### 数学代写|随机过程统计代写Stochastic process statistics代考|MTH7090

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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

## 数学代写|随机过程统计代写Stochastic process statistics代考|Continuity and Convexity

For multivariable optimization, two important concepts are continuity and convexity. The functions to be optimized are desired to have those properties, although, even if they are neither continuous nor convex, the functions can be optimized with certain limitations. In this section, continuity and convexity of functions are described, and the importance of such properties on optimization is stressed.

A given function $\mathrm{f}(\bar{x})$ is continuous in a point $\bar{x}{0}$ if the following equality is true: $$\mathrm{f}\left(\bar{x}{0}\right)=\lim {\bar{x} \rightarrow \bar{x}{0}} \mathrm{f}(\bar{x})$$
If Equation $2.7$ is true for any value of $\bar{x}$ in the domain of the function, where $\bar{x} \in R^{n}$, then the function is continuous in the entire domain. An example of a continuous function is shown in Figure 2.4. It can be observed that the function is defined for any value of $\bar{x}$, and it does not exhibit any disruption. The limits for the function can be evaluated for any $\bar{x}$, and they are equal to the value of the function. Thus, it is continuous.

A noncontinuous function is presented in Figure 2.5. The function is defined for almost any value of $\bar{x}$. Nevertheless, when $\bar{x}$ is close to the point

$\bar{x}{0}=\left[\begin{array}{ll}0 & 0\end{array}\right]^{T}$, the function grows, and it will reach infinity at the point $\bar{x}{0}$. Thus, the function is not defined at $\bar{x}_{0}$, and it is noncontinuous.

Because most of the deterministic optimization methods are based on the calculation of derivatives, dealing with continuous functions ensures that the derivatives exist for all feasible regions. For noncontinuous functions, if the solution is close to a discontinuity point, the derivative will not exist and problems will arise with the optimization algorithm. Nevertheless, through a proper analysis of the functions and a good selection of the limits of the variables, it is possible to avoid the discontinuities in several cases.

## 数学代写|随机过程统计代写Stochastic process statistics代考|Unconstrained Optimization

The simplest case of deterministic, nonlinear optimization occurs when the problem has no constraints, i.e., the objective function must be optimized for any $\bar{x} \in R^{n}$ on its domain. For linear programming, constraints must always exist, because a linear function continues increasing (or decreasing) its value when the decision variables change; thus, no optimal solution can be obtained for a linear objective function without constraints. For nonlinear optimization, most of the solution methods are based on the calculation of derivatives to perform a search for stationary points. A given point $\bar{x}^{}$ is a stationary point of the function $\mathrm{f}(\bar{x})$ if it complies with the following condition: $$\nabla f\left(\bar{x}^{}\right)=0$$
Equation $2.15$ is known as the first-order necessary condition for optimality. A point $\vec{x}$ complying this condition could be at optimum, but not necessarily, because it could also be a saddle point. To ensure that $\bar{x}^{}$ is at least a local minimum, $\bar{H}\left(\bar{x}^{}\right)$ must be positive definite or positive semidefinite. On the other hand, to ensure that $\bar{x}^{}$ is at least a local maximum, $\bar{H}\left(\bar{x}^{}\right)$ must be negative definite or negative semidefinite.

To solve an unconstrained optimization problem, a gradient-based approach can be used. Such methods basically take an initial solution and start a search for regions where the gradient is reduced. To do that, a search direction and the step size must be determined. The first one indicates in which direction the movement will be performed, and the second one indicates how large the movement will be. The objective is to find a solution for which the gradient is zero, which represents a stationary point, which can be a minimum or a maximum if it complies with the conditions mentioned in the previous paragraph. The general algorithm for a gradient-based method is as follows:

1. Select an initial solution, $\bar{x}{0}=\left[\begin{array}{llll}x{1}^{0} & x_{2}^{0} & \ldots & x_{n}^{0}\end{array}\right]^{\mathrm{T}}$, and evaluate the objective function at $\bar{x}_{0}$.
2. Determine the gradient of the objective function, $\nabla f(\bar{x})=\left[\frac{\partial f}{\partial x_{1}} \frac{\partial f}{\partial x_{2}}\right.$ $\left.\frac{\partial \mathrm{f}}{\partial x_{\mathrm{n}}}\right]^{\mathrm{T}}$, and, if necessary, the Hessian matrix for the objective function.

## 数学代写|随机过程统计代写Stochastic process statistics代考|Equality-Constrained Optimization

Most of the process engineering optimization problems are, indeed, constrained. Thus, it is important to understand how to deal with such situations. In this section, a couple of methods for equality-constrained optimization are discussed. A general way to represent an equality-constrained optimization problem can be obtained by simplifying Equation $1.8$ : Here,the main concern is to obtain an optimal solution for $z(\bar{x})$, which also complies with the set of equality constraints. Two strategies to ensure that are presented here: the method of Lagrange multipliers and the generalized reduced gradient method.

In this method, the optimization problem presented in Equation $2.19$ is reformulated to obtain an objective function that involves the original objective, $z(\bar{x})$, and the entire set of equality constraints, $h_{i}(\bar{x})=0$, where $i=1,2, \ldots, n$. The resultant expression is known as the Lagrangian function and is expressed as follows:
$$\text { optimize } L=z(\bar{x})+\sum_{i=1}^{m} \lambda_{i} h_{i}(\bar{x})$$

where the variables $\lambda_{i}$ are known as the Lagrange multipliers. Solutions of the optimization problem expressed by Equation $2.20$ can be obtained through the necessary conditions for the Lagrangian function:
$$\frac{\partial L}{\partial \bar{x}}=\nabla z(\bar{x})+\sum_{i=1}^{m} \lambda_{i} \nabla h_{i}(\bar{x})=0$$
From the necessary conditions, a system of equations of $\mathbf{M} \times \mathbf{M}$ is obtained. If a solution for the system can be obtained, that solution will represent a stationary point. To ensure the obtained solution is at least a local minimum, $H\left[L^{}\right]$ should be positive definite or positive semidefinite. On the other hand, if $H\left[L^{}\right]$ is negative definite or negative semidefinite, the solution is at least a local maximum. If the Hessian matrix is indefinite, the solution is a saddle point.

## 数学代写|随机过程统计代写Stochastic process statistics代考|Continuity and Convexity

F(X¯0)=林X¯→X¯0F(X¯)

X¯0=[00]吨，函数增长，在该点达到无穷大X¯0. 因此，该函数未定义在X¯0, 并且是不连续的。

## 数学代写|随机过程统计代写Stochastic process statistics代考|Unconstrained Optimization

∇F(X¯)=0

1. 选择一个初始解决方案，X¯0=[X10X20…Xn0]吨，并评估目标函数X¯0.
2. 确定目标函数的梯度，∇F(X¯)=[∂F∂X1∂F∂X2 ∂F∂Xn]吨，以及，如果需要，目标函数的 Hessian 矩阵。

## 数学代写|随机过程统计代写Stochastic process statistics代考|Equality-Constrained Optimization

优化 大号=和(X¯)+∑一世=1米λ一世H一世(X¯)

∂大号∂X¯=∇和(X¯)+∑一世=1米λ一世∇H一世(X¯)=0

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## MATLAB代写

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