## 数学代写|最优化理论作业代写optimization theory代考|CSE535

statistics-lab™ 为您的留学生涯保驾护航 在代写最优化理论optimization theory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。我们的专家在代写最优化理论optimization theory代写方面经验极为丰富，各种代写最优化理论optimization theory相关的作业也就用不着说。

## 数学代写|最优化理论作业代写optimization theory代考|Running time; the class P

Let $A$ denote a finite alphabet.
Definition 19.1.1 (Running time) Let $M$ be a Turing machine and $w \in$ $A^*$ an input. The running time $T_M(w)$ of $M$ applied to $w$ is the number of applications of the transition function of $M$ until $M$ enters a final state (where $M$ is starting from the configuration $q_0 w$ ). If $M$ does not halt we define $T_M(w):=\infty$.

Definition 19.1.2 (Time boundedness) Let $t: \mathbb{N} \rightarrow \mathbb{N}$ and $M$ be a Turing machine. We say that $M$ is $t(n)$ time bounded if for all $w \in A^*$ we have $T_M(w) \leq t(|w|)$. That is, for every input $w$ machine $M$ halts on $w$ in a number of steps which is bounded by the value of $t$ on the length of $w$.
Remark 19.1.3 Similarly the space used by a Turing machine as well as space-bounded computations can be defined (see [11]). This plays a major role in complexity theory. However, here we restrict ourselves to running time considerations only (however, see Exercise 19.4.7).

Exercise 19.1.4 Determine the running time of the Turing machines in Example 18.2.6 and Exercise 18.2.7.

It has turned out that a good formalization of what an efficient algorithm should be is the requirement that its running time is bounded by a polynomial. This idea leads to one of the most important notions in complexity theory.

Definition 19.1.5 (Polynomial time computability) Let two finite alphabets $\Sigma_1$ and $\Sigma_2$ be given. A function $f: \Sigma_1^* \rightarrow \Sigma_2^$ is computable in polynomial time if there exists a Turing machine $M$ over $\Sigma \supseteq \Sigma_1 \cup \Sigma_2$ and a polynomial $p$ such that for all $w \in \Sigma_1^$ we have $\phi_M(w)=f(w)$ and $T_M(w) \leq p(|w|)$

## 数学代写|最优化理论作业代写optimization theory代考|Some important decision problems

We shall now introduce a first group of decision problems which we want to study with respect to the complexity of algorithms solving them. In each case a size function is added.
Definition 19.2.1 (Decision Problems I)
a) Hamiltonian Circuit. INSTANCE: A graph $G=(V, E)$
QUESTION: Does $G$ contain a Hamiltonian circuit (cf. Chapter 14)?
SIZE: $|V|$
b) Maximum Matching. INSTANCE: A graph $G=(V, E)$ and a natural number $k \leq \frac{|V|}{2}$.
QUESTION: Does $G$ contain a matching of cardinality $k$ ?
SIZE: $|V|$ (note that $k<|V|$ )
c) Traveling Salesman. INSTANCE: A graph $G=(V, E)$, where the vertices are $V=\left{v_1, \ldots, v_n\right}$, together with weights $d_{i j} \in \mathbb{Z}{+}$indicating the distance of $v_i$ to $v_j$; a bound $B \in \mathbb{Z}{+}$satisfying $B \leq \sum_{i, j} d_{i j}$.

QUESTION: Is there a permutation $\pi \in S_n$ such that
\begin{aligned} & \sum_{i=1}^{n-1} d_{\pi(i), \pi(i+1)}+d_{\pi(n), \pi(1)} \leq B ? \ & \text { SIZE: } n+\max {i, j}\left{\left\lceil\log _2\left(d{i j}\right)+1\right\rceil\right} \end{aligned}
d) Linear Programming. INSTANCE: Numbers $n, m \in \mathbb{N}$; an integer matrix $A:=\left[a_{i j}\right] \in \mathbb{Z}^{m \times n}$, vectors $c \in \mathbb{Z}^n, b \in \mathbb{Z}^m$ and a $B \in \mathbb{Z}$.
QUESTION: Is there an $x \in \mathbb{Q}^n$ such that $A \cdot x \leq b$ and $c^T \cdot x \leq B$ ?
\begin{aligned} & \text { SIZE: } \sum_{i=1}^m \sum_{j=1}^n\left(\left\lceil\log 2\left(\left|a{i j}\right|+1\right)\right\rceil+1\right)+\sum_{i=1}^m\left\lceil\log 2\left(\left|b_i\right|+1\right)\right\rceil+ \ & \sum{i=1}^n\left\lceil\log 2\left(\left|c_i\right|+1\right)\right\rceil+\left\lceil\log _2(|B|+1)\right\rceil \end{aligned} e) Quadratic Programming. INSTANCE: A natural number $n \in \mathbb{N}$; a symmetric integer matrix $A:=\left[a{i j}\right] \in \mathbb{Z}^{n \times n}$, a vector $b \in \mathbb{Z}^n$ and a $c \in \mathbb{Z}$.
QUESTION: Is there an $x \in \mathbb{H}^n$ (i.e. $x \geq 0$ ) such that $\frac{1}{2} \cdot x^T \cdot A \cdot x+$ $b^T \cdot x+c \leq 0$ ?
SIZE: $n \cdot m+\max$ {bit-length of entries in $A, b$, and $c$ }
The problem could also be defined with more general linear side constraints and an arbitrary bound on the function value. For our purposes the above definition is sufficiently general.
f) Integer Linear Programming. INSTANCE: Numbers $n, m \in \mathbb{N}$; an integer matrix $A:=\left[a_{i j}\right] \in \mathbb{Z}^{m \times n}$ and a vector $b \in \mathbb{Z}^m$.
QUESTION: Is there an $x \in \mathbb{Z}^n$ such that $A \cdot x \leq b$.
SIZE: $n \cdot m+\max {$ bit-length of entries in $A$ and $b}$
g) 0-1 Linear Programming. INSTANCE: Numbers $n, m \in \mathbb{N}$; an integer matrix $A:=\left[a_{i j}\right] \in \mathbb{Z}^{m \times n}$ and a vector $b \in \mathbb{Z}^m$.
QUESTION: Is there an $x \in{0,1}^n$ such that $A \cdot x \leq b$.
SIZE: $n \cdot m+\max {$ bit-length of entries in $A$ and $b}$

## 数学代写|最优化理论作业代写optimization theory代考|Running time; the class P

SIZE: $n \cdot m+\max {$$A和b}中表项的位长 g) 0-1线性规划。实例:数字n, m \in \mathbb{N};一个整数矩阵A:=\left[a_{i j}\right] \in \mathbb{Z}^{m \times n}和一个矢量b \in \mathbb{Z}^m。 问题:有没有x \in{0,1}^n这样的A \cdot x \leq b。 SIZE: n \cdot m+\max {$$A$和中条目的位长度 $b}$

## 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|最优化理论作业代写optimization theory代考|CAAM560

statistics-lab™ 为您的留学生涯保驾护航 在代写最优化理论optimization theory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。我们的专家在代写最优化理论optimization theory代写方面经验极为丰富，各种代写最优化理论optimization theory相关的作业也就用不着说。

## 数学代写|最优化理论作业代写optimization theory代考|Integral polyhedra

A polyhedron $M$ is called rational if it can be defined as $M=\left{x \in \mathbb{R}^n\right.$ : $A x \leq b}$ for some matrix $A$ and vector $b$ with rational entries. $M$ is called integral if it is the convex hull of all its integral points:
$$M=M_{\text {int }}:=\mathcal{C}\left(x \mid x \in M \cap \mathbb{Z}^n\right) .$$
Lemma 17.3.1 Let $M \subseteq \mathbb{R}^n$ be a rational polyhedron. Then $\Sigma \cap \mathbb{Q}^n$ is dense in each stratum $\Sigma$ of $M$.

Proof. Assume $\Sigma=\Sigma(\bar{x})$ is any stratum of $M={x \mid A x \leq b}, \quad A \in$ $\mathbb{Q}^{n \times n}, \quad b \in \mathbb{Q}^n$. Then $\Sigma=\left{x \in M \mid J_0(x)=J_0(\bar{x})\right}=\left{x \mid \overline{a_j^T}=b_j(j \in\right.$ $J_0(\bar{x}), a_j^T x<b_j\left(j \notin J_0(\bar{x})\right}$. But it is clear by the usual Gaussian elimination procedure that $\mathbb{Q}^n$ is dense in the affine subspace $\left{x \in \mathbb{R}^n: a_j^T x=b_j(j \in\right.$ $\left.\left.J_0(\bar{x})\right)\right}=: W$. Now $\Sigma$ is relatively open in $W$ and the result follows.

Recall from section 5.3 that each polyhedron $M$ can be written as
$$M=\mathcal{C}\left(a_1, \ldots, a_m\right)+\mathcal{K}\left(b_1, \ldots, b_r\right)$$
with suitable points $a_i, b_j, 1 \leq i \leq m, 1 \leq j \leq r$. By using the previous lemma and reconsidering the proofs in section 5.3, it is easy to see that $M$ is rational if and only if the $a_i$ and $b_j$ in the above representation can be chosen as rational vectors. Since $\mathcal{K}\left(b_1, \ldots, b_r\right)$ is a cone, $b_1, \ldots, b_r$ may even be chosen as integral vectors.
The following theorem is due to Meyer [168].
Theorem 17.3.2 $M_{\text {int }}$ is a rational polyhedron for each rational polyhedron $M$.

Proof. Let $M=P+\mathcal{K}\left(b_1, \ldots, b_r\right)$ where $P$ is a polytope and $b_1, \ldots, b_r$ are integral vectors.
Consider $P+L$ where
$$L:=\left{\sum_{i=1}^r \lambda_i b_i \mid 0 \leq \lambda_i \leq 1,1 \leq i \leq r\right}$$
$P+L$ is a bounded convex set and thus contains only finitely many integral points. It follows that $Q:=\mathcal{C}\left(x \mid x \in(P+L) \cap \mathbb{Z}^n\right)$ is an integral polytope. We want to show that
$$M_{\text {int }}=Q+\mathcal{K}\left(b_1, \ldots, b_r\right)$$
which clearly implies the result.

## 数学代写|最优化理论作业代写optimization theory代考|Finite Alphabets

The Turing machine, introduced by Alan Turing in 1936, is one among the theoretical concepts which were developed in the first half of the 20th century in order to formalize a computability notion for problems defined over finite alphabets. The latter means that all the data which specify a problem instance can be expressed via a string (word) of letters which belong to a finite set. Before explaining the ideas of a Turing machine we thus first introduce some elementary facts about finite alphabets.
Definition 18.1.1 (Finite alphabets)
a) A finite set $A$ is called a finite alphabet.
b) A finite string $a_1 \ldots a_s$ which is built using letters $a_i \in A$ is a word over $A$. We also consider the unique string consisting of no letter of $A$ as a word over $A$ and denote it by $e$. It is called the empty word over $A$.
c) The set of all words over $A$ is denoted by $A^*:={w \mid w$ is a word over $A}$. Sometimes, we also use $A^{+}:=A \backslash{e}$.

Whenever we deal with a finite alphabet we suppose that none of its letters decomposes into a sequence of other letters. This general assumption is necessary for the following obvious definition of the length of a word.
Definition 18.1.2 (Length of a word) Let $A$ be a finite alphabet and $w=w_1 \ldots w_n \in A^*$. The length or size $|w|$ of $w$ is defined to be the number $n$. The empty word has length $|e|:=0$.

The operation of combining words will frequently occur in the following.
Definition 18.1.3 (Concatenation) For a finite alphabet $A$ as above and words $x=x_1 x_2 \ldots x_n, y=y_1 y_2 \ldots y_m$ in $A^$ the concatenation $x y$ of $x$ and $y$ is the word $x_1 x_2 \ldots x_n y_1 \ldots y_m \in A^$. Obviously, $|x y|=|x|+|y|$. If $x$ has the form $x=u v$ for $u, v \in A^$ we call $u$ a prefix of $x$ and $v$ a suffix of $x$. Exercise 18.1.4 a) Let $\Sigma_1$ and $\Sigma_2$ be finite alphabets with cardinalities at least 2. Show that there is an injective function $\phi: \Sigma_2^ \rightarrow \Sigma_1^$ such that $\forall w \in \Sigma_2^|\phi(w)| \leq C \cdot|w|$, where $C=\left\lceil\log {\left|\Sigma_1\right|}\left|\Sigma_2\right|\right\rceil$. b) Show by a simple counting argument that if $\left|\Sigma_1\right|=1$, then for any injective function $\phi$ as in a) there is an infinite sequence $\left{w_i\right}{i \in \mathbb{N}}, w_i \in$ $\Sigma_2^*$ such that $\left|\phi\left(w_i\right)\right| \geq\left|\Sigma_2\right| w_i \mid$.

## 数学代写|最优化理论作业代写optimization theory代考|Integral polyhedra

$$M=M_{\text {int }}:=\mathcal{C}\left(x \mid x \in M \cap \mathbb{Z}^n\right) .$$

## 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|最优化理论作业代写optimization theory代考|CSCI8955

statistics-lab™ 为您的留学生涯保驾护航 在代写最优化理论optimization theory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。我们的专家在代写最优化理论optimization theory代写方面经验极为丰富，各种代写最优化理论optimization theory相关的作业也就用不着说。

## 数学代写|最优化理论作业代写optimization theory代考|König’s Theorem

We can easily deduce König’s Theorem from the Max-flow Min-cut Theorem as follows: Suppose $G=(V, E)$ is a bipartite graph with bipartition $V=$ $U \cup W$. Construct a network $D=\left(V^{\prime}, A\right)$ as follows: $V^{\prime}=V \cup{s, t}$ with two new elements $s$ and $t$ and
\begin{aligned} A:={(s, u) \mid u \in U} \quad & \cup((u, w) \mid u \in U, w \in W,{u, w} \in E} \ \cup & {(w, t) \mid w \in W} \cup{(t, s)} . \end{aligned}
As a capacity function, we let $c(s, u)=c(w, t)=1, c(u, w)=|U|+1$ for $(u, w) \in A, u \in U, w \in W, c(t, s)$ large, e.g. $|A|(|U|+|W|)$. the integrality of the capacities, we have an integral max flow $x$.Clearly, $x_a \leq 1$ for all arcs $a \in A$. It follows that the set
$$\left{{u, w} \in E \mid u \in U, w \in W, x_{(u, w)}=1\right}$$
is a matching in $G$ whose cardinality equals $\operatorname{val}(x)$.
On the other hand, suppose that $C$ is a cut satisfying $\operatorname{cap}(C)=\operatorname{val}(x) \leq$ $\min (|U|,|W|)$. Then no arc can join some $u \in C \cap U$ to some $w \in W \backslash \bar{C}$ since $c(u, w)>\operatorname{cap}(C)$. It follows, that $(U \backslash C) \cup(W \cap C)$ is a vertex cover for $G$ with cardinality equal to $\operatorname{cap}(C)$.
This proves König’s Theorem and gives us another algorithm to compute maximum matchings in bipartite graphs.

## 数学代写|最优化理论作业代写optimization theory代考|Dilworth’s Theorem

We are now going to consider a famous theorem about chain decompositions of partially ordered sets (posets) and need a few definitions:

Definition 16.3.1 Assume that $(P, \prec)$ is a poset, i.e., ” $\prec$ ” is irreflexive and transitive.
A chain in $P$ is a subset $K \subseteq P$ such that any two elements $p \neq p^{\prime}$ in $K$

are comparable, i.e. either $p \prec p^{\prime}$ or $p^{\prime} \prec p$ holds. An antichain in $P$ is a subset $L \subseteq H$ such that no two elements $p, p^{\prime}$ in $L$ are comparable, i.e., $p \prec p^{\prime}$ never holds. A chain decomposition of $P$ is a set partition $\mathcal{K}$ of $P, \mathcal{K}=\left{K_1, \ldots, K_{\ell}\right}, \bigcup_{i=1}^{\ell} K_i=P, K_i \cap K_j=\emptyset(i \neq j)$, such that each $K_i$ is a chain, $1 \leq i \leq \ell$.

Theorem 16.3.2 (Dilworth (1950)) For each finite poset $(P, \prec)$ we have $\max {|L| \mid L \subset P$ antichain $}=\min {|\mathcal{K}| \mid \mathcal{K}$ chain decomposition of $P}$.

Proof. We use König’s Theorem. Construct a bipartite graph $G=(P \cup$ $\left.P^{\prime}, E\right)$ as follows: The colour classes are the poset $P$ and some disjoint copy $P^{\prime}=\left{p^{\prime} \mid p \in P\right}$ of $P$. The edge set is $E:=\left{\left{p, q^{\prime}\right} \mid p, q \in P, p \prec q\right}$.
(i) Assume that $M$ is a matching in $G$. We obtain a chain decomposition $\mathcal{K}$ of $P$ with $|M|+|\mathcal{K}|=|P|$ as follows: Let $N:=\left{\left{p, p^{\prime}\right} \mid p \in P\right}$. Enumerate by $p_1, \ldots, p_{\ell}$ the elements of $P$ such that $p_i^{\prime}$ is not an endpoint of some edge in $M$. Let $K_i:=\left{p \in P \mid p\right.$ and $p_i$ are in the same component of $\left.G^{\prime}:=\left(P \cup P^{\prime}, M \cup N\right)\right}, 1 \leq i \leq \ell$.
It is easy to check that $\mathcal{K}:=\left{K_1, \ldots, K_{\ell}\right}$ is indeed a chain decomposition of $P$ and that the component of $p_i$ in $G^{\prime}$ contains exactly $\left|K_i\right|-1$ edges from $M$, hence
$$|P|=\sum_{i=1}^{\ell}\left|K_i\right|=\ell+\sum_{i=1}^{\ell}\left(\left|K_i\right|-1\right)=\ell+|M| .$$
(ii) Now let $W \subseteq P \cup P^{\prime}$ be some vertex cover for $G$. We construct an antichain $L$ in $P$ with $|W|+|L| \geq|P|$ as follows:
Let $W_0:=\left{p \in P \mid p \in W\right.$ or $\left.p^{\prime} \in W\right}$.
Then $\left|W_0\right| \leq|W|$ and $P \backslash W_0$ is an antichain by definition of $W$, hence
$$|W|+\left|P \backslash W_0\right| \geq\left|W_0\right|+\left|P \backslash W_0\right|=|P| .$$
(iii) By König’s Theorem, $\nu(G)=\tau(G)$, and we may choose some matching $\tilde{M}$ and some vertex cover $\bar{W}$ of $G$ with $|\bar{M}|=|\bar{W}|$.
Using (i) and (ii), we obtain a chain decomposition $\overline{\mathcal{K}}$ and an antichain $\bar{L}$ satisfying
$$|\overline{\mathcal{K}}|=|P|-|\bar{M}|=|P|-|\bar{W}| \leq|\bar{L}|,$$
hence
$$\max {|L| \mid L \text { antichain }} \geq \min {|\mathcal{K}| \mid \mathcal{K} \text { chain decomposition }}$$

The reverse inequality is, however, trivial: Each element of an antichain must occur in a chain decomposition and no two elements of an antichain can occur in the same chain. The theorem is proved.

## 数学代写|最优化理论作业代写optimization theory代考|Dilworth’s Theorem

16.3.1假设$(P, \prec)$是一个偏序集，即” $\prec$ “是非自反的和传递的。
$P$中的链是一个子集$K \subseteq P$，使得任意两个元素$p \neq p^{\prime}$在 $K$

(i)假设$M$是$G$的匹配项。我们用$|M|+|\mathcal{K}|=|P|$得到$P$的链分解$\mathcal{K}$如下:设$N:=\left{\left{p, p^{\prime}\right} \mid p \in P\right}$。通过$p_1, \ldots, p_{\ell}$枚举$P$的元素，使$p_i^{\prime}$不是$M$中某些边的端点。设$K_i:=\left{p \in P \mid p\right.$和$p_i$在$\left.G^{\prime}:=\left(P \cup P^{\prime}, M \cup N\right)\right}, 1 \leq i \leq \ell$的同一个组件中。

$$|P|=\sum_{i=1}^{\ell}\left|K_i\right|=\ell+\sum_{i=1}^{\ell}\left(\left|K_i\right|-1\right)=\ell+|M| .$$
(ii)现在设$W \subseteq P \cup P^{\prime}$为$G$的某个顶点覆盖。我们用$|W|+|L| \geq|P|$在$P$中构造一个反链$L$如下:

$$|W|+\left|P \backslash W_0\right| \geq\left|W_0\right|+\left|P \backslash W_0\right|=|P| .$$
(iii)根据König定理$\nu(G)=\tau(G)$，我们可以选择$G$与$|\bar{M}|=|\bar{W}|$的某个匹配$\tilde{M}$和某个顶点覆盖$\bar{W}$。

$$|\overline{\mathcal{K}}|=|P|-|\bar{M}|=|P|-|\bar{W}| \leq|\bar{L}|,$$

$$\max {|L| \mid L \text { antichain }} \geq \min {|\mathcal{K}| \mid \mathcal{K} \text { chain decomposition }}$$

## 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|最优化理论作业代写optimization theory代考|CSE276

statistics-lab™ 为您的留学生涯保驾护航 在代写最优化理论optimization theory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。我们的专家在代写最优化理论optimization theory代写方面经验极为丰富，各种代写最优化理论optimization theory相关的作业也就用不着说。

## 数学代写|最优化理论作业代写optimization theory代考|The Simplex Method (Nelder–Mead)

We recall the concept of a simplex. Let $\xi_1, \ldots, \xi_m \in \mathbb{R}^n$ be linearly independent vectors. With $x^0 \in \mathbb{R}^n$ and $x^i=x^0+\xi_i, i=1, \ldots, m$, the convex hull $\mathcal{C}\left(x^0, x^1, \ldots, x^m\right)$ is called an $m$-simplex in $\mathbb{R}^n$. The segment connecting $x^i$ and $x^j, i \neq j$, is called an edge; its length is equal to $\left|x^i-x^j\right|$. If all edges in a simplex have equal length, the simplex is called regular.

A regular $n$-simplex in $\mathbb{R}^n$ with edge length 1 is easy to construct. In fact, consider the following $(n, n+1)$ matrix whose columns are supposed to be the vertices of the regular simplex:
$$\left(\begin{array}{ccccc} 0 & \alpha & \beta & \cdots & \beta \ 0 & \beta & \alpha & \cdots & \beta \ \vdots & & & & \vdots \ 0 & \beta & \beta & \cdots & \alpha \end{array}\right)$$
Clearly, $\alpha$ and $\beta$ in (12.2.1) should satisfy the following relations:
$$\alpha^2+(n-1) \beta^2=1 \text { and } 2(\alpha-\beta)^2=1 .$$
It follows:
$$\left{\begin{array}{l} \alpha=\frac{1}{n \sqrt{2}}(n-1+\sqrt{n+1}), \ \beta=\frac{1}{n \sqrt{2}}(-1+\sqrt{n+1}) . \end{array}\right.$$

## 数学代写|最优化理论作业代写optimization theory代考|One–Dimensional Minimization

In this chapter we consider the minimization of a function $f$ of one variable. In fact, many optimization methods rely on on successive line searches, i.e. one-dimensional optimization steps. We distinguish two types of methods:
I. Interpolation Methods. Here, the function $f$ is successively interpolated by means of polynomials of degree two or three; the minima of the latter polynomials produce new approximations for the desired minimum. In general, these methods are favourable with respect to smooth functions. We will describe the following interpolations:
Hermite-interpolation (first derivatives are required)
II. Search Methods. They are based on successively shrinking a certain interval in which the function $f$ is unimodal; the function $f$ is called unimodal in $[x, y]$ if $f$ has a unique local optimum in the open interval $(x, y)$. We will mention the following:
Golden Section Method
Fibonacci-Search
Armijo’s Rule

## 数学代写|最优化理论作业代写optimization theory代考|The Simplex Method (Nelder–Mead)

$$\left(\begin{array}{ccccc} 0 & \alpha & \beta & \cdots & \beta \ 0 & \beta & \alpha & \cdots & \beta \ \vdots & & & & \vdots \ 0 & \beta & \beta & \cdots & \alpha \end{array}\right)$$

$$\alpha^2+(n-1) \beta^2=1 \text { and } 2(\alpha-\beta)^2=1 .$$

$$\left{\begin{array}{l} \alpha=\frac{1}{n \sqrt{2}}(n-1+\sqrt{n+1}), \ \beta=\frac{1}{n \sqrt{2}}(-1+\sqrt{n+1}) . \end{array}\right.$$

## 数学代写|最优化理论作业代写optimization theory代考|One–Dimensional Minimization

1 .插值方法。在这里，函数$f$是通过二阶或三阶多项式相继插值的;后一种多项式的最小值产生了期望最小值的新近似值。一般来说，这些方法对于光滑函数是有利的。我们将描述以下插值:

2搜索方法。它们是基于连续缩小某一区间，该区间内的函数$f$是单峰的;如果$f$在开放区间$(x, y)$具有唯一的局部最优，则在$[x, y]$中称为$f$单峰函数。我们将提到以下几点:

## 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|最优化理论作业代写optimization theory代考|CSC591

statistics-lab™ 为您的留学生涯保驾护航 在代写最优化理论optimization theory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。我们的专家在代写最优化理论optimization theory代写方面经验极为丰富，各种代写最优化理论optimization theory相关的作业也就用不着说。

## 数学代写|最优化理论作业代写optimization theory代考|Introduction, Steepest Descent

The simplex method for solving linear optimization problems stops after a finite number of steps. In general, however, algorithms for solving optimization problems will generate an infinite sequence $\left(x^k\right) \subset \mathbb{R}^n$, and one hopes that, as $k$ tends to infinity, an acceptable solution will be produced. In case of convergence, it is natural to ask how fast the sequence acturally converges. There are several orders of convergence, and we will discuss linear, superlinear and quadratic convergence. We will always assume that $x^k \neq \bar{x}$ for a sequence $\left(x^k\right) \subset \mathbb{R}^n$ that converges to $\bar{x}$.

Definition 9.1.1 Let $\left(x^k\right) \subset \mathbb{R}^n$ be a sequence converging to $\bar{x}$. With regard to the order of convergence, we define:
$$\begin{array}{ll} \text { Linear Convergence } & : \text { if } \operatorname{lim~}{\sup {k \rightarrow \infty}} \frac{\left|x^{k+1}-\bar{x}\right|}{\left|x^k-\bar{x}\right|} \leq L<1, \ \text { Superlinear Convergence } & : \text { if } L=0, \ \text { Quadratic Convergence } & : \text { if } \lim \sup _{k \rightarrow \infty} \frac{\left|x^{k+1}-\bar{x}\right|}{\left|x^k-\bar{x}\right|^2}<\infty . \end{array}$$
As a first optimization method we discuss the method of steepest descent. To this aim let $f \in C^1\left(\mathbb{R}^n, \mathbb{R}\right)$. For minimizing $f$, one might proceed as follows. Starting at a point $\bar{x} \in \mathbb{R}^n$ with $D f(\bar{x}) \neq 0$, minimize the following function $\varphi(t)$ of one variable $t$ for $t \geq 0$ :
$$\varphi(t)=f\left(\bar{x}-t D^{\top} f(\bar{x})\right)$$
Let $\bar{t} \geq 0$ be a point at which $\varphi(t)$ is minimized for $t \geq 0$. Then, $\bar{x}$ is replaced by the point $\bar{x}-\bar{t} D^{\top} f(\bar{x})$, and the procedure is repeated. The name steepest descent method comes from the fact that the directional derivative of $f$ at $\bar{x}$ is minimized in the direction of $-D^{\top} f(\bar{x})$, i.e. $\xi=$ $-D^{\top} f(\bar{x}) /\left|D^{\top} f(\bar{x})\right|$ solves the following problem (exercise):
Minimize $D f(\bar{x}) \cdot \xi$ subject to $|\xi|=1$

## 数学代写|最优化理论作业代写optimization theory代考|Search for Zeros of Mappings, Newton’s Method

The search for a local minimum of $f \in C^1\left(\mathbb{R}^n, \mathbb{R}\right)$ might be weakened by searching for points satisfying the necessary optimality condition of first order: ” $D f(\bar{x})=0$ “. This leads to the determination of zeros of the associated mapping $x \mapsto D^{\top} f(x)$.

We will study iterative methods for finding zeros in a more general framework. Let $g \in C^r\left(\mathbb{R}^n, \mathbb{R}\right), r \geq 1$, be a mapping for which we are interested to find the zeros. We consider iterative methods of the following form:
$$x^{k+1}:=x^k-H_k \cdot g\left(x^k\right),$$
where $H_k$ is a nonsingular $(n, n)$-matrix, depending on $k$ (steering matrix). In our convergence considerations we tacitly assume that $x^k \neq \bar{x}$ when $x^k$ coverges to $\bar{x}$. For an $(n, n)$-matrix $A$, let $|A|$ be again the induced matrix norm, i.e. $|A|=\max _{|x| \leq 1}|A x|$.

## 数学代写|最优化理论作业代写optimization theory代考|Introduction, Steepest Descent

$$\begin{array}{ll} \text { Linear Convergence } & : \text { if } \operatorname{lim~}{\sup {k \rightarrow \infty}} \frac{\left|x^{k+1}-\bar{x}\right|}{\left|x^k-\bar{x}\right|} \leq L<1, \ \text { Superlinear Convergence } & : \text { if } L=0, \ \text { Quadratic Convergence } & : \text { if } \lim \sup _{k \rightarrow \infty} \frac{\left|x^{k+1}-\bar{x}\right|}{\left|x^k-\bar{x}\right|^2}<\infty . \end{array}$$

$$\varphi(t)=f\left(\bar{x}-t D^{\top} f(\bar{x})\right)$$

## 数学代写|最优化理论作业代写optimization theory代考|Search for Zeros of Mappings, Newton’s Method

$$x^{k+1}:=x^k-H_k \cdot g\left(x^k\right),$$

## 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|最优化理论作业代写optimization theory代考|MS&E213

statistics-lab™ 为您的留学生涯保驾护航 在代写最优化理论optimization theory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。我们的专家在代写最优化理论optimization theory代写方面经验极为丰富，各种代写最优化理论optimization theory相关的作业也就用不着说。

## 数学代写|最优化理论作业代写optimization theory代考|The Determination of an Initial Vertex

In order to start the simplex method, we need to determine an initial vertex. For some problem this is obvious. For example, let the constraints be $A x \leq b$, $x \geq 0$. With “slack variables” $y$ we obtain the feasible set in standard form:
$$A x+y=b, \quad x \geq 0, y \geq 0$$
In case that $b \geq 0$, the point $(x, y)=(0, b)$ is a vertex.
Now, consider the standard feasible set described by: $A x=b, x \geq 0$, where $A$ is an $(m, n)$-matrix with $\operatorname{rank}(A)=m$. Without loss of generality we may assume $b \geq 0$. Consider the following auxiliary problem (AP):
$$\text { (AP) }\left{\begin{array}{l} \text { Minimize } \sum_{i=1}^m y_i \ \left{\begin{array}{l} A x+y=b \ x \geq 0, y \geq 0 . \end{array}\right. \end{array}\right.$$
The point $(x, y)=(0, b)$ is a vertex for (AP). Start the simplex method for (AP) and let $(\bar{x}, \bar{y})$ be the solution vertex for (AP). If $\bar{y} \neq 0$, then the system $A x=b, x \geq 0$, has no solution. If $\bar{y}=0$, then $\bar{x}$ is a vertex for the system $A x=b, x \geq 0$. In case that $(\bar{x}, 0)$ is a degenerate vertex for (AP), the set of vectors in the basis corresponding to components of $\bar{x}$ can be completed with further columns of $A$ in order to obtain a basis at the vertex $\bar{x}$ for the system $A x=b, x \geq 0$.

## 数学代写|最优化理论作业代写optimization theory代考|Running Time Analysis

For all known pivoting strategies it is possible to construct an example in $\mathbb{R}^n$ for which the simplex iteration needs an exponential (in $n$ ) number of steps in order to reach the optimal vertex. These examples are constructed by means of deformations of the $n$-dimensional unit cube together with an appropriate objective function (cf. [140]). It then turns out that the simplex method runs through all vertices. The number of vertices of a unit cube is $2^n$. For further details we refer to [199].

We will give the geometric idea in two dimensions, using the strategy “choose the smallest $p_j$ ” (cf. (6.2.3)). In Figure 6.5 we see that the simplex method runs through all 4 vertices (the last vertex is situated outside of the picture).

Although the exponential feature appears in the “worst case”, the simplex method performs in practice rather good. For a probabilistic consideration of the latter phenomenon we refer to [29].

## 数学代写|最优化理论作业代写optimization theory代考|The Determination of an Initial Vertex

$$A x+y=b, \quad x \geq 0, y \geq 0$$

$$\text { (AP) }\left{\begin{array}{l} \text { Minimize } \sum_{i=1}^m y_i \ \left{\begin{array}{l} A x+y=b \ x \geq 0, y \geq 0 . \end{array}\right. \end{array}\right.$$

## 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|最优化理论作业代写optimization theory代考|Determination of Optimal Trajectories by Using Gradient Projection

statistics-lab™ 为您的留学生涯保驾护航 在代写最优化理论optimization theory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。我们的专家在代写最优化理论optimization theory代写方面经验极为丰富，各种代写最优化理论optimization theory相关的作业也就用不着说。

## 数学代写|最优化理论作业代写optimization theory代考|Determination of Optimal Trajectories by Using Gradient Projection

Let us now discuss a technique, also due to Rosen, $\dagger$ for solving optimal control problems by using the gradient projection algorithm. The problem is to find an admissible control history $\mathbf{u}^$ that causes the system $$\dot{\mathbf{x}}(t)=\mathbf{a}(\mathbf{x}(t), \mathbf{u}(t))$$ with known initial state $\mathbf{x}\left(t_0\right)=\mathbf{x}0$ to follow an admissible trajectory $\mathbf{x}^$ that minimizes the performance measure
$$J=h\left(\mathbf{x}\left(t_f\right)\right)+\int{t_0}^{t_f} g(\mathbf{x}(t), \mathbf{u}(t)) d t$$
For simplicity of notation, we shall assume that time does not appear explicitly in either the state equations or the performance measure; the solution of time-varying problems requires only straightforward modifications of the procedure to be described. It is also assumed that the final time $t_f$ is specified, and since the equations are time-invariant, we can let $t_0=0$. Although the technique to be presented applies to problems involving general linear constraints among the state and control variables, we shall restrict our discussion to problems with constraints of the form
$$\begin{array}{rlrlrl} M_{i-} & \leq u_i(t) \leq M_{i+}, & t \in\left[0, t_f\right], & & i & =1,2, \ldots, m \ S_{i-} & \leq x_i(t) \leq S_{i+}, \quad t \in\left[0, t_f\right], & & i & =1,2, \ldots, n \ x_i\left(t_j\right) & =T_{i j}, \quad t_j \text { specified, } & & i=1,2, \ldots, n . \end{array}$$
$M_{i-}$ and $M_{i+}$ denote the lower and upper bounds on the $i$ th control component, $S_{i-}$ and $S_{i+}$ are the lower and upper bounds on the $i$ th state component, and $T_{i j}$ is the required value of the state component $x_i$ at the time $t_j$.

## 数学代写|最优化理论作业代写optimization theory代考|The Minimum Principle

Applying the minimum principle, or the calculus of variations, to determine optimal controls generally leads to a nonlinear two-point boundarysolution. As noted previously, these iterative algorithms determine optimal controls in open-loop form.

If the state equations of a process are linear (or have been linearized), and the performance measure is a quadratic form, the optimal control law can be determined by numerically integrating a matrix differential equation of the Riccati type.

An important feature of the variational approach is that the form of optimal controls can be determined; hence, it is necessary only to consider the subset of controls having the appropriate form; this is a significant conceptual and computational advantage.

Dynamic Programming
Dynamic programming is essentially a clever way of examining all of the candidates for an optimal control law. To do this by direct enumeration of all the possibilities is a horrendous task, but by using the principle of optimality a multiple-stage decision process can be reduced to a sequence of singlestage decision processes, and a feasible computational algorithm is obtained. The algorithm consists of solving the functional recurrence equation
\begin{aligned} J_{N-K, N}^(\mathbf{x}(N-K))= & \min {\mathbf{u}(N-K)}\left{g_D(\mathbf{x}(N-K), \mathbf{u}(N-K))\right. \ & \left.+J{N-(K-1), N}^\left(\mathbf{a}_D(\mathbf{x}(N-K), \mathbf{u}(N-K))\right)\right} \end{aligned}
by a direct search among the admissible control values. The presence of state and control constraints generally complicates the application of variational techniques; however, in dynamic programming, state and control constraints reduce the range of values to be searched and thereby simplify the solution. Another desirable feature of the dynamic programming approach is that the computational procedure determines the optimal control law. Moreover, since the algorithm makes a direct comparison of the performance measure values associated with all optimal control law candidates, it is ensured that the global, or absolute, optimal control law is obtained. The primary limitation of the dynamic programming approach is the need for large storage capacity in the digital computer when solving problems involving high-order systems-this is the “curse of dimensionality.”

## 数学代写|最优化理论作业代写optimization theory代考|Determination of Optimal Trajectories by Using Gradient Projection

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

$$\begin{array}{rlrlrl} M_{i-} & \leq u_i(t) \leq M_{i+}, & t \in\left[0, t_f\right], & & i & =1,2, \ldots, m \ S_{i-} & \leq x_i(t) \leq S_{i+}, \quad t \in\left[0, t_f\right], & & i & =1,2, \ldots, n \ x_i\left(t_j\right) & =T_{i j}, \quad t_j \text { specified, } & & i=1,2, \ldots, n . \end{array}$$
$M_{i-}$和$M_{i+}$分别表示控制组件$i$的上下限，$S_{i-}$和$S_{i+}$分别表示状态组件$i$的上下限，$T_{i j}$为状态组件$x_i$在$t_j$时刻所需值。

## 数学代写|最优化理论作业代写optimization theory代考|The Minimum Principle

\begin{aligned} J_{N-K, N}^(\mathbf{x}(N-K))= & \min {\mathbf{u}(N-K)}\left{g_D(\mathbf{x}(N-K), \mathbf{u}(N-K))\right. \ & \left.+J{N-(K-1), N}^\left(\mathbf{a}_D(\mathbf{x}(N-K), \mathbf{u}(N-K))\right)\right} \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|最优化理论作业代写optimization theory代考|One Iteration of the Numerical Procedure

statistics-lab™ 为您的留学生涯保驾护航 在代写最优化理论optimization theory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。我们的专家在代写最优化理论optimization theory代写方面经验极为丰富，各种代写最优化理论optimization theory相关的作业也就用不着说。

## 数学代写|最优化理论作业代写optimization theory代考|One Iteration of the Numerical Procedure

The method of quasilinearization consists of solving a sequence of linearized two-point boundary-value problems. We now know how to:

1. Linearize nonlinear differential equations.
2. Solve linear two-point boundary-value problems.
The following example illustrates how these two steps go together to constitute one iteration of the quasilinearization algorithm.

Example 6.4-2. A nonlinear first-order system is described by the differential equation
$$\dot{x}(t)=x^2(t)+u(t) .$$
The initial condition is $x(0)=3.0$, and the performance measure to be minimized is
$$J=\int_0^1\left[2 x^2(t)+u^2(t)\right] d t$$
From the Hamiltonian
$$\mathscr{H}(x(t), u(t), p(t))=2 x^2(t)+u^2(t)+p(t) x^2(t)+p(t) u(t),$$
the costate equation is
$$\dot{p}(t)=-\frac{\partial \mathscr{H}}{\partial x}=-4 x(t)-2 p(t) x(t)$$
The algebraic relationship that must be satisfied is
$$\frac{\partial \mathscr{H}}{\partial u}=0=2 u(t)+p(t)$$

## 数学代写|最优化理论作业代写optimization theory代考|The Continuous Stirred-Tank Chemical Reactor Problem

For comparison with the methods of steepest descent and variation of extremals, let us again solve the stirred-tank reactor problem discussed in Sections 6.2 and 6.3, this time using the quasilinearization algorithm.

Example 6.4-3. The problem statement is given in Example 6.2-2. The reduced differential equations are given by Eqs. (6.3-29) and (6.3-30). Linearizing these nonlinear differential equations, using (6.4-29), we obtain
\begin{aligned} {\left[\begin{array}{c} \dot{\mathbf{x}}^{(i+1)}(t) \ \cdots \dot{\mathbf{p}}^{(i+1)}(t) \end{array}\right]=} & \mathbf{A}(t)\left[\begin{array}{l} \mathbf{x}^{(i+1)}(t) \ \hdashline\left(\mathbf{p}^{(t+1)}(t)\right. \end{array}\right]+\left[\begin{array}{c} \mathbf{a}\left(\mathbf{x}^{(t)}(t), \mathbf{p}^{(i)}(t), t\right) \ \hdashline-\frac{\partial \mathscr{H}}{\partial \mathbf{x}}\left(\mathbf{x}^{(i)}(t), \mathbf{p}^{(i)}(t), t\right) \end{array}\right] \ & -\mathbf{A}(t)\left[\begin{array}{l} \mathbf{x}^{(t)}(t) \ \cdots \ \mathbf{p}^{(t)}(t) \end{array}\right] \end{aligned}
$\mathbf{A}(t)$ denotes the $2 n \times 2 n$ matrix
$$\mathbf{A}(t)=\left[\begin{array}{c:c:c:c} -2-10 p_1 \alpha_5+\alpha_4 & \alpha_1 & -5 \alpha_5^2 & 0 \ -\alpha_4 & -1-\alpha_1 & 0 & 0 \ -2+\alpha_2 \alpha_6+5 p_1^2 & \alpha_3 \alpha_6 & 2+10 p_1 \alpha_5-\alpha_4 & \alpha_4 \ \alpha_3 \alpha_6 & -2 & -\alpha_1 & 1+\alpha_1 \end{array}\right]_{\mathbf{x}(t)(t), \mathrm{p}^{(1)}(t)}$$
where
\begin{aligned} & \alpha_1 \triangleq \exp \left[\frac{25 x_1}{x_1+2}\right] \ & \alpha_2 \triangleq \frac{100\left[x_2+0.5\right]\left[23-x_1\right] \alpha_1}{\left[x_1+2\right]^4} \ & \alpha_3 \triangleq \frac{50 \alpha_1}{\left[x_1+2\right]^2} \ & \alpha_4 \triangleq\left[x_2+0.5\right] \alpha_3 . \ & \alpha_5 \triangleq x_1+0.25 \ & \alpha_6 \triangleq p_2-p_1 \end{aligned}

## 数学代写|最优化理论作业代写optimization theory代考|One Iteration of the Numerical Procedure

$$\dot{x}(t)=x^2(t)+u(t) .$$

$$J=\int_0^1\left[2 x^2(t)+u^2(t)\right] d t$$

$$\mathscr{H}(x(t), u(t), p(t))=2 x^2(t)+u^2(t)+p(t) x^2(t)+p(t) u(t),$$

$$\dot{p}(t)=-\frac{\partial \mathscr{H}}{\partial x}=-4 x(t)-2 p(t) x(t)$$

$$\frac{\partial \mathscr{H}}{\partial u}=0=2 u(t)+p(t)$$

## 数学代写|最优化理论作业代写optimization theory代考|The Continuous Stirred-Tank Chemical Reactor Problem

\begin{aligned} {\left[\begin{array}{c} \dot{\mathbf{x}}^{(i+1)}(t) \ \cdots \dot{\mathbf{p}}^{(i+1)}(t) \end{array}\right]=} & \mathbf{A}(t)\left[\begin{array}{l} \mathbf{x}^{(i+1)}(t) \ \hdashline\left(\mathbf{p}^{(t+1)}(t)\right. \end{array}\right]+\left[\begin{array}{c} \mathbf{a}\left(\mathbf{x}^{(t)}(t), \mathbf{p}^{(i)}(t), t\right) \ \hdashline-\frac{\partial \mathscr{H}}{\partial \mathbf{x}}\left(\mathbf{x}^{(i)}(t), \mathbf{p}^{(i)}(t), t\right) \end{array}\right] \ & -\mathbf{A}(t)\left[\begin{array}{l} \mathbf{x}^{(t)}(t) \ \cdots \ \mathbf{p}^{(t)}(t) \end{array}\right] \end{aligned}
$\mathbf{A}(t)$表示$2 n \times 2 n$矩阵
$$\mathbf{A}(t)=\left[\begin{array}{c:c:c:c} -2-10 p_1 \alpha_5+\alpha_4 & \alpha_1 & -5 \alpha_5^2 & 0 \ -\alpha_4 & -1-\alpha_1 & 0 & 0 \ -2+\alpha_2 \alpha_6+5 p_1^2 & \alpha_3 \alpha_6 & 2+10 p_1 \alpha_5-\alpha_4 & \alpha_4 \ \alpha_3 \alpha_6 & -2 & -\alpha_1 & 1+\alpha_1 \end{array}\right]_{\mathbf{x}(t)(t), \mathrm{p}^{(1)}(t)}$$

\begin{aligned} & \alpha_1 \triangleq \exp \left[\frac{25 x_1}{x_1+2}\right] \ & \alpha_2 \triangleq \frac{100\left[x_2+0.5\right]\left[23-x_1\right] \alpha_1}{\left[x_1+2\right]^4} \ & \alpha_3 \triangleq \frac{50 \alpha_1}{\left[x_1+2\right]^2} \ & \alpha_4 \triangleq\left[x_2+0.5\right] \alpha_3 . \ & \alpha_5 \triangleq x_1+0.25 \ & \alpha_6 \triangleq p_2-p_1 \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|最优化理论作业代写optimization theory代考|A Continuous Stirred-Tank Chemical Reactor

statistics-lab™ 为您的留学生涯保驾护航 在代写最优化理论optimization theory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。我们的专家在代写最优化理论optimization theory代写方面经验极为丰富，各种代写最优化理论optimization theory相关的作业也就用不着说。

## 数学代写|最优化理论作业代写optimization theory代考|A Continuous Stirred-Tank Chemical Reactor

Example 6.2-2. The state equations for a continuous stirred-tank chemical reactor are given below [L-5]. The flow of a coolant through a coil inserted in the reactor is to control the first-order, irreversible exothermic reaction taking place in the reactor. The states of the plant are $x_1(t)=T(t)$ (the deviation from the steady-state temperature) and $x_2(t)=C(t)$ (the deviation from the steady-state concentration). $u(t)$, the normalized control variable, represents the effect of coolant flow on the chemical reaction. The state equations are
\begin{aligned} \dot{x}_1(t)= & -2\left[x_1(t)+0.25\right]+\left[x_2(t)+0.5\right] \exp \left[\frac{25 x_1(t)}{x_1(t)+2}\right] \ & -\left[x_1(t)+0.25\right] u(t) \ \dot{x}_2(t)= & 0.5-x_2(t)-\left[x_2(t)+0.5\right] \exp \left[\frac{25 x_1(t)}{x_1(t)+2}\right] \end{aligned}
with initial conditions $\mathbf{x}(0)=\left[\begin{array}{ll}0.05 & 0\end{array}\right]^T$. The performance measure to be minimized is
$$J=\int_0^{0.78}\left[x_1^2(t)+x_2^2(t)+R u^2(t)\right] d t,$$
indicating that the desired objective is to maintain the temperature and concentration close to their steady-state values without expending large amounts of control effort. $R$ is a weighting factor that we shall select (arbitrarily) as 0.1 . The costate equations are determined from the Hamiltonian,
\begin{aligned} \mathscr{H}(\mathbf{x}(t) & u(t), \mathbf{p}(t))=x_1^2(t)+x_2^2(t)+R u^2(t) \ & +p_1(t)\left[-2\left[x_1(t)+0.25\right]+\left[x_2(t)+0.5\right] \exp \left[\frac{25 x_1(t)}{x_1(t)+2}\right]\right. \ & \left.-\left[x_1(t)+0.25\right] u(t)\right]+p_2(t)\left[0.5-x_2(t)\right. \ & \left.-\left[x_2(t)+0.5\right] \exp \left[\frac{25 x_1(t)}{x_1(t)+2}\right]\right] \end{aligned}
as
\begin{aligned} \dot{p}_1(t)= & -\frac{\partial \mathscr{H}}{\partial x_1}=-2 x_1(t)+2 p_1(t) \ & -p_1(t)\left[x_2(t)+0.5\right]\left[\frac{50}{\left[x_1(t)+2\right]^2}\right] \exp \left[\frac{25 x_1(t)}{x_1(t)+2}\right] \ & +p_1(t) u(t)+p_2(t)\left[x_2(t)+0.5\right]\left[\frac{50}{\left[x_1(t)+2\right]^2}\right] \exp \left[\frac{25 x_1(t)}{x_1(t)+2}\right] \ \dot{p}_2(t)= & -\frac{\partial \mathscr{H}}{\partial x_2}=-2 x_2(t)-p_1(t) \exp \left[\frac{25 x_1(t)}{x_1(t)+2}\right] \ & +p_2(t)\left[1+\exp \left[\frac{25 x_1(t)}{x_1(t)+2}\right]\right] . \end{aligned}

## 数学代写|最优化理论作业代写optimization theory代考|Features of the Steepest Descent Algorithm

To conclude our discussion of the steepest descent method, let us summarize the important features of the algorithm.

Initial Guess. A nominal control history, $\mathbf{u}^{(0)}(t), t \in\left[t_0, t_f\right]$, must be selected to begin the numerical procedure. In selecting the nominal control we utilize whatever physical insight we have about the problem.

Storage Requirements. The current trial control $\mathbf{u}^{(i)}$, the corresponding state trajectory $\mathbf{x}^{(i)}$, and the gradient history $\partial \mathscr{H}^{(i)} / \partial \mathbf{u}$, are stored. If storage must be conserved, the state values needed to determine $\partial \mathscr{H}^{(t)} / \partial \mathbf{u}$ can be obtained by reintegrating the state equations with the costate equations. If this is done $\mathbf{x}^{(t)}$ does not need to be stored; however, the computation time will increase. Generating the required state values in this manner may make the results of the backward integration more accurate, because the piecewise-constant approximation for $\mathbf{x}^{(l)}$ need not be used.

Convergence. The method of steepest descent is generally characterized by ease of starting – the initial guess for the control is not usually crucial. On the other hand, as a minimum is approached, the gradient becomes small and the method has a tendency to converge slowly.

Computations Required. In each iteration numerical integration of $2 n$ firstorder ordinary differential equations is required. In addition, the time history of $\partial \mathscr{H}^{(t)} / \partial \mathrm{u}$ at the times $t_k, k=0,1, \ldots, N-1$, must be evaluated. To speed up the iterative procedure, a single variable search may be used to determine the step size for the change in the trial control.

Stopping Criterion. The iterative procedure is terminated when a criterion such as $\left|\partial \mathscr{H}^{(i)} / \partial \mathbf{u}\right|<\gamma_1$ or $\left|J^{(i)}-J^{(i+1)}\right|<\gamma_2$ is satisfied; $\gamma_1$ and $\gamma_2$ are preselected positive numbers.

## 数学代写|最优化理论作业代写optimization theory代考|A Continuous Stirred-Tank Chemical Reactor

\begin{aligned} \dot{x}_1(t)= & -2\left[x_1(t)+0.25\right]+\left[x_2(t)+0.5\right] \exp \left[\frac{25 x_1(t)}{x_1(t)+2}\right] \ & -\left[x_1(t)+0.25\right] u(t) \ \dot{x}_2(t)= & 0.5-x_2(t)-\left[x_2(t)+0.5\right] \exp \left[\frac{25 x_1(t)}{x_1(t)+2}\right] \end{aligned}

$$J=\int_0^{0.78}\left[x_1^2(t)+x_2^2(t)+R u^2(t)\right] d t,$$

\begin{aligned} \mathscr{H}(\mathbf{x}(t) & u(t), \mathbf{p}(t))=x_1^2(t)+x_2^2(t)+R u^2(t) \ & +p_1(t)\left[-2\left[x_1(t)+0.25\right]+\left[x_2(t)+0.5\right] \exp \left[\frac{25 x_1(t)}{x_1(t)+2}\right]\right. \ & \left.-\left[x_1(t)+0.25\right] u(t)\right]+p_2(t)\left[0.5-x_2(t)\right. \ & \left.-\left[x_2(t)+0.5\right] \exp \left[\frac{25 x_1(t)}{x_1(t)+2}\right]\right] \end{aligned}
as
\begin{aligned} \dot{p}_1(t)= & -\frac{\partial \mathscr{H}}{\partial x_1}=-2 x_1(t)+2 p_1(t) \ & -p_1(t)\left[x_2(t)+0.5\right]\left[\frac{50}{\left[x_1(t)+2\right]^2}\right] \exp \left[\frac{25 x_1(t)}{x_1(t)+2}\right] \ & +p_1(t) u(t)+p_2(t)\left[x_2(t)+0.5\right]\left[\frac{50}{\left[x_1(t)+2\right]^2}\right] \exp \left[\frac{25 x_1(t)}{x_1(t)+2}\right] \ \dot{p}_2(t)= & -\frac{\partial \mathscr{H}}{\partial x_2}=-2 x_2(t)-p_1(t) \exp \left[\frac{25 x_1(t)}{x_1(t)+2}\right] \ & +p_2(t)\left[1+\exp \left[\frac{25 x_1(t)}{x_1(t)+2}\right]\right] . \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|最优化理论作业代写optimization theory代考|MINIMUM CONTROL-EFFORT PROBLEMS

statistics-lab™ 为您的留学生涯保驾护航 在代写最优化理论optimization theory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。我们的专家在代写最优化理论optimization theory代写方面经验极为丰富，各种代写最优化理论optimization theory相关的作业也就用不着说。

## 数学代写|最优化理论作业代写optimization theory代考|MINIMUM CONTROL-EFFORT PROBLEMS

In the preceding section we considered problems in which the objective was to transfer a system from an arbitrary initial state to a specific target set as quickly as possible. Let us now consider problems in which control effort required, rather than elapsed time, is the criterion of optimality. Such problems arise frequently in aerospace applications, where often there are limited control resources available for achieving desired objectives.

The class of problems we will discuss is the following: Find a control $\mathbf{u}^*(t)$ satisfying constraints of the form
$$M_{i_{-}} \leq u_i(t) \leq M_{i+}, \quad i=1,2, \ldots, m$$
which transfers a system described by
$$\dot{\mathbf{x}}(t)=\mathbf{a}(\mathbf{x}(t), \mathbf{u}(t), t)$$
from an arbitrary initial state $x_0$ to a specified target set $S(t)$ with a minimum expenditure of control effort.

As measures of control effort we shall consider the two performance indices
$$J_1(\mathbf{u})=\int_{t_0}^{t_s}\left[\sum_i^m \beta_i\left|u_i(t)\right|\right] d t$$
and
$$J_2(\mathbf{1})=\int_{t_0}^{t r}\left[\sum_{i=1}^m r_i u_i^2(t)\right] d t,$$
where $\beta_i$ and $r_i, i=1, \ldots, m$, are nonnegative weighting factors. As discussed in Chapter 2 , the fuel consumed by a mass-expulsion thrusting system is often expressed by an integral of the form (5.5-3); thus, if a performance measure to be minimized has the form given by $J_1$, we shall refer to the problem as a minimum-fuel problem. The total electrical energy supplied to a network of resistors by several voltage and current sources is given by an integral of the form (5.5-4); hence, if a performance measure of this form is to be minimized, we shall say that we wish to solve a minimum-energy problem. The reader must be cautioned that in a particular problem (5.5-3) may not represent fuel expenditure, or control energy required may not be given by (5.5-4); therefore, the results obtained in this section will apply to the performance measure $J_1$ or $J_2$, not necessarily to the problems of minimizing fuel or energy consumption.

## 数学代写|最优化理论作业代写optimization theory代考|Minimum-Fuel Problems

In our discussion of minimum-time problems in Section 5.4 the concept of reachable states was introduced. Recall that $R(t)$ was used to denote the set of states that can be reached at time $t$ by starting from an initial state $\mathbf{x}_0$ at time $t_0$. Minimum-fuel problems may also be visualized in terms of reachable states; that is, the minimum-fuel solution is given by the intersection of the target set $S(t)$ with the set of reachable states $R(t)$, which requires the smallest amount of consumed fuel. To represent this idea geometrically we could use a state-time-consumed-fuel coordinate system and determine the intersections (if any) of $S(t)$ and $R(t)$. Unfortunately, although such a geometric representation is helpful as a conceptual device, it is of limited value in actually obtaining solutions. Instead of pursuing this avenue further, we shall approach minimum control-effort problems by starting with the necessary conditions provided by Pontryagin’s minimum principle.

The Form of the Optimal Control for a Class of Minimum-Fuel Problems. Let us assume that the state equations of a system are of the form
$$\dot{\mathbf{x}}(t)=\mathbf{a}(\mathbf{x}(t), t)+\mathbf{B}(\mathbf{x}(t), t) \mathbf{u}(t),$$
where $\mathbf{B}$ is an $n \times m$ array that may be explicitly dependent on the states and time. The performance measure to be minimized is
$$J(\mathbf{u})=\int_{t_0}^{t_t}\left[\sum_{i=1}^m\left|u_i(t)\right|\right] d t$$
and the admissible controls are to satisfy the constraints
$$-1 \leq u_i(t) \leq+1, \quad i=1,2, \ldots, m, \quad t \in\left[t_0, t_f\right]$$

## 数学代写|最优化理论作业代写optimization theory代考|MINIMUM CONTROL-EFFORT PROBLEMS

$$M_{i_{-}} \leq u_i(t) \leq M_{i+}, \quad i=1,2, \ldots, m$$

$$\dot{\mathbf{x}}(t)=\mathbf{a}(\mathbf{x}(t), \mathbf{u}(t), t)$$

$$J_1(\mathbf{u})=\int_{t_0}^{t_s}\left[\sum_i^m \beta_i\left|u_i(t)\right|\right] d t$$

$$J_2(\mathbf{1})=\int_{t_0}^{t r}\left[\sum_{i=1}^m r_i u_i^2(t)\right] d t,$$

## 数学代写|最优化理论作业代写optimization theory代考|Minimum-Fuel Problems

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

$$J(\mathbf{u})=\int_{t_0}^{t_t}\left[\sum_{i=1}^m\left|u_i(t)\right|\right] d t$$

$$-1 \leq u_i(t) \leq+1, \quad i=1,2, \ldots, m, \quad t \in\left[t_0, t_f\right]$$

## 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。