### 数学代写|数学生态学作业代写Mathematical Ecology代考| Optimization Problems with Constraints

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Optimization Problems with Constraints

Solving an optimization problem with constraints is more difficult compared to the unconstrained optimization. One of powerful techniques for finding local maxima and minima of a function subject to constraints is the method of Lagrange multipliers [5]. To illustrate it for a problem with equality-constraints, let us consider the following simple discrete optimization problem: find $x_{1}, x_{2}$ that
$$\text { maximize } f\left(x_{1}, x_{2}\right)$$
subject to the equality-constraint
$$f\left(x_{1}, x_{2}\right)=c,$$
where both functions $f$ and $g$ have continuous first partial derivatives.
The Lagrange function (or Lagrangian) of the optimization problem (1.42)-(1.43) is determined as
$$L\left(x_{1}, x_{2}, \lambda\right)=f\left(x_{1}, x_{2}\right)+\lambda\left(g\left(x_{1}, x_{2}\right)-c\right),$$
where the unknown variable $\lambda$ is called the Lagrange multiplier (dual variable, or adjoint variable). The second term in (1.44) is zero along a solution to the problem. Thus in order to solve (1.42)-(1.43), we can find the maximum of (1.44). The maximization of the Lagrangian (1.44) includes one more unknown variable but does not involve the equality-constraint (1.43). By construction of (1.44), if $\left(x_{10}\right.$, $x_{20}$ ) brings a maximum to the original problem (1.42)-(1.43), then there exists $\lambda_{0}$ such that $\left(x_{10}, x_{20}, \lambda_{0}\right)$ is a stationary point $(\partial L / \partial \lambda=0)$ of the Lagrange function (1.38). Note that $\partial L / \partial \lambda=0$ implies (1.37).

The method of Lagrange multipliers yields necessary conditions for optimality. Sufficient conditions for optimality are also possible but are more difficult to obtain. The Lagrangian can be reformulated in the terms of Hamiltonian for many specific optimization problems. In particular, the method of Lagrange multipliers can be used to derive the maximum principle for the optimal control of differential equations provided in Sect. 2.4.

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Continuous Optimization

Optimization problems in the continuous models of Sect. $1.2 .2$ are known as continuous-time optimization problems or the optimal control problems. The control variables in such problems are scalar-or vector-valued functions of a continuous independent variable and the objective function is a functional that depends on the control variables.

Historically, calculus of variations is the first classic technique for the continuous-time optimization developed over 200 years mainly for geometric and physical applications. A variational problem minimizes a certain functional on a set of smooth functions in an open domain. Further extension of the variational techniques to the non-smooth unknown functions and closed domains leads to the modern optimal control theory and its main tools, the principle of maximum of $\mathrm{L}$. Pontryagin and the dynamic programming method of R. Bellman.

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Aggregate Models of Economic Dynamics

This chapter explores aggregate optimization models of the neoclassic economic growth theory, which are based on the concept of production functions. The models are described by ordinary differential equations and involve static and dynamic optimization. Section $2.1$ analyzes production functions with several inputs, their fundamental characteristics, and major types (Cobb-Douglas, CES, Leontief, and linear). Special attention is given to two-factor production functions and their use in the neoclassic models of economic growth. Sections $2.2$ and $2.3$ describe and analyze the well-known Solow-Swan and Solow-Ramsey models. Section $2.4$ contains maximum principles used to analyze dynamic optimization problems in this and other chapters.

最大化 F(X1,X2)

F(X1,X2)=C,

## 数学代写|数学生态学作业代写Mathematical Ecology代考|Continuous Optimization

Sect 连续模型中的优化问题。1.2.2被称为连续时间优化问题或最优控制问题。此类问题中的控制变量是连续自变量的标量或向量值函数，目标函数是取决于控制变量的函数。

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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。