{0}$ are the given numbers.
The quality of the approximate solution is characterized in the general case by the global error $\left(E\left(I_{n}(f), X, Y\right)\right)$, i.e., the sum of the three components: $E_{H}\left(I_{0}, I_{n}(f), Y\right)$ are the errors that are caused by inaccurate input information; $E_{\mu}\left(I_{0}, I_{n}(f), X\right)$ are the errors of the method; and $E_{z}\left(I_{n}(f), X, Y\right)$ are the errors through rounding $[106,114]$. Computations are often considered in the absence of some or all components of global error. All these can be some real computing situations or the results of idealization of computing conditions to simplify the research [106].
Thus, in the general case, it is needed to compute an approximate solving problem $f \in F$ using the model $I_{0}, I_{n}(f), X, Y$ under constraints (2.1), (2.2), and (2.3).

Further on, we will assume (if nothing other is not expected) that memory $M$ can be increased to the necessary volume; in other words, the constrain (2.3) can be removed but, apparently, by increasing the characteristic of $T$ (process time). This can be done, for example, by increasing a share of “slow” (disk) memory in the general structure of computer memory. Considering that within $\varepsilon \rightarrow 0, M_{0}(\varepsilon) \rightarrow \infty$ (for example, when it comes to rounding errors or errors in the method in stepwise algorithms), we will assume that $\varepsilon \geq \varepsilon_{0}>0$, where $\varepsilon_{0}$ is a given number.
Consider the problem of -solution finding (2.1), (2.2), and (2.3) [285].
Let $A(\varepsilon, X)(A=A(\varepsilon, X) \subseteq A(X))$ be a multitude of CA for which the condition (2.1) is used; in other words the algorithms for $\varepsilon$-solution computation for the given conditions. CA $A\left(\varepsilon, T_{0}\right)$ for which the conditions (2.1), (2.2) are used will be called $T$-effective, and $\left(A\left(\varepsilon, T_{0}\right) \subseteq A(\varepsilon, X)\right)$ is a multitude of $T$-effective CA.

数学代写|优化算法代写optimization algorithms代考|Theories of Computational Complexity

MS Bakhvalov, PS Bondarenko, VV Voievodin, H. Vozhniakovsky, VV Ivanov, MP Komeichuk, IM Molchanov, SM Nikolski, A. Sard, IV Sergienko, SL Sobolev, J. 特劳布等人。这些结果允许估计e.

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。

数学代写|优化算法代写optimization algorithms代考|Improvement of the Lower Estimate of the Accuracy

statistics-lab™ 为您的留学生涯保驾护航 在代写优化算法optimization 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optimization algorithms代写方面经验极为丰富，各种代写优化算法optimization 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

数学代写|优化算法代写optimization algorithms代考|Approximate Solving Problem by the Choice

Let $\widetilde{I}$ be any class of informational operators [285]. Assume that the class creates the informational operators of one type with different sets of functionals. For example, if a set of values of function is used, then their number or set of nodes can change their number or even the value of function is computed within constant $N$ (or both). Informational operators of different types (the value of the function and its derivatives, the coefficient of the factorize by certain basis, etc.) create different classes. It is possible to introduce the characteristics:
\begin{aligned} &\rho(\Pi, A, \widetilde{I})=\inf {I{N}(f) \in I} \rho\left(\Pi, A, I_{N}(P)\right)\left(\rho\left(\Pi, A, I_{N}(P)\right) \equiv \rho(\Pi, A)\right) \ &\rho(\Pi, \Lambda, \widetilde{I})=\inf _{A \in \Lambda} \rho(\Pi, A, \widetilde{I}) \end{aligned}
where $\rho=(\Pi, A, \tilde{I})$ is a lower boundary of the error of the algorithm $A \in \Lambda$ in the problem class $\Pi$ using information from class $\tilde{I}$, and $\rho(\Pi, \Lambda, \tilde{I})$ is a lower bound of the error of algorithms in the computing model $(\Pi, \Lambda, \widehat{I})$.

Information $I_{N}^{0}(P) \in \widetilde{I}$, for which the condition $\rho\left(\Pi, A, I_{N}^{0}(P)\right)=\rho(\Pi, A, \widetilde{I})$ is performed, is called an optimal in classes $\Pi, \tilde{I}$ by using the algorithm $A \in \Lambda$. If $\rho\left(\Pi, A^{0}, I_{N}^{0}(P)\right)=\rho(\Pi, \Lambda, \widetilde{I})$, then the algorithm $A^{0} \in \Lambda$ and the information $I_{N}^{0}(P) \in \widetilde{I}$ are called optimal in this computational model $(\Pi, \Lambda, \widetilde{I})$.

Likewise, it is possible to introduce the definition of complexity for the problem $P$ and the problem of class $\Pi$ and their characteristics:

• $T(\Pi, A, \widetilde{I}, \varepsilon)=\inf {I{N}(P) \in \widetilde{I}} T\left(\Pi, A, I_{N}(P), \varepsilon\right)$ is $\varepsilon$-complexity of the algorithm $A \in \Lambda(\varepsilon)$ in the problem of class $\Pi$ within the use of information $\tilde{I}$.
• $T(\Pi, \Lambda(\varepsilon), \tilde{I})=\inf _{A \in \Lambda(\varepsilon)} T(\Pi, A, \tilde{I}, \varepsilon)$ is $\varepsilon$-complexity of the problem in this computation model $(\Pi, \Lambda(\varepsilon), \widetilde{I})$.
• $T(P, A, \widetilde{I}, \varepsilon)$ is the $\varepsilon$-complexity of the algorithm $A \in \Lambda(\varepsilon)$ when the problem $P \in \Pi$ is solved using information $\tilde{I}$.
• $T(P, \Lambda(\varepsilon), \widetilde{I})$ is the $\varepsilon$-complexity of the problem $P$ by using the algorithms $\Lambda(\varepsilon)$ and information $\tilde{I}$, as well as the definition of complexity optimal algorithm and optimal information.

It is possible to introduce an optimization of nodes in numerical integrating as an example of such optimization: by optimization with accuracy for a fixed $N$ by computing $\varepsilon$-solution with $N=O\left(\varepsilon^{-1 / q}\right), q$ is the index of the smoothness of the subintegral function.

This case is about the optimization of choosing the functionals within the constrain of the same type of informational operator that is a set of values of the subintegral function.

Examples of the value optimization of the characteristics $E(\rho(\cdot))$ and $T$ moving to another class of informational operators are contained in [298].

Approximate Information There is known information (approximated) $I_{N \sigma}(P)$ instead of information (exact) $I_{N}(P)$ where $\sigma \geq 0$ characterizes the deviation of the approximate information from the exact one. It is possible to consider the characteristics for the approximate information $I_{N \sigma}(P)$ that are similar to those that were given above for $I_{N}(P)$ assuming that information $I_{N \sigma}(P)$ can be adjusted considering $I_{0}$-information about the problem of the class $\Pi$. Thus, the central algorithm [270] in this case decreases the effect of error of the information $I_{N o}(P)$ on the approximate solution. Examples of constructing these algorithms are given in $[33,106]$.

数学代写|优化算法代写optimization algorithms代考|Basic Approaches to Constructing the Accuracy

Consider the problem of the computation of the integral that looks
\begin{aligned} &I_{1}(\omega)=\int_{a}^{b} f(x) e^{-i \omega x} d x \ &I_{2}(\omega)=\int_{a}^{b} f(x) \sin \omega x d x \ &I_{3}(\omega)=\int_{a}^{b} f(x) \cos \omega x d x \end{aligned}
assuming that $f(x) \in F(F)$ is a certain class of functions, and $\omega$ is a certain real number $(\omega \mid \geq 2 \pi(b-a))$.

Let the information about $f(x)$ be given by $N$ values at nodes $\left{x_{i}\right}_{0}^{N-1}$ from its definition domain: $\left{f_{i}\right}_{0}^{N-1}=\left{f\left(x_{i}\right)\right}_{0}^{N-1}$, $\varepsilon_{i}$ characterizes the accuracy of the problem $f\left(x_{i}\right)=f_{i}:\left|\tilde{f}{i}-f{i}\right| \leq \varepsilon_{i}, i=\overline{0, N-1}$.

We concretize the general definition of the accuracy optimal algorithm that is given in the par. $1.4$ for the problem of the approximate computation $I(\omega)$ (we will understand one of the integrals $(1.20,1.21$, and $1.22)$ under $I(\omega))$ ).

Mark $R=R\left(f, A,\left{x_{i}\right}_{0}^{N-1},\left{\varepsilon_{i}\right}_{0}^{N-1}, \omega\right)$ as the result of the approximate computation $I(\omega)$ with quadrature formula $A$.
Introduce the characteristics
\begin{aligned} &V\left(f, A,\left{x_{i}\right}_{0}^{N-1},\left{\varepsilon_{i}\right}_{0}^{N-1}, \omega\right)=\rho(I(\omega), R) \ &V\left(F, A,\left{\varepsilon_{i}\right}_{0}^{N-1}, \omega\right)=\sup {f \in F} V\left(f, A,\left{x{i}\right}_{0}^{N-1},\left{\varepsilon_{i}\right}_{0}^{N-1}, \omega\right) \ &V=V\left(F,\left{\varepsilon_{i}\right}_{0}^{N-1}, \omega\right)=\inf {A} V\left(F, A,\left{\varepsilon{i}\right}_{0}^{N-1}, \omega\right) \ &V(F, \omega)=V(F, 0, \omega) \end{aligned}

数学代写|优化算法代写optimization algorithms代考|The function f(x)

Definition 1.1 The function $f^{\pm}(x)$ is called majorizing (minorant) class of functions $F_{N}$ that are defined in some domain $D$ if:

1. $f^{+}(x) \geq f(x)\left(f^{-}(x) \leq f(x)\right)$ for all $m$,
2. $f^{+}(x) \in F_{N}\left(f^{-}(x) \in F_{N}\right)$.
The Chebyshev center $\left(y_{1}, \ldots, y_{N}\right)$ and the Chebyshev radius $\rho^{}(\omega)$ of domain of uncertainty of solving the problem $(1.20,1.21$, and $1.22$ ) can be defined as follows [102]: $$\left(y_{1}, \ldots, y_{m}\right),\left(y_{1}, \ldots, y_{m}\right)=F\left(x_{1}, \ldots, x_{n}\right) \ldots$$ The quadrature formula that computes $I^{}(\omega)$ will be called accuracy optimal, and $\rho^{}(\omega)$ is the error of introduction of the value domain of the integral $I(\omega)$ using $I^{}(\omega)$ or the optimal estimate of the error of numerical integration $I(\omega)$ on the class $F_{N}\left(\delta=\rho^{}(\omega)\right)$. The quadrature formula $R(\omega)$ of the computation $I(\omega)$ for which $$\sup {f \in F{N}}|R(\omega)-I(\omega)| \leq \rho^{}(\omega)+\eta, \eta>0 \text { and } \eta=o\left(\rho^{}\right), O\left(\rho^{}\right)$$
$\left(y_{1}, \ldots, y_{N}\right)$ is called asymptotically optimal or accuracy order optimal.
Within given information about the problem, any quadrature formula can’t give an accuracy less than $\rho^{}(\omega)$. For interpolation classes $\left(y_{1}, \ldots, y_{m}\right)=F\left(x_{1}, \ldots, x_{n}\right)$, the Chebyshev radius $\rho^{}(\omega)$ ) coincides with an optimal estimate $V_{1}$.

The use of the limiting function method for the estimate $V$ is based on the following statement [293].

Theorem $1.3$ Let $f(x) \in F$ ( $F$ is a class of limiting functions) on $f(x)$ the information about its value in $N$ nodes of a random grid, and there is at least one quadrature formula $A \in M$ such as that $I^{+}(\omega) \leq I(\omega) \leq \Gamma(\omega)$. Then the next estimate is valid for $V_{1}$ :

$$V_{1} \geq \sup {F{N} \in F} \rho^{}(\omega)$$ It follows from the definition of the estimates $V$ and $V_{1}$ : $$V \geq V_{1}$$ In the case of $F \equiv F_{N}$, we have $V=\rho^{}(\omega)$.
Remark 1.1 Similar statements are colligated on n-dimensional case [293, 298], and they are used to construct optimal error estimates and prove some optimal cubature formulae of computation of multidimensional integrals from highoscillating functions of the form
\begin{aligned} I_{1}^{n}(\omega) &=\underbrace{\int_{0}^{1} \ldots \int_{0}^{1}}{n} f\left(x{1}, \ldots, x_{n}\right) \sin \omega x_{1} \cdot \ldots \cdot \sin \omega x_{n} d x_{1} \ldots d x_{n}, \ I_{2}^{n}(\omega) &=\underbrace{\int_{0}^{1} \ldots \int_{0}^{1}}{n} f\left(x{1}, \ldots, x_{n}\right) \cos \omega x_{1} \ldots \ldots \cos \omega x_{n} d x_{1} \ldots d x_{n} \end{aligned}
in the case when $n>1, f(X)$ is a known function, $f(X)=f\left(x_{1}, \ldots, x_{n}\right) \in F(F$ is a certain class of functions $X=\left{x_{1}, \ldots, x_{n}\right}, \omega$ is a certain real number $(|\omega| \geq 2 \pi)$, and information about $f(X)$ is given by $N$ values in node points $\left{X_{i}\right}_{0}^{N-1}$ from its domain of definition: $\left{f_{i}\right}_{0}^{N-1}=\left{f\left(X_{i}\right)\right}_{0}^{N-1}$.

数学代写|优化算法代写optimization algorithms代考|Approximate Solving Problem by the Choice

\begin{aligned} &\rho(\Pi, A, \widetilde{I})=\inf {I {N}(f) \in I} \rho\left( \Pi, A, I_{N}(P)\right)\left(\rho\left(\Pi, A, I_{N}(P)\right) \equiv \rho(\Pi, A)\right ) \ &\rho(\Pi, \Lambda, \widetilde{I})=\inf _{A \in \Lambda} \rho(\Pi, A, \widetilde{I}) \end{aligned}

• $T(\Pi, A, \widetilde{I}, \varepsilon)=\inf {I {N}(P) \in \widetilde{I}} T\left(\Pi, A, I_{N}( P), \varrepsilon\right)一世s\伐普西隆−C这米pl和X一世吨是这F吨H和一种lG这r一世吨H米一个 \in \Lambda(\varepsilon)一世n吨H和pr这bl和米这FCl一种ss\π在一世吨H一世n吨H和在s和这F一世nF这r米一种吨一世这n\波浪号{I}$。
• 吨(圆周率,Λ(e),一世~)=信息一种∈Λ(e)吨(圆周率,一种,一世~,e)是e-此计算模型中问题的复杂性(圆周率,Λ(e),一世~).
• 吨(磷,一种,一世~,e)是个e- 算法的复杂性一种∈Λ(e)当问题磷∈圆周率使用信息解决一世~.
• 吨(磷,Λ(e),一世~)是个e- 问题的复杂性磷通过使用算法Λ(e)和信息一世~，以及复杂度最优算法和最优信息的定义。

数学代写|优化算法代写optimization algorithms代考|Basic Approaches to Constructing the Accuracy

\begin{aligned} &V\left(f, A,\left{x_{i}\right}_{0}^{N-1},\left{\varepsilon_{i}\right} _{0}^{N-1}, \omega\right)=\rho(I(\omega), R) \ &V\left(F, A,\left{\varepsilon_{i}\right}_{ 0}^{N-1}, \omega\right)=\sup {f \in F} V\left(f, A,\left{x {i}\right}_{0}^{N-1 },\left{\varepsilon_{i}\right}_{0}^{N-1}, \omega\right) \ &V=V\left(F,\left{\varepsilon_{i}\right}_ {0}^{N-1}, \omega\right)=\inf {A} V\left(F, A,\left{\varepsilon {i}\right}_{0}^{N-1} , \omega\right) \ &V(F, \omega)=V(F, 0, \omega) \end{aligned}

数学代写|优化算法代写optimization algorithms代考|The function f(x)

1. F+(X)≥F(X)(F−(X)≤F(X))对全部米,
2. F+(X)∈Fñ(F−(X)∈Fñ).
切比雪夫中心(是1,…,是ñ)和切比雪夫半径ρ(ω)解决问题的不确定性域(1.20,1.21， 和1.22) 可以定义如下[102]：(是1,…,是米),(是1,…,是米)=F(X1,…,Xn)…计算的求积公式一世(ω)将被称为精度最优，并且ρ(ω)是积分值域引入的误差一世(ω)使用一世(ω)或数值积分误差的最优估计一世(ω)在课堂上Fñ(d=ρ(ω)). 求积公式R(ω)计算的一世(ω)为此支持F∈Fñ|R(ω)−一世(ω)|≤ρ(ω)+这,这>0 和 这=这(ρ),这(ρ)
(是1,…,是ñ)称为渐近最优或精度阶最优。
在有关问题的给定信息内，任何求积公式的准确度都不能低于ρ(ω). 对于插值类(是1,…,是米)=F(X1,…,Xn), 切比雪夫半径ρ(ω)) 与最优估计一致在1.

\begin{aligned}形式的高振荡函数计算多维积分的一些最优容积公式 I_{1}^{n}(\omega) &=\underbrace{\int_{0}^{1} \ldots \int_{0}^{1}} {n} f\left(x {1}, \ldots, x_{n}\right) \sin \omega x_{1} \cdot \ldots \cdot \sin \omega x_{n} d x_{1} \ldots d x_{n}, \ I_{2} ^{n}(\omega) &=\underbrace{\int_{0}^{1} \ldots \int_{0}^{1}} {n} f\left(x {1}, \ldots, x_ {n}\right) \cos \omega x_{1} \ldots \ldots \cos \omega x_{n} d x_{1} \ldots d x_{n} \end{aligned}

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。

数学代写|优化算法代写optimization algorithms代考|Identification and Clarification of A Priori Information

statistics-lab™ 为您的留学生涯保驾护航 在代写优化算法optimization 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optimization algorithms代写方面经验极为丰富，各种代写优化算法optimization 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

数学代写|优化算法代写optimization algorithms代考|Identification and Clarification of A Priori Information

Input data about the problem and its quality is very important in many aspects. Mention some of them:

• The more qualitative information on the problem is, the more qualitative approximate solution we can compute.
• Maximum use of all available information on the problem allows to limit the class of solved problems, and thus, it increases the “potential ability” of the NM; the more accurate input data is, the more accurate estimates of error and the less uncertainly range of the approximate solving problem.
• The computer technology of solving problem with the fixed values of quality with accuracy and fast speed is based on the analysis of error estimates.

We stop on some aspects of identification and clarification of a priori information on a problem.

An appropriate a priori information on the problem is required for obtaining a problem solution of a high quality, for example, the order of the derivative, constants that constrain it, the Hölder constant, and the corresponding mark-for the problems of function recovery and functionals. Useful information may also be about geometric properties-convexity, monotonies, number of extremums, etc. Such information is necessary to obtain an error estimate of the finding solution. If this information is given with sufficient low accuracy, then the conclusions on the quality of solving the problem will be inaccurate.

Consequently, obtaining qualitative a priori information is important in solving applied problems. Such information can be obtained from specialists who have a good knowledge of the physical phenomenon that we are studying. This information can also be obtained by using algorithms for identifying and clarifying a priori information.

For example, if the function is approximated from Lipschitz interpolational class, $F \equiv C_{L, N,} \varepsilon$ [287], and not only $L$ and $\varepsilon$ are known, but an only approximation to them is known. In such cases, it is advisable to use methods of residual and quasisolutions for approximating function [203].

For the class $F \equiv C_{L, N, e}$ the approximating function is the solution of the problem:
$$\min {f \in F} \max {i} \varepsilon_{i^{*}}$$
Otherwise, the method of quasisolutions involves finding a function that deviates less from the given set of points $\left(x_{i}, \tilde{f}{i}\right), \tilde{f}{i}=f_{i}+\varepsilon_{i}, i=\overline{0, N-1}$.

The solving problem $(1.15)$ is a line spline $S(x, L)$ in which the maximal deviation from the given points $\left(x_{i}, \tilde{f}{i}\right), i=\overline{0, N-1}$ is the minimum [203]: \begin{aligned} &S(x, L)=\widehat{f}{i}+\frac{x-x_{i}}{x_{i+1}-x_{i}}\left(\widetilde{f}{i+1}-\tilde{f}{i}\right), x \in\left[x_{i}, x_{i+1}\right], i=\overline{0, N-1}, \ &\widehat{f}{i}=\frac{\tilde{f}{i}-\tilde{f}{i}}{2}, \tilde{f}{i}^{\pm}=\max {1 \leq j \leq N}\left[\pm\left(\tilde{f}{j} \mp L\left|x_{j}-x_{i}\right|\right)\right], i=\overline{0, N-1} \end{aligned}
It often happens that the quantitative a priori information that is used to define a class $F$ is given in the form of constraints on some functional. A uniform norm of the derivative is used a functional $\Phi(f)$ itself for classes $C_{L, N}$ and $C_{L, N,}$. We will approximate the function $f(x)$ by a function that is the solution of the following problem:
$$\min _{f \in F} \Phi(f)$$

数学代写|优化算法代写optimization algorithms代考|Accuracy Optimal Computational Algorithms

In the due form, the concept of the optimality of the solution or the algorithm is determined by some criteria. Such criteria can be a requirement of the solution to have a given error of a method or the algorithm to have the highest possible rate of convergence. Specific content of criteria has an important value for its use.

One of the main criteria for the optimality of approximate solving problem can be the requirement of its maximum accuracy (or minimum error) by the given resources that can be used in the solution process. The concept of the resource includes the amount and accuracy of input data of the problem, free use of computer memory, limit the time of computing on this computing machinery, the available supply of mathematical software of computing machinery, etc.

In such a statement, it is natural to consider the question of the “potential ability” of NM at the beginning of the study, in other words, on that maximum accuracy of the solution that can be achieved for this given input information on the problem.
Every CA of solving a certain problem uses only a finite number of input data on the problem, and thus, it automatically is a CA of solving the class of all those problems that have the same input data. On this multitude of problems, there are always two problems in the solution of which the worst and best limits of the optimized characteristics are achieved. Therefore, every, including the optimal one, CA of solving a problem that concerns us will have some “potential ability.” If, for example, there are two problems with the same input data, accurate solutions of which $x_{1}$ and $x_{2}$ are the elements of the metric space, moreover, the distance between them are
$$\rho\left(x_{1}, x_{2}\right) \geq d>0$$
Then a solution $x$ obtains for each CA their solving that have a property
$$\max {i=1,2} \rho\left(x, x{i}\right) \geq \frac{d}{2}$$
This means that there is no CA that would give a solution to the considered problem with a guaranteed accuracy of less than $d / 2$. If there is a need to improve the accuracy of solving the problem, it should be included by some additional information about it. Then the problem will belong to a new more “narrow” class of problems, and the CA of which solution will have a new more powerful “potential ability.” Similar considerations are valid for any other index (characteristics) of CA and problems.

数学代写|优化算法代写optimization algorithms代考|Time Optimal Computational Algorithms

The computational complexity of the problem essentially depends on the computing model (computer architecture). Assume that it is possible to use a computing model for the building $E$-solution that is based on the use of input information of $I$, informational operator of $L_{M}(I)$ that can be introduced, for example, in the form of some multitude of functionals, as well as on a multitude of parameters $X, Y$.

Define the time-optimal algorithms and similar to them algorithms. Let the problem $P(I) L_{N}(I)=I=\left(i_{1}, i_{2}, \ldots, i_{N}\right), I \in$ 田 be input data, and $M$ is a multitude of algorithms $A$ of solving problem with a given accuracy $\varepsilon, \varepsilon>0$, on fixed computing machinery, and $Q(A, I, \varepsilon)$ is a number of arithmetic operations that are required for this. Consider the characteristics
\begin{aligned} &Q_{N}(A, \varepsilon)=\sup {I \in \mathcal{J}} Q(A, I, \varepsilon) \ &Q{N}(\varepsilon)=\inf {A \in M} Q{N}(A, \varepsilon) \end{aligned}
Call an algorithm a time-optimal one on which $Q_{N}(\varepsilon)$ is achieved. If $Q_{N}\left(A^{}, \varepsilon\right)=Q_{N}(\varepsilon)+\xi, \xi>0$, then $A^{}$ is called time-optimal with accuracy within $\xi$. If $\xi=o\left[Q_{N}(\varepsilon)\right]$ or $\xi=O\left[Q_{N}(\varepsilon)\right]$, then it is asymptotically optimal or time-optimal in order algorithm, correspondingly.

The purpose of constructing a lower complexity estimate $Q_{N}(\varepsilon)$ is to prove that none of the algorithms in this computational model has a less complexity of computation than the current function $Q_{N}(\varepsilon)$. Unfortunately, the well-known “high” (nontrivial) lower estimates are most likely an exception to the rule.

The scheme of upper estimates of complexity constructing is like this. CA $A^{}$ is built based on a certain method of solving the problem in the current computing model, and it is proved that the computational complexity does not exceed $Q_{N}\left(A^{}, \varepsilon\right)$ within the input data from the class. $Q_{N}\left(A^{}, \varepsilon\right)$ is called the upper estimate of the computational complexity of CA $A^{}$ of solving problem obtaining.

数学代写|优化算法代写optimization algorithms代考|Identification and Clarification of A Priori Information

• 关于问题的定性信息越多，我们可以计算的定性近似解就越多。
• 最大限度地利用有关问题的所有可用信息可以限制已解决问题的类别，从而增加 NM 的“潜在能力”；输入数据越准确，误差估计越准确，近似求解问题的不确定性范围越小。
• 准确、快速地解决质量固定值问题的计算机技术是基于对误差估计的分析。

$$\min {f \in F} \max {i} \varepsilon_{i^{*}}$$

ρ(X1,X2)≥d>0

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。

数学代写|优化算法代写optimization algorithms代考|Elements of the Computing Theory

statistics-lab™ 为您的留学生涯保驾护航 在代写优化算法optimization 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optimization algorithms代写方面经验极为丰富，各种代写优化算法optimization 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

数学代写|优化算法代写optimization algorithms代考|Elements of the Computing Theory

Since every year the complexity of scientific and applied problems is increasing, there is an uprise of the need of a large amount of process of input data with the use of computing machinery for the numerical computation, and optimization of computations by all rational methods and means is a topical and important question. As a rule, the solution of the applied problems reduces to solving the typical problems of computational mathematics. Computational mathematics is a science about the methods of numerical solving problems of applied mathematics using modern computing machinery. The subject of computational mathematics is numerical methods (NM) or, that is the same thing, the multitude of computational algorithms (CA) and the question of their substantiation: the convergence and speed of NM convergence, their stability and error, the optimality on different criterion, the implementation time on the computing machinery, the necessary memory of computing machinery, etc.

The purpose of this chapter is to explain the main concepts and some results of computational mathematics, which are repeatedly used in the monograph, and at the same time, they have an independent value. The presentation of this information, in our opinion, is necessary for a confident possession of the given material.
In the first chapter:

• Certain supporting data in the theory of computing is given for determining the accuracy of the approximate solving problem, the quality of the used algorithm, and the comparative study of various algorithms according to some criteria.
• The principal features of the CA are determined (accuracy, execution time, necessary memory of the computing machinery), and the methods of their estimate are provided.
• There is a definition of optimal (inaccuracy and high speed), asymptotically optimal, and optimal in the order algorithms.
• The questions of identifying and clarifying a priori information on the problem are considered in cases when a priori information is not incorrectly set in the order to improve the quality of the obtained approximate solutions of a problem.

数学代写|优化算法代写optimization algorithms代考|Theory of Error Computations

One of the main directions of development of modern computational mathematics is the improvement of the theory of error computations, the research of comparative study of CA, and the optimal algorithm development in the solving problems of computational and applied mathematics.

The solution of the majority of the problems with modern computing tools is based on the computational experiment (CE), which seamlessly unites the mathematical model (MM), the computational algorithm (CA), the computations on the computing machinery, and the experiment. CE is necessary for obtaining an adequate quantitative description of the studied phenomenon or process itself with the necessary accuracy for the least possible machine time. The process of observations and comparing them with the matched model of MM is conducted on the computing machinery to check the validity of MM. The implementation of MM is performed with the use of numerical methods (NM), which approximate the input model and make it suitable for practical use. The errors of this approximation, as well as the rounding errors in the implementation of NM on the computing machinery and the errors of measurement or observation of implementation of the studied process, must be considered in determining errors of MM or the adequacy of MM and in the process itself. The errors of input MM must be considered in the process of definition of the requirements for the accuracy of the solution of various problems that are solved within this model.

In this monograph, the most important computational aspects of the determination of the error of MM and constructing its optimal implementation are analyzed. The multifaceted approach is based on the analysis of the three main characteristics of the corresponding NM-accuracy, implementation time, and necessary memory on computing machinery. According to these characteristics, the comparative study and the optimization of the corresponding NM are performed.

Main Characteristics of CA Let the problem $P(I)$ is solved by the algorithm $A(X)$ on the computing machinery $C(Y)$, where $I, X, Y$ are finite multitudes (vectors) of parameters from which essentially $P, A, C$ are, respectively, dependent on. The components of the vector $I$ may include data on a priori qualities of solving a problem, for example, constant that constrains the absolute values of the order of the derivatives from the given functions, data on the accuracy of the input values, etc. A number of iterations of the algorithm, the degree of approximation, the step of the grid, etc., can be the vector $X$ components. Vector $Y$ can contain the number of cell digits of the computing machinery memory, the total volume of its RAM, the run time of the computing machinery, the quantity of the computing machinery

processors, the characteristics of the used operational systems and compiler programs, etc.

In the practice of computational solving problems on the computing machinery, the following are the characteristics of problems, algorithms, and computing machinery: $E(I, X, Y)$ is a global error of solution $E$ of the problem $P$ on the computing machinery $C$ with algorithm $A, T(I, X, Y)$ is the time that is required to obtain a solution of the problem, $M(I, X, Y)$ is a required computing machinery memory, and fef is a coefficient of technical and economic efficiency. Explain the characteristics of $T, M$, and fef..

Total time $T$ – The period of time from setting the problem $P(I)$ to its solution of CA $A(X)$ on computing machinery $C(Y)$ can be estimated as follows:
$$T=T_{1}+T_{2}+T_{3}+T_{4}$$
where $T_{1}$ is the time to set the problem and develop or select CA $A$ and computing machinery $C ; T_{2}$ is the programming time, transmission motion, and debugging $\mathrm{CA}$ $A ; T_{3}$ is the implementation time of $\mathrm{CA}, A$ on the computing machinery $C$; and $T_{4}$ is the time of interpretation of the obtained solution and its comprehension. Practically when estimating $T_{3}$, only the essential operators are often considered to the number and time of the operation performing of the computing machinery. In further detail, the given characteristic Tis described below and in the works $[2,85,97,114]$.

数学代写|优化算法代写optimization algorithms代考|Problem Statement of Optimization of Computation

Optimization of computations lies in the optimization of one of the introduced characteristics (in a general way on $I, X, Y$ ) in accordance with certain constraints on others $[118,111]$.

The first primary problem is to minimize time $T(I, X, Y)$ while upholding the real (Re) constraints on $M, E$, and $f e f$ :
$$\begin{gathered} T(I, X, Y)=\min {I, X, Y} \ M(I, X, Y) \leq M{\mathrm{Re}}, E(I, X, Y) \leq E_{\mathrm{Re}}, f e f(I, X, Y) \leq f e f_{\mathrm{Re}} . \end{gathered}$$

The second primal problem is to minimize the global error $E(I, X, Y)$ with constraints on $M, T$, and $f e f$.
$$\begin{gathered} E(I, X, Y)=\min {l, X, Y} \ M(I, X, Y) \leq M{\mathrm{Re},} \quad T(I, X, Y) \leq T_{\mathrm{Re}}, \quad f e f(I, X, Y) \leq f e f_{\mathrm{Re}} \end{gathered}$$
The third primal problem is to maximize a coefficient of the technical and economic effectiveness $f e f(I, X, Y)$ :
$$\begin{gathered} f e f(I, X, Y)=\max {I, X, Y} \ M(I, X, Y) \leq M{\mathrm{Re}}, \quad T(I, X, Y) \leq T_{\mathrm{Re}}, \quad E(I, X, Y) \leq E_{\mathrm{Re}} \end{gathered}$$
We make some remarks on the mentioned statements of the primal problems. Let the computing machinery $C(Y)$ be fixed. Then $T, M, E$, and fef depend only on $I, X$. It is convenient to consider $I$ to be a random value and consider the probabilistic characteristics of the values $T, M, E$, and $f e f$, which will also be the characteristics of CA $A$ and will depend only on $X$. We designate each of the characteristics $T, M, E$, and fef through $H(I, X)$ and designate the frequency distribution accordingly to $H, I$ through $P(H), P(I)$. The essential characteristics of $\mathrm{CA} A(X)$ is the mathematical expectation $M_{H}(X)$ and the dispersion $D_{H}(X)$ :
\begin{aligned} &M_{H}(X)=\int_{G} H P(I) d I=\int_{-\infty}^{\infty} H P(H) d H \ &D_{H}(X)=\int_{G}\left(H-M_{H}\right)^{2} P(I) d I=\int_{-\infty}^{\infty}\left(H-M_{H}\right)^{2} P(H) d H \end{aligned}
where $G$ is a domain of possible values of $I$.

数学代写|优化算法代写optimization algorithms代考|Elements of the Computing Theory

• 给出了计算理论中的一定支持数据，用于确定近似求解问题的准确性、所用算法的质量，以及按一定标准对各种算法进行比较研究。
• 确定了 CA 的主要特征（准确性、执行时间、计算机的必要内存），并提供了它们的估计方法。
• 在顺序算法中有最优（不准确和高速）、渐近最优和最优的定义。
• 在没有错误设置先验信息的情况下，考虑识别和澄清有关问题的先验信息的问题，以提高获得的问题近似解的质量。

数学代写|优化算法代写optimization algorithms代考|Theory of Error Computations

CA的主要特点让问题磷(一世)由算法解决一种(X)在计算机器上C(是)， 在哪里一世,X,是是有限数量的参数（向量），它们本质上是磷,一种,C分别依赖于。向量的组成部分一世可能包括有关解决问题的先验质量的数据，例如，约束给定函数的导数阶的绝对值的常数、有关输入值准确性的数据等。 算法的多次迭代，逼近的程度，网格的步长等，可以是向量X组件。向量是可以包含计算机内存的单元位数，其RAM的总容量，计算机的运行时间，计算机的数量

数学代写|优化算法代写optimization algorithms代考|Problem Statement of Optimization of Computation

F和F(一世,X,是)=最大限度一世,X,是 米(一世,X,是)≤米R和,吨(一世,X,是)≤吨R和,和(一世,X,是)≤和R和

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。

数学代写|优化算法作业代写optimisation algorithms代考| Nature’s Resum ´ e

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|Optimisation by Natural Selection

If we momentarily restrict our attention to the biological branch of nature, we can highlight some of the useful characteristics of this plentiful supplier of inspiration. Undoubtedly the most important contribution to modern biology was made by Charles Darwin with his Theory of Evolution by Natural Selection. Observing the achievements of animal husbandry, he writes:
Why, if man can by patience select variations most useful to himself, should nature fail in selecting variations useful, under changing conditions of life, to her living products … I can see no limit to this power, in slowly and beautifully adapting each form to the most complex relations of life. [17]
The immense explanatory power of a relatively simple set of rules-reproduction, mutation and selection-has often earned Darwin’s theory the title of the most significant scientific discovery of the 19th century. As the evolutionary biologist Theodosius Dobzhansky writes, “nothing in biology makes sense except in the light of evolution.” [19]

The period between the birth of an organism and the birth of its offspring can be decades. The optimality of its behaviour during this period will influence the

likelihood of its genes being propagated. We should not then be surprised to find evolution producing numerous ‘optimisation sub-processes’ suited to different timescales. To achieve this, organisms use various mechanisms to interact with their environment, which may be of use to an algorithm designer. Not only is natural selection itself a source of much inspiration, but it is also key to the existence of all the biological problem-solving mechanisms we find in nature. Accordingly, a solid understanding of evolution by natural selection is of use to any researcher interested in nature-inspired techniques.

If we are to extract an optimisation method from nature, it seems appropriate to ask exactly what nature was using it for. What is being optimised by natural selection? Is there some approximation to an objective function? How is the problem constrained? How can we measure success?

Ants make up 10 percent of the biomass of all animals in the Amazon rain forest [53], but that does not necessarily mean they are a superior solution. Should we measure success by the longevity of the gene? the individual? or perhaps the species? Maybe the efficiency of energy use is important? In The Diversity of Life, E. O. Wilson writes,
The hallmark of life is this: a struggle among an immense variety of organisms weighing next to nothing for a vanishingly small amount of energy.

The natural world is not a stagnant place; meteorological events, tidal forces, plate tectonics, and all the biological activities. Evolution by natural selection is a dynamic process, where the fitness landscape is always changing. As individuals and populations search for new ways to exploit their environment, the environment changes. For example, if a species becomes too skilled at hunting a certain prey, the food supply may run out. To survive, organisms must be able to cope with changing environmental conditions. This change can occur over millennia, a few generations, an individual’s lifetime, or in an instant.

Some organisms have the ability to withstand large variations in the environment. This approach can be thought of as change tolerance, or robustness. Other organisms respond to change more dynamically, using a process called adaptation.
In the most general sense, adaptation is a feedback process in which external changes in an environment are mirrored by compensatory internal changes in an adaptive system. [23]
Nature has been observed to achieve this adaptive ability in many ways, and biologists will undoubtedly continue to discover new mechanisms in the future. An important feature of any adaptive process is some form of memory, either implicit or explicit. Memory allows previous experience to influence future actions.

Closely related to memory is the concept of a learning mechanism. Learning mechanisms process experience and store it in memory. This ability is clearly seen in the human brain, although the mechanism is still poorly understood [38]. A less obvious example is the human immune system, which is capable of recognising and combating infectious foreign elements with specialised responses based on previous exposure .

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

Many of the nature-inspired algorithms currently in use are being applied to a wide range of problems. This puts them in the category of metaheuristics, where little or no problem specific information is used in the design of the algorithm. But is this kind of generality found in nature, or is it a human innovation?

Generality is related to the concept of adaptability. Some problem-solving mechanisms found in nature can be viewed as hierarchic algorithms. A successful high level algorithm will often use various adaptive subroutines. For example, ants build nests in many different environments, using the most suitable available materials. As generations pass they may adjust to better collect local materials, but the general rules of assembly are retained. This can be tied back to the use of diversity

as a means of preservation. A species which survives only in a very small niche is far more likely to suffer extinction when the environment changes. On the other hand some degree of specialisation will be advantageous, especially during periods of stability. As such, natural selection must find a balance between generality and specialisation.

Natural selection itself is certainly a widespread process in nature, capable of finding novel and elaborate solutions to a huge number of problems. Accordingly, it is not surprising that the evolutionary algorithms have been so broadly and successfully applied [27].

It is interesting to consider natural algorithms in terms of the No Free Lunch Theorems [54]. Since all problem-solving techniques found in nature are to some extent specialised to real problems, there is at least an intuitive reason to think they will perform better than random search on the set of problems arising from real world situations.

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。

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.

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。

数学代写|优化算法作业代写optimisation algorithms代考| Multi-objective Optimization

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|Multi-objective Optimization

Many optimization problems in the real world have $k$ possibly contradictory objectives $f_{i}$ which must be optimized simultaneously. Furthermore, the solutions must satisfy $m$ inequality constraints $g$ and $p$ equality constraints $h$. A solution candidate $x$ is feasible, if and only if $g_{i}(x) \geq 0 \forall i=1,2, \ldots, m$ and $h_{i}(x)=0 \forall i=1,2, . ., p$ holds. A multi-objective optimization problem (MOP) can then be formally defined as follows:

Definition 7 (MOP). Find a solution candidate $x^{\star}$ in $\mathbb{X}$ which minimizes (or maximizes) the vector function $\mathbf{f}\left(x^{\star}\right)=\left(f_{i}\left(x^{\star}\right), f_{2}\left(x^{\star}\right), \ldots, f_{k}\left(x^{\star}\right)\right)^{T}$ and is feasible, (i.e., satisfies the $m$ inequality constraints $g_{i}\left(x^{\star}\right) \geq 0 \forall i=1,2, . ., m$, the $p$ equality constraints $\left.h_{i}\left(x^{\star}\right)=0 \forall i=1,2, . ., p\right)$.

As in single-objective optimization, nature-inspired algorithms are popular techniques to solve such problems. The fact that there are two or more objective functions implies additional difficulties. Due to the contradictory feature of the functions in a MOP and the fact that there exists no total order in $\mathbb{R}^{n}$ for $n>1$, the notions of “better than” and “optimum” have to be redefined. When comparing any two solutions $x_{1}$ and $x_{2}$, solution $x_{1}$ can have a better value in objective $f_{i}$, i.e., $f_{i}\left(x_{1}\right)<f_{i}\left(x_{2}\right)$, while solution $x_{2}$ can have a better value in objective $f_{j}$. The concepts commonly used here are Pareto dominance and Pareto optimality.

Definition 8 (Pareto Dominance). In the context of multi-objective global optimization, a solution candidate $x_{1}$ is said to dominate another solution candidate $x_{2}$ (denoted by $\left.x_{1} \preccurlyeq x_{2}\right)$ if and only if $\mathbf{f}\left(x_{1}\right)$ is partially less than $\mathbf{f}\left(x_{2}\right)$, i.e., $\forall i \in{1, \ldots, k} f_{i}\left(x_{1}\right) \leq f_{i}\left(x_{2}\right) \wedge \exists j \in{1, \ldots, k}: f_{j}\left(x_{1}\right)<f_{j}\left(x_{2}\right)$.

The dominance notion allows us to assume that if solution $x_{1}$ dominates solution $x_{2}$, then $x_{1}$ is preferable to $x_{2}$. If both solution are non-dominated (such as candidate (1) and (2) in Fig. 12), some additional criteria have to be used to choose one of them.

Definition 9 (Pareto Optimality). A feasible point $x^{\star} \in \mathbb{X}$ is Paretooptimal if and only if there is no feasible $x_{b} \in \mathbb{X}$ with $x_{b} \preccurlyeq x^{*}$.

This definition states that $x^{\star}$ is Pareto-optimal if there is no other feasible solution $x_{b}$ which would improve some criterion without causing a simultaneous worsening in at least one other criterion. The solution to a MOP.

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

In order to obtain an accurate approximation to the true Pareto front, many nature-inspired multi-objective algorithms apply a fitness assignment scheme based on the concept of Pareto dominance, as commented before. For example, NSGA-II [61, 62], the most well-known multi-objective technique, assigns to each solution a rank depending on the number of solutions dominating it. Thus, solutions with rank 1 are non-dominated, solutions with rank 2 are dominated by one solution, and so on. Other algorithms, such as SPEA2 $[247,248]$ introduce the concept of strength, which is similar to the ranking but also considers the number of dominated solutions.

While the use of Pareto-based ranking methods allows the techniques to search in the direction of finding approximations with good convergence, additional strategies are needed to promote spread. The most commonly adopted approach is to include a kind of density estimator in order to select those solutions which are in the less crowded regions of the objective space. Thus, NSGA-II employs the crowding distance [61] and SPEA2 the distance to the $\mathrm{k}$-nearest neighbor [62].

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

How the constraints mentioned in Definition 7 are handled is a whole research area in itself with roots in single-objective optimization. Maybe one of the most popular approach for dealing with constraints goes back to Courant [48] who introduced the idea of penalty functions $[73,44,201]$ in 1943: Consider, for instance, the term $f^{\prime}(x)=f(x)+v[h(x)]^{2}$ where $f$ is the original objective

function, $h$ is an equality constraint, and $v>0$. If $f^{\prime}$ is minimized, an infeasible individual will always have a worse fitness than a feasible one with the same objective values.

Besides such static penalty functions, dynamic terms incorporating the generation counter $[111,157]$ or adaptive approaches utilizing additional population statistics $[95,199]$ have been proposed. Rigorous discussions on penalty functions have been contributed by Fiacco and McCormick [73] and Smith and Coit [201].

During the last fifteen years, many approaches have been developed which incorporate constraint handling and multi-objectivity. Instead of using penalty terms, Pareto ranking can also be extended by additionally comparing individuals according to their feasibility, for instance. Examples for this approach are the Method of Inequalities (MOI) of Zakian [245] as used by Pohlheim [164] and the Goal Attainment method defined in [76]. Deb $[56,58]$ even suggested to simply turn constraints into objective functions in his MOEA version of Goal Programming.

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。