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## 数学代写|最优化作业代写optimization theory代考|Conjugate Direction, Variable Metric

As a motivation we consider the minimization of a function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ having the following special form:
$$f\left(x_1, \ldots, x_n\right)=\sum_{i=1}^n f_i\left(x_i\right)$$
Note that each function $f_i$ in (10.1.1) is a function of only one variable. Then, it is easily seen that $\bar{x} \in \mathbb{R}^n$ minimizes $f$ iff the component $\bar{x}i$ minimizes $f_i, i=1, \ldots, n$. Consequently, the minimization of $f$ can be achieved by successively minimizing along the coordinate axes. Next, consider a quadratic function $f$ : $$f(x)=\frac{1}{2} x^{\top} A x+b^{\top} x$$ where $A$ is a symmetric, positive definite $(n, n)$-matrix. Let $v_1, \ldots, v_n \in \mathbb{R}^n$ be a basis for $\mathbb{R}^n$. Putting $x=\sum{i=1}^n \nu_i v_i$, we obtain:
\begin{aligned} f(x)=\varphi(\nu)=\sum_{i=1}^n \underbrace{\left(b^{\top} v_i\right)}{\beta_i} \nu_i & +\frac{1}{2} \sum{i=1}^n \underbrace{\left(v_i^{\top} A v_i\right)}{\alpha_i} \nu_i^2 \ & +\frac{1}{2} \sum{\substack{i, j \ i \neq j}}\left(v_i^{\top} A v_j\right) \nu_i \nu_j . \end{aligned}
If $v_i^{\top} A v_j=0, i \neq j$, then it follows:
$$f(x)=\varphi(\nu)=\sum_{i=1}^n\left(\frac{1}{2} \alpha_i \nu_i^2+\beta_i \nu_i\right)=: \sum_{i=1}^n \varphi_i\left(\nu_i\right),$$
and, hence, $\varphi(\nu)$ is a function of the type (10.1.1). This gives rise to (or motivates) the following definition.

## 数学代写|最优化作业代写optimization theory代考|Conjugate Gradient-, DFP-, BFGS-Method

For practical applications of the idea of conjugate directions it is important to construct algorithms that automatically generate new conjugate directions from the data known at a specific step in the optimization procedure. This will be studied in the present section.

Lemma 10.2.1 According to Algorithm $\mathcal{A}$, let $x^1, x^2, \ldots, x^{\ell}, \ell \leq n$, be generated, where $v_1, v_2, \ldots, v_{\ell} \in \mathbb{R}^n \backslash{0}$ are pairwise conjugate with respect to A. Then, it holds:
$$D f\left(x^{\ell}\right) v_i=0, \quad i=1,2, \ldots, \ell .$$

Proof. Obviously, we have $x^{\ell}=x^r+\sum_{j=r+1}^{\ell} \lambda_j v_j$. It follows:
$$D f\left(x^{\ell}\right)-D f\left(x^r\right)=\left(x^{\ell}-x^r\right)^{\top} A=\sum_{j=r+1}^{\ell} \lambda_j v_j^{\top} A .$$
Let $r \geq 1$. Since $x^r$ minimizes $f\left(x^{r-1}+\lambda v_r\right)$, we have $D f\left(x^r\right) v_r=0$. Hence, it follows for $1 \leq r<\ell$ :
$$D f\left(x^{\ell}\right) v_r=\left[D f\left(x^{\ell}\right)-D f\left(x^r\right)\right] v_r=\sum_{j=r+1}^{\ell} \lambda_j\left(v_j^{\top} A v_r\right)=0 .$$
Finally, the equation $D f\left(x^{\ell}\right) v_{\ell}=0$ follows from the fact that $x^{\ell}$ minimizes the function $f\left(x^{\ell-1}+\lambda v_{\ell}\right)$.

## 数学代写|最优化作业代写optimization theory代考|Conjugate Direction, Variable Metric

$$f\left(x_1, \ldots, x_n\right)=\sum_{i=1}^n f_i\left(x_i\right)$$

\begin{aligned} f(x)=\varphi(\nu)=\sum_{i=1}^n \underbrace{\left(b^{\top} v_i\right)}{\beta_i} \nu_i & +\frac{1}{2} \sum{i=1}^n \underbrace{\left(v_i^{\top} A v_i\right)}{\alpha_i} \nu_i^2 \ & +\frac{1}{2} \sum{\substack{i, j \ i \neq j}}\left(v_i^{\top} A v_j\right) \nu_i \nu_j . \end{aligned}

$$f(x)=\varphi(\nu)=\sum_{i=1}^n\left(\frac{1}{2} \alpha_i \nu_i^2+\beta_i \nu_i\right)=: \sum_{i=1}^n \varphi_i\left(\nu_i\right),$$

## 数学代写|最优化作业代写optimization theory代考|Conjugate Gradient-, DFP-, BFGS-Method

$$D f\left(x^{\ell}\right) v_i=0, \quad i=1,2, \ldots, \ell .$$

$$D f\left(x^{\ell}\right)-D f\left(x^r\right)=\left(x^{\ell}-x^r\right)^{\top} A=\sum_{j=r+1}^{\ell} \lambda_j v_j^{\top} A .$$

$$D f\left(x^{\ell}\right) v_r=\left[D f\left(x^{\ell}\right)-D f\left(x^r\right)\right] v_r=\sum_{j=r+1}^{\ell} \lambda_j\left(v_j^{\top} A v_r\right)=0 .$$

## 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。

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## 数学代写|最优化作业代写optimization theory代考|Geometric Interpretation of Karmarkar’s Algorithm

Recall Karmarkar’s Algorithm and suppose that the iterate $x^k \in \stackrel{o}{\Sigma}^{\Sigma}$ has been generated.

The simplex $\Sigma$ will be transformed into itself and the point $x^k$ is shifted into the barycenter, all by means of the transformation $T_k$ :
$$T_k\left(x_1, \ldots, x_n\right)=\frac{n}{\sum_i\left(x_i / z_i\right)}\left(x_1 / z_1, \ldots, x_n / z_n\right), \text { with } z:=x^k$$

Note that $T_k$ maps each stratum of $\Sigma$ into itself; in particular, all vertices of $\Sigma$ are fixed points of $T_k$ (exercise).

The inverse of $T_k$ is easily computed (instead of dividing by $z_i$ we now have to multiply by $z_i$ ):
$$T_k^{-1}\left(y_1, \ldots, y_n\right)=\frac{n}{\sum_i y_i z_i}\left(y_1 z_1, \ldots, y_n z_n\right), \text { with } z:=x^k .$$
The equation $A x=0$ becomes in the $y$-variables: $A T_k^{-1}(y)=0$. From (8.2.2) it follows (after multiplication with $\sum_i y_i z_i / n$ ):
$$A x=0 \text { iff } A D_k y=0,$$
with $D_k$ as defined in (8.1.4).
The function $\left(x_1, \ldots, x_n\right) \longmapsto x_1$ to be minimized, becomes a nonlinear function in the $y$-coordinates:
$$\left(y_1, \ldots, y_n\right) \longmapsto y_1 \cdot\left(\frac{n z_1}{\sum_i y_i z_i}\right) \text {, with } z:=x^k$$

## 数学代写|最优化作业代写optimization theory代考|Proof of Theorem 8.1.2 (Polynomiality)

In order to prove Theorem 8.1.2 we have to estimate how much the objective function decreases in each step of the algorithm. Recall that the linear function $\left(x_1, \ldots, x_n\right) \longmapsto x_1$ transforms awkwardly under the transformation $T_k$

(cf. (8.2.4)). Therefore, a comparable function $f$ is chosen that transforms nicely under $T_k$ :
$$\left.f\left(x_1, \ldots, x_n\right)=x_1^n / x_1 \cdot x_2 \cdots x_n \quad \text { (homogeneous of degree } 0\right) .$$
The function $f$ is well defined on $\stackrel{o}{\Sigma}$. On $\stackrel{o}{\Sigma}$ we obviously have $x_1 \cdot x_2 \cdots x_n \leq 1$, and, consequently,
$$x_1^n \leq f(x), \quad x \in \stackrel{o}{\Sigma}$$
For $f$ the following interesting transformation formula holds:
Lemma 8.3.1 For $x, y \in \stackrel{o}{\Sigma}$ it holds:
$$\frac{f\left(T_k(x)\right)}{f\left(T_k(y)\right)}=\frac{f(x)}{f(y)} .$$
Proof. (Exercise)
Note that $f(1,1, \ldots, 1)=1$, and, hence,
$$f\left(x^{k+1}\right) / f\left(x^k\right)=f(\widetilde{x})$$
where $\tilde{x}$ is the minimizer in Step 1 of Karmarkar’s Algorithm in the $(k+1)$-th iteration. If $\alpha=\frac{1}{2}$, then we will show:
$$f(\widetilde{x}) \leq 2 e^{-1}$$

## 数学代写|最优化作业代写optimization theory代考|Geometric Interpretation of Karmarkar’s Algorithm

$$T_k\left(x_1, \ldots, x_n\right)=\frac{n}{\sum_i\left(x_i / z_i\right)}\left(x_1 / z_1, \ldots, x_n / z_n\right), \text { with } z:=x^k$$

$T_k$的倒数很容易计算(我们现在要乘以$z_i$而不是除以$z_i$):
$$T_k^{-1}\left(y_1, \ldots, y_n\right)=\frac{n}{\sum_i y_i z_i}\left(y_1 z_1, \ldots, y_n z_n\right), \text { with } z:=x^k .$$

$$A x=0 \text { iff } A D_k y=0,$$

$$\left(y_1, \ldots, y_n\right) \longmapsto y_1 \cdot\left(\frac{n z_1}{\sum_i y_i z_i}\right) \text {, with } z:=x^k$$

## 数学代写|最优化作业代写optimization theory代考|Proof of Theorem 8.1.2 (Polynomiality)

(参见(8.2.4))。因此，选择一个类似的函数$f$，它可以很好地在$T_k$下进行转换:
$$\left.f\left(x_1, \ldots, x_n\right)=x_1^n / x_1 \cdot x_2 \cdots x_n \quad \text { (homogeneous of degree } 0\right) .$$

$$x_1^n \leq f(x), \quad x \in \stackrel{o}{\Sigma}$$

$$\frac{f\left(T_k(x)\right)}{f\left(T_k(y)\right)}=\frac{f(x)}{f(y)} .$$

$$f\left(x^{k+1}\right) / f\left(x^k\right)=f(\widetilde{x})$$

$$f(\widetilde{x}) \leq 2 e^{-1}$$

## 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。

Posted on Categories:Optimization Theory, 优化理论, 数学代写, 最优化

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## 数学代写|最优化作业代写optimization theory代考|Parametric Aspects: The Unconstrained Case

In this section we study the dependence of local minima and their corresponding functional values on additional parameters. The appearance of parameters may represent perturbations of an optimization problem. The crucial tools in such investigations are theorems on implicit functions. For a basic reference see [65].

We start with unconstrained optimization problems. Let $f \in C^2\left(\mathbb{R}^n \times\right.$ $\left.\mathbb{R}^r, \mathbb{R}\right)$. The general point $z \in \mathbb{R}^n \times \mathbb{R}^r$ will be represented as $z=(x, t)$, where $x$ is the state variable and where $t$ plays the role of a parameter. In this way we may regard $f$ as being an $r$-parametric family of functions of $n$ variables. Let $\bar{x} \in \mathbb{R}^n$ be a local minimum for $f(\cdot, \bar{t})$. The necessary optimality condition of first order reads
$$D_x f(\bar{x}, \bar{t})=0$$
where $D_x f$ denotes the row vector of first partial derivatives with respect to $x$.

Formula (3.1.1) represents $n$ equations with $n+r$ variables. In case that the Jacobian matrix $D D_x^{\top} f(\bar{x}, \bar{t})$, an $(n, n+r)$-matrix, has full rank $(=n)$, in virtue of the implicit function theorem we can choose $n$ variables such that the equation $D_x f=0$ defines these variables as an implicit function of the remaining $r$ variables. With respect to the chosen $n$ variables the corresponding $(n, n)$-submatrix of $D D_x^{\top} f(\bar{x}, \bar{t})$ should be nonsingular. For example, let $\bar{x} \in \mathbb{R}^n$ be a local minimum for $f(\cdot, \bar{t})$ which is nondegenerate, i.e. $D_x^2 f(\bar{x}, \bar{t})$ is nonsingular (and, hence, positive definite). Then, the implicit function theorem yields the existence of open neighborhoods $\mathcal{O}, \mathcal{V}$ of $(\bar{x}, \bar{t}), \bar{t}$, and a mapping $x(\cdot) \in C^1\left(\mathcal{V}, \mathbb{R}^n\right)$ such that for all $(x, t) \in \mathcal{O}$ we have:
$$D_x f(x, t)=0 \text { iff } x=x(t) .$$

## 数学代写|最优化作业代写optimization theory代考|Parametric aspects: The Constrained Case

In this section we take constraints into account and again we define the concept of a nondegenerate local minimum. This yields a (stable) system of nonlinear equations which enables us to study the sensitivity of a local minimum with regard to data pertubations.

Let $k \geq 2$ and $f, h_i, g_j \in C^k\left(\mathbb{R}^n \times \mathbb{R}^r, \mathbb{R}\right), i \in I, j \in J$ and $|I|+|J|<\infty$. For each $t \in \mathbb{R}^r$ we have $f(\cdot, t)$ as an objective function and $M(t)$ as a feasible set, where
$$M(t)=\left{x \in \mathbb{R}^n \mid h_i(x, t)=0, i \in I, g_j(x, t) \geq 0, j \in J\right} .$$
Definition 3.2.1 Let $f, h_i, g_j$ be as above. A (feasible) point $\bar{x} \in M(\bar{t})$ is called nondegenerate local minimum for $f(\cdot, \bar{t}){\mid M(\bar{t})}$ if the following conditions are satisfied: (1) LICQ holds at $\bar{x}$. (2) The point $\bar{x}$ is a critical point for $f(\cdot, \bar{t}){\mid M(\bar{t})}$.
Let $\bar{\lambda}i, \bar{\mu}_j, i \in I, j \in J_0(\bar{x}, \bar{t}):=\left{j \in J \mid g_j(\bar{x}, \bar{t})=0\right}$ be the corresponding Lagrange multipliers and $L$ the Lagrange function, i.e. \begin{aligned} D_x f & =\sum{i \in I} \bar{\lambda}i D_x h_i+\sum{j \in J_0(\bar{x}, \bar{t})} \bar{\mu}j D_x g{j \mid(\bar{x}, \bar{t})} \ L(x, t) & =f-\sum_{i \in I} \bar{\lambda}i h_i-\sum{j \in J_0(\bar{x}, \bar{t})} \bar{\mu}j g{j \mid(x, t)} \end{aligned}
(3) $\bar{\mu}j>0, j \in J_0(\bar{x}, \bar{t})$. (4) $D_x^2 L(\bar{x}, \bar{t})$ is positive definite on $T{\bar{x}} M(\bar{t})$, where (cf. (2.1.6))

## 数学代写|最优化作业代写optimization theory代考|Parametric Aspects: The Unconstrained Case

$$M(t)=\left{x \in \mathbb{R}^n \mid h_i(x, t)=0, i \in I, g_j(x, t) \geq 0, j \in J\right} .$$
3.2.1如上所述$f, h_i, g_j$。如果满足以下条件，则称为$f(\cdot, \bar{t}){\mid M(\bar{t})}$的(可行)点$\bar{x} \in M(\bar{t})$为非退化局部最小值:(1)LICQ保持在$\bar{x}$。(2) $\bar{x}$点是$f(\cdot, \bar{t}){\mid M(\bar{t})}$的临界点。

(3) $\bar{\mu}j>0, j \in J_0(\bar{x}, \bar{t})$。(4) $D_x^2 L(\bar{x}, \bar{t})$在$T{\bar{x}} M(\bar{t})$上是肯定的，其中(cf. (2.1.6))

## 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。

Posted on Categories:Optimization Theory, 优化理论, 数学代写, 最优化

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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## 数学代写|最优化作业代写optimization theory代考|CONTINUOUS LINEAR REGULATOR PROBLEMS

Problems like Example 3.11-1 with linear plant dynamics and quadratic performance criteria are referred to as linear regulator problems. In this section we investigate the use of the Hamilton-Jacobi-Bellman equation as a means of solving the general form of the continuous linear regulator problem.t
The process to be controlled is described by the state equations
$$\dot{\mathbf{x}}(t)=\mathbf{A}(t) \mathbf{x}(t)+\mathbf{B}(t) \mathbf{u}(t)$$
and the performance measure to be minimized is
$$J=\frac{1}{2} \mathbf{x}^T\left(t_f\right) \mathbf{H} \mathbf{x}\left(t_f\right)+\int_{t_0}^{t_f} \frac{1}{2}\left[\mathbf{x}^T(t) \mathbf{Q}(t) \mathbf{x}(t)+\mathbf{u}^T(t) \mathbf{R}(t) \mathbf{u}(t)\right] d t$$
$\mathbf{H}$ and $\mathbf{Q}$ are real symmetric positive semi-definite matrices, $\mathbf{R}$ is a real, symmetric positive definite matrix, the initial time $t_0$ and the final time $t_f$ are specified, and $\mathbf{u}(t)$ and $\mathbf{x}(t)$ are not constrained by any boundaries.

To use the Hamilton-Jacobi-Bellman equation, we first form the Hamiltonian:
$$\begin{array}{cc} \mathscr{H}\left(\mathbf{x}(t), \mathbf{u}(t), J_{\mathbf{x}}^, t\right)= & \frac{1}{2} \mathbf{x}^T(t) \mathbf{Q}(t) \mathbf{x}(t)+\frac{1}{2} \mathbf{u}^T(t) \mathbf{R}(t) \mathbf{u}(t)+J_{\mathbf{x}}^{ T}(\mathbf{x}(t), t) \ & \cdot[\mathbf{A}(t) \mathbf{x}(t)+\mathbf{B}(t) \mathbf{u}(t)] \end{array}$$

## 数学代写|最优化作业代写optimization theory代考|THE HAMILTON-JACOBI-BELLMAN EQUATION-SOME OBSERVATIONS

We have derived the Hamilton-Jacobi-Bellman equation and used it to solve two examples of the linear regulator type. Let us now make some observations concerning the H-J-B functional equation.
Boundary Conditions
In our derivation we have assumed that $t_f$ is fixed; however, the results still apply if $t_f$ is free. For example, if $S$ represents some hypersurface in the state space and $t_f$ is defined as the first time the system’s trajectory intersects $S$, then the boundary condition is
$$J^\left(\mathbf{x}\left(t_f\right), t_f\right)=h\left(\mathbf{x}\left(t_f\right), t_f\right)$$ A Necessary Condition The results we have obtained represent a necessary condition for optimality; that is, the minimum cost function $J^(\mathbf{x}(t), t)$ must satisfy the Hamilton-Jacobi-Bellman equation.

A Sufficient Condition
Although we have not derived it here, it is also true that if there is a cost function $J^{\prime}(\mathbf{x}(t), t)$ that satisfies the Hamilton-Jacobi-Bellman equation, then $J^{\prime}$ is the minimum cost function; i.e.,
$$J^{\prime}(\mathbf{x}(t), t)=J^*(\mathbf{x}(t), t)$$
Rigorous proofs of the necessary and sufficient conditions embodied in the H-J-B equation are given in [K-5] and also in [A-2], which contains several examples.
Solution of the Hamilton-Jacobi-Bellman Equation
In both of the examples that we considered, a solution was obtained by guessing a form for the minimum cost function. Unfortunately, we are normally unable to find a solution so easily. In general, the H-J-B equation must be solved by numerical techniques-see [F-1], for example. Actually, a numerical solution involves some sort of a discrete approximation to the exact optimization relationship [Eq. (3.11-10)]; alternatively, by solving the recurrence relation [Eq. (3.7-18)] we obtain the exact solution to a discrete approximation of the Hamilton-Jacobi-Bellman functional equation.

## 数学代写|最优化作业代写optimization theory代考|CONTINUOUS LINEAR REGULATOR PROBLEMS

$$\dot{\mathbf{x}}(t)=\mathbf{A}(t) \mathbf{x}(t)+\mathbf{B}(t) \mathbf{u}(t)$$

$$J=\frac{1}{2} \mathbf{x}^T\left(t_f\right) \mathbf{H} \mathbf{x}\left(t_f\right)+\int_{t_0}^{t_f} \frac{1}{2}\left[\mathbf{x}^T(t) \mathbf{Q}(t) \mathbf{x}(t)+\mathbf{u}^T(t) \mathbf{R}(t) \mathbf{u}(t)\right] d t$$
$\mathbf{H}$和$\mathbf{Q}$是实对称正半定矩阵，$\mathbf{R}$是实对称正定矩阵，指定了初始时间$t_0$和最终时间$t_f$, $\mathbf{u}(t)$和$\mathbf{x}(t)$不受任何边界约束。

$$\begin{array}{cc} \mathscr{H}\left(\mathbf{x}(t), \mathbf{u}(t), J_{\mathbf{x}}^, t\right)= & \frac{1}{2} \mathbf{x}^T(t) \mathbf{Q}(t) \mathbf{x}(t)+\frac{1}{2} \mathbf{u}^T(t) \mathbf{R}(t) \mathbf{u}(t)+J_{\mathbf{x}}^{ T}(\mathbf{x}(t), t) \ & \cdot[\mathbf{A}(t) \mathbf{x}(t)+\mathbf{B}(t) \mathbf{u}(t)] \end{array}$$

## 数学代写|最优化作业代写optimization theory代考|THE HAMILTON-JACOBI-BELLMAN EQUATION-SOME OBSERVATIONS

$$J^\left(\mathbf{x}\left(t_f\right), t_f\right)=h\left(\mathbf{x}\left(t_f\right), t_f\right)$$必要条件我们得到的结果是最优性的必要条件;即最小代价函数$J^(\mathbf{x}(t), t)$必须满足Hamilton-Jacobi-Bellman方程。

$$J^{\prime}(\mathbf{x}(t), t)=J^*(\mathbf{x}(t), t)$$
[K-5]和[A-2]给出了H-J-B方程所包含的充分必要条件的严格证明，并给出了几个例子。
Hamilton-Jacobi-Bellman方程的解

## 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。

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## 数学代写|最优化作业代写optimization theory代考APPLICATION OF THE PRINCIPLE OF OPTIMALITY TO DECISION-MAKING

The following example illustrates the procedure for making a single optimal decision with the aid of the principle of optimality.

Consider a process whose current state is $b$. The paths resulting from all allowable decisions at $b$ are shown in Fig. 3-2(a). The optimal paths from $c, d$, and $e$ to the terminal point $f$ are shown in Fig. 3-2(b). The principle of

optimality implies that if $b-c$ is the initial segment of the optimal path from $b$ to $f$, then $c-f$ is the terminal segment of this optimal path. The same reasoning applied to initial segments $b-d$ and $b-e$ indicates that the paths in Fig. 3-2(c) are the only candidates for the optimal trajectory from $b$ to $f$. The optimal trajectory that starts at $b$ is found by comparing
\begin{aligned} & C_{b c f}^=J_{b c}+J_{c f}^ \ & C_{b d f}^=J_{b d}+J_{d f}^ \ & C_{b e f}^=J_{b e}+J_{e f}^ . \end{aligned}
The minimum of these costs must be the one associated with the optimal decision at point $b$.

Dynamic programming is a computational technique which extends the above decision-making concept to sequences of decisions which together define an optimal policy and trajectory. The optimal routing problem in the next section illustrates the procedure.

## 数学代写|最优化作业代写optimization theory代考|DYNAMIC PROGRAMMING APPLIED TO A ROUTING PROBLEM

A motorist wishes to know how to minimize the cost of reaching some destination $h$ from his current location. He can only travel (one-way as indicated) on the streets shown on his map (Fig. 3-3), and at the intersectionto-intersection costs given.

Instead of trying all allowable paths leading from each intersection to $h$ and selecting the one with lowest cost (an exhaustive search), consider the application of the principle of optimality. In this problem, “state” refers to the intersection and a “decision” is the choice of heading (control) elected by the driver when he leaves an intersection.

Suppose the motorist is at $c$; from there he can go, only to $d$ or $f$, and then on to $h$. Let $J_{c d}$ denote the cost of moving from $c$ to $d$ and $J_{c f}$ the cost from $c$ to $f$. Assume that the motorist already knows the minimum costs, $J_{d h}^$ and $J_{f h}^$, to reach the final destination $h$ from $d$ and $f$. (In this example, $J_{d h}^=10$ and $J_{f h}^=5$.) Then the minimum cost $J_{c h}^$ to reach $h$ from $c$ is the smaller of $$C_{c d h}^=J_{c d}+J_{d h}^=\text { minimum cost to reach } h \text { from } c \text { via } d$$ and $$C_{c f h}^=J_{c f}+J_{f h}^=\text { minimum cost to reach } h \text { from } c \text { via } f .$$ Thus, \begin{aligned} J_{c h}^ & =\min \left{C_{c d h}^, C_{c f h}^\right} \ & =\min {15,8} \ & =8 \end{aligned}
and the optimal decision at $c$ is to go to $f$.

## 数学代写|最优化作业代写optimization theory代考APPLICATION OF THE PRINCIPLE OF OPTIMALITY TO DECISION-MAKING

\begin{aligned} & C_{b c f}^=J_{b c}+J_{c f}^ \ & C_{b d f}^=J_{b d}+J_{d f}^ \ & C_{b e f}^=J_{b e}+J_{e f}^ . \end{aligned}

## 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。

Posted on Categories:Optimization Theory, 优化理论, 数学代写, 最优化

## avatest™帮您通过考试

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## 数学代写|最优化作业代写optimization theory代考|CONCLUDING REMARKS

In control system design, the ultimate objective is to obtain a controller that will cause a system to perform in a desirable manner. Usually, other factors, such as weight, volume, cost, and reliability also influence the controller design, and compromises between performance requirements and implementation considerations must be made. Classical design procedures are best suited for linear, single-input, single-output systems with zero initial conditions. Using simulation, mathematical analysis, or graphical methods, the designer evaluates the effects of inserting various physical devices into the system. By trial and error either an acceptable controller design is obtained, or the designer concludes that the performance requirements cannot be satisfied.

Many complex aerospace problems that are not amenable to classical techniques have been solved by using optimal control theory. However, we are forced to admit that optimal control theory does not, at the present time, constitute a generally applicable procedure for the design of simple controllers. The optimal control law, if it can be obtained, usually requires a digital computer for implementation (an important exception is the linear regulator problem discussed in Section 5.2), and all of the states must be available for feedback to the controller. These limitations may preclude implementation of the optimal control law; however, the theory of optimal control is still useful, because

1. Knowing the optimal control law may provide insight helpful in designing a suboptimal, but easily implemented controller.
2. The optimal control law provides a standard for evaluating proposed suboptimal designs. In other words, by knowing the optimal control law we have a quantitative measure of performance degradation caused by using a suboptimal controller.

## 数学代写|最优化作业代写optimization theory代考|PERFORMANCE MEASURES FOR OPTIMAL CONTROL PROBLEMS

The “optimal control problem” is to find a control $\mathbf{u}^* \in U$ which causes the system
$$\dot{\mathbf{x}}(t)=\mathbf{a}(\mathbf{x}(t), \mathbf{u}(t), t)$$

to follow a trajectory $\mathbf{x}^* \in X$ that minimizes the performance measure
$$J=h\left(\mathbf{x}\left(t_f\right), t_f\right)+\int_{t_0}^{t_s} g(\mathbf{x}(t), \mathbf{u}(t), t) d t .$$
Let us now discuss some typical control problems to provide some physical motivation for the selection of a performance measure.
Minimum-Time Problems
Problem: To transfer a system from an arbitrary initial state $\mathbf{x}\left(t_0\right)=\mathbf{x}0$ to a specified target set $S$ in minimum time. The performance measure to be minimized is \begin{aligned} J & =t_f-t_0 \ & =\int{t_0}^{t s} d t, \end{aligned}
with $t_f$ the first instant of time when $\mathbf{x}(t)$ and $S$ intersect. The automobile example discussed in Section 1.1 is a minimum-time problem. Other typical examples are the interception of attacking aircraft and missiles, and the slewing mode operation of a radar, or gun system.

## 数学代写|最优化作业代写optimization theory代考|PERFORMANCE MEASURES FOR OPTIMAL CONTROL PROBLEMS

“最优控制问题”是找到一个控制$\mathbf{u}^* \in U$，使系统
$$\dot{\mathbf{x}}(t)=\mathbf{a}(\mathbf{x}(t), \mathbf{u}(t), t)$$

$$J=h\left(\mathbf{x}\left(t_f\right), t_f\right)+\int_{t_0}^{t_s} g(\mathbf{x}(t), \mathbf{u}(t), t) d t .$$

## 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。