## 数学代写|模拟和蒙特卡洛方法作业代写simulation and monte carlo method代考|AEM4060

2022年10月17日

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## 数学代写|模拟和蒙特卡洛方法作业代写simulation and monte carlo method代考|Duality

The aim of duality is to provide an alternative formulation of an optimization problem that is often more computationally efficient or has some theoretical significance (see [7], page 219). The original problem (1.66) is referred to as the primal problem, whereas the reformulated problem, based on Lagrange multipliers, is referred to as the dual problem. Duality theory is most relevant to convex optimization problems. It is well known that if the primal optimization problem is (strictly) convex, then the dual problem is (strictly) concave and has a (unique) solution from which the optimal (unique) primal solution can be deduced.

Definition 1.16.5 (Lagrange Dual Program) The Lagrange dual program of the primal program (1.66), is
\begin{aligned} \max {\boldsymbol{\alpha}, \boldsymbol{\beta}} & \mathcal{L}^(\boldsymbol{\alpha}, \boldsymbol{\beta}) \ \text { subject to: } & \boldsymbol{\alpha} \geqslant 0, \end{aligned} where $\mathcal{L}^$ is the Lagrange dual function:
$$\mathcal{L}^(\boldsymbol{\alpha}, \boldsymbol{\beta})=\inf {\mathbf{x} \in \mathscr{X}} \mathcal{L}(\mathbf{x}, \boldsymbol{\alpha}, \boldsymbol{\beta}) .$$
It is not difficult to see that if $f^$ is the minimal value of the primal problem, then $\mathcal{L}^(\boldsymbol{\alpha}, \boldsymbol{\beta}) \leqslant f^$ for any $\boldsymbol{\alpha} \geqslant 0$ and any $\boldsymbol{\beta}$. This property is called weak duality. The Lagrangian dual program thus determines the best lower bound on $f^$. If $d^$ is the optimal value for the dual problem, then $d^$. The difference $f^-d^$ is called the duality gap.

The duality gap is extremely useful for providing lower bounds for the solutions of primal problems that may be impossible to solve directly. It is important to note that for linearly constrained problems, if the primal is infeasible (does not have a solution satisfying the constraints), then the dual is either infeasible or unbounded. Conversely, if the dual is infeasible, then the primal has no solution. Of crucial importance is the strong duality theorem, which states that for convex programs (1.66) with linear constrained functions $h_i$ and $g_i$ the duality gap is zero, and any $\mathrm{x}^$ and $\left(\boldsymbol{\alpha}^, \boldsymbol{\beta}^*\right)$ satisfying the KKT conditions are (global) solutions to the primal and dual programs, respectively. In particular, this holds for linear and convex quadratic programs (note that not all quadratic programs are convex).

For a convex primal program with $C^1$ objective and constraint functions, the Lagrangian dual function (1.70) can be obtained by simply setting the gradient (with respect to $\mathbf{x})$ of the Lagrangian $\mathcal{L}(\mathbf{x}, \boldsymbol{\alpha}, \boldsymbol{\beta})$ to zero. One can further simplify the dual program by substituting into the Lagrangian the relations between the variables thus obtained.

## 数学代写|模拟和蒙特卡洛方法作业代写simulation and monte carlo method代考|RANDOM NUMBER GENERATION

In the early days of simulation, randomness was generated by manual techniques, such as coin flipping, dice rolling, card shuffling, and roulette spinning. Later on, physical devices, such as noise diodes and Geiger counters, were attached to computers for the same purpose. ‘I’he prevailing belief held that only mechanical or electronic devices could produce truly random sequences. Although mechanical devices are still widely used in gambling and lotteries, these methods were abandoned by the computer-simulation community for several reasons: (1) mechanical methods were too slow for general use, (2) the generated sequences could not be reproduced, and (3) it was found that the generated numbers exhibit both bias and dependence. Although certain modern physical generation methods are fast and would pass most statistical tests for randomness (e.g., those based on the universal background radiation or the noise of a PC chip), their main drawback remains their lack of repeatability. Most of today’s random number generators are not based on physical devices but on simple algorithms that can be easily implemented on a computer. They are fast, require little storage space, and can readily reproduce a given sequence of random numbers. Importantly, a good random number generator captures all the important statistical properties of true random sequences, even though the sequence is generated by a deterministic algorithm. For this reason these generators are sometimes called pseudorandom.

Most computer languages already contain a built-in pseudorandom number generator. The user is typically requested only to input the initial seed, $X_0$, and upon invocation the random number generator produces a sequence of independent, uniform $(0,1)$ random variables. We therefore assume in this book the availability of such a “black box” that is capable of producing a stream of pseudorandom numbers. In Matlab, for example, this is provided by the rand function. The “seed” of the random number generator, which can be set by the rng function, determines which random stream is used, and this is very useful for testing purposes.

# 模拟和蒙特卡洛方法代写

## 数学代写|模拟和蒙特卡洛方法作业代写模拟和蒙特卡罗方法代考|对偶性

\begin{aligned} \max {\boldsymbol{\alpha}, \boldsymbol{\beta}} & \mathcal{L}^(\boldsymbol{\alpha}, \boldsymbol{\beta}) \ \text { subject to: } & \boldsymbol{\alpha} \geqslant 0, \end{aligned}，其中$\mathcal{L}^$是拉格朗日对偶函数:
$$\mathcal{L}^(\boldsymbol{\alpha}, \boldsymbol{\beta})=\inf {\mathbf{x} \in \mathscr{X}} \mathcal{L}(\mathbf{x}, \boldsymbol{\alpha}, \boldsymbol{\beta}) .$$

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## 有限元方法代写

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## 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。