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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|组合优化代写Combinatorial optimization代考|Implementation of the Simplex Algorithm

The previous description of the SIMPLEX ALGORITHM is simple but not suitable for an efficient implementation. As we will see, it is not necessary to solve a linear equation system in each iteration. To motivate the main idea, we start with a proposition (which is actually not needed later): for LPs of the form $\max {c x: A x=b, x \geq 0}$, vertices can be represented not only by subsets of rows but also by subsets of columns.

For a matrix $A$ and a set $J$ of column indices we denote by $A^J$ the submatrix consisting of the columns in $J$ only. Consequently, $A_I^J$ denotes the submatrix of $A$ with rows in $I$ and columns in $J$. Sometimes the order of the rows and columns is important: if $J=\left(j_1, \ldots, j_k\right)$ is a vector of row (column) indices, we denote by $A_J\left(A^J\right)$ the matrix whose $i$-th row (column) is the $j_i$-th row (column) of $A$ $(i=1, \ldots, k)$.

Proposition 3.15. Let $P:={x: A x=b, x \geq 0}$, where $A$ is a matrix and $b$ is a vector. Then $x$ is a vertex of $P$ if and only if $x \in P$ and the columns of $A$ corresponding to positive entries of $x$ are linearly independent.

Proof: Let $A$ be an $m \times n$-matrix. Let $X:=\left(\begin{array}{cc}-I & 0 \ A & I\end{array}\right)$ and $b^{\prime}:=\left(\begin{array}{l}0 \ b\end{array}\right)$. Let $N:=$ ${1, \ldots, n}$ and $M:={n+1, \ldots, n+m}$. For an index set $J \subseteq N \cup M$ with $|J|=n$ let $\bar{J}:=(N \cup M) \backslash J$. Then $X_J^N$ is nonsingular iff $X_{M \cap J}^{N \cap J}$ is nonsingular iff $X_M^J$ is nonsingular.

If $x$ is a vertex of $P$, then – by Proposition $3.9$ – there exists a set $J \subseteq N \cup M$ such that $|J|=n, X_J^N$ is nonsingular, and $X_J^N x=b_J^{\prime}$. Then the components of $x$ corresponding to $N \cap J$ are zero. Moreover, $X_M^J$ is nonsingular, and hence the columns of $A^{N \cap \bar{J}}$ are linearly independent.

Conversely, let $x \in P$, and let the set of columns of $A$ corresponding to positive entries of $x$ be linearly independent. By adding suitable unit column vectors to these columns we obtain a nonsingular submatrix $X_M^B$ with $x_i=0$ for $i \in N \backslash B$. Then $X_{\bar{B}}^N$ is nonsingular and $X_{\bar{B}}^N x=b_{\bar{B}}^{\prime}$. Hence, by Proposition $3.9, x$ is a vertex of $P$.

数学代写|组合优化代写Combinatorial optimization代考|Convex Hulls and Polytopes

In this section we collect some more facts on polytopes. In particular, we show that polytopes are precisely those sets that are the convex hull of a finite number of points. We start by recalling some basic definitions:

Definition 3.30. Given vectors $x_1, \ldots, x_k \in \mathbb{R}^n$ and $\lambda_1, \ldots, \lambda_k \geq 0$ with $\sum_{i=1}^k \lambda_i=1$, we call $x=\sum_{i=1}^k \lambda_i x_i a$ convex combination of $x_1, \ldots, x_k$. A set $X \subseteq \mathbb{R}^n$ is convex if $\lambda x+(1-\lambda) y \in X$ for all $x, y \in X$ and $\lambda \in[0,1]$. The convex hull $\operatorname{conv}(X)$ of a set $X$ is defined as the set of all convex combinations of points in $X$. An extreme point of a set $X$ is an element $x \in X$ with $x \notin \operatorname{conv}(X \backslash{x})$.

So a set $X$ is convex if and only if all convex combinations of points in $X$ are again in $X$. The convex hull of a set $X$ is the smallest convex set containing $X$. Moreover, the intersection of convex sets is convex. Hence polyhedra are convex. Now we prove the “finite basis theorem for polytopes”, a fundamental result which seems to be obvious but is not trivial to prove directly:

Theorem 3.31. (Minkowski [1896], Steinitz [1916], Weyl [1935]) A set P is a polytope if and only if it is the convex hull of a finite set of points.

Proof: (Schrijver [1986]) Let $P=\left{x \in \mathbb{R}^n: A x \leq b\right}$ be a nonempty polytope. Obviously,
$$P=\left{x:\left(\begin{array}{l} x \ 1 \end{array}\right) \in C\right} \text {, where } C=\left{\left(\begin{array}{l} x \ \lambda \end{array}\right) \in \mathbb{R}^{n+1}: \lambda \geq 0, A x-\lambda b \leq 0\right} .$$
$C$ is a polyhedral cone, so by Theorem $3.29$ it is generated by finitely many nonzero vectors, say by $\left(\begin{array}{l}x_1 \ \lambda_1\end{array}\right), \ldots,\left(\begin{array}{l}x_k \ \lambda_k\end{array}\right)$. Since $P$ is bounded, all $\lambda_i$ are nonzero; w.l.o.g. all $\lambda_i$ are 1 . So $x \in P$ if and only if
$$\left(\begin{array}{l} x \ 1 \end{array}\right)=\mu_1\left(\begin{array}{c} x_1 \ 1 \end{array}\right)+\cdots+\mu_k\left(\begin{array}{c} x_k \ 1 \end{array}\right)$$
for some $\mu_1, \ldots, \mu_k \geq 0$. In other words, $P$ is the convex hull of $x_1, \ldots, x_k$.
Now let $P$ be the convex hull of $x_1, \ldots, x_k \in \mathbb{R}^n$. Then $x \in P$ if and only if $\left(\begin{array}{c}x \ 1\end{array}\right) \in C$, where $C$ is the cone generated by $\left(\begin{array}{c}x_1 \ 1\end{array}\right), \ldots,\left(\begin{array}{c}x_k \ 1\end{array}\right)$. By Theorem $3.29, C$ is polyhedral, so
$$C=\left{\left(\begin{array}{l} x \ \lambda \end{array}\right): A x+b \lambda \leq 0\right} .$$
We conclude that $P=\left{x \in \mathbb{R}^n: A x+b \leq 0\right}$.

组合优化代考

数学代写|组合优化代写组合优化代考|凸壳和Polytopes

$$P=\left{x:\left(\begin{array}{l} x \ 1 \end{array}\right) \in C\right} \text {, where } C=\left{\left(\begin{array}{l} x \ \lambda \end{array}\right) \in \mathbb{R}^{n+1}: \lambda \geq 0, A x-\lambda b \leq 0\right} .$$
$C$是一个多面体锥，因此根据定理$3.29$它是由有限个非零向量生成的，比如$\left(\begin{array}{l}x_1 \ \lambda_1\end{array}\right), \ldots,\left(\begin{array}{l}x_k \ \lambda_k\end{array}\right)$。由于$P$是有界的，所有$\lambda_i$都是非零的;W.L.O.G.所有$\lambda_i$都是1。所以$x \in P$当且仅当
$$\left(\begin{array}{l} x \ 1 \end{array}\right)=\mu_1\left(\begin{array}{c} x_1 \ 1 \end{array}\right)+\cdots+\mu_k\left(\begin{array}{c} x_k \ 1 \end{array}\right)$$

$$C=\left{\left(\begin{array}{l} x \ \lambda \end{array}\right): A x+b \lambda \leq 0\right} .$$

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MATLAB代写

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assignmentutor™您的专属作业导师
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