## 数学代写|凸优化作业代写Convex Optimization代考|MATH4071

2023年2月1日

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## 数学代写|凸优化作业代写Convex Optimization代考|Convex hull description of polyhedra

The convex hull of the finite set $\left{v_1, \ldots, v_k\right}$ is
$$\operatorname{conv}\left{v_1, \ldots, v_k\right}=\left{\theta_1 v_1+\cdots+\theta_k v_k \mid \theta \succeq 0, \mathbf{1}^T \theta=1\right}$$
This set is a polyhedron, and bounded, but (except in special cases, e.g., a simplex) it is not simple to express it in the form (2.5), i.e., by a set of linear equalities and inequalities.
A generalization of this convex hull description is
$$\left{\theta_1 v_1+\cdots+\theta_k v_k \mid \theta_1+\cdots+\theta_m=1, \theta_i \geq 0, i=1, \ldots, k\right},$$
where $m \leq k$. Here we consider nonnegative linear combinations of $v_i$, but only the first $m$ coefficients are required to sum to one. Alternatively, we can interpret (2.9) as the convex hull of the points $v_1, \ldots, v_m$, plus the conic hull of the points $v_{m+1}, \ldots, v_k$. The set $(2.9)$ defines a polyhedron, and conversely, every polyhedron can be represented in this form (although we will not show this).

The question of how a polyhedron is represented is subtle, and has very important practical consequences. As a simple example consider the unit ball in the $\ell_{\infty}-$ norm in $\mathbf{R}^n$
$$C=\left{x|| x_i \mid \leq 1, i=1, \ldots, n\right} .$$
The set $C$ can be described in the form (2.5) with $2 n$ linear inequalities $\pm e_i^T x \leq 1$, where $e_i$ is the $i$ th unit vector. To describe it in the convex hull form (2.9) requires at least $2^n$ points:
$$C=\operatorname{conv}\left{v_1, \ldots, v_{2^n}\right},$$
where $v_1, \ldots, v_{2^n}$ are the $2^n$ vectors all of whose components are 1 or $-1$. Thus the size of the two descriptions differs greatly, for large $n$.

## 数学代写|凸优化作业代写Convex Optimization代考|Linear-fractional functions

A linear-fractional function is formed by composing the perspective function with an affine function. Suppose $g: \mathbf{R}^n \rightarrow \mathbf{R}^{m+1}$ is affine, i.e.,
$$g(x)=\left[\begin{array}{c} A \ c^T \end{array}\right] x+\left[\begin{array}{l} b \ d \end{array}\right]$$
where $A \in \mathbf{R}^{m \times n}, b \in \mathbf{R}^m, c \in \mathbf{R}^n$, and $d \in \mathbf{R}$. The function $f: \mathbf{R}^n \rightarrow \mathbf{R}^m$ given by $f=P \circ g$, i.e.,
$$f(x)=(A x+b) /\left(c^T x+d\right), \quad \operatorname{dom} f=\left{x \mid c^T x+d>0\right},$$
is called a linear-fractional (or projective) function. If $c=0$ and $d>0$, the domain of $f$ is $\mathbf{R}^n$, and $f$ is an affine function. So we can think of affine and linear functions as special cases of linear-fractional functions.

Like the perspective function, linear-fractional functions preserve convexity. If $C$ is convex and lies in the domain of $f$ (i.e., $c^T x+d>0$ for $x \in C$ ), then its image $f(C)$ is convex. This follows immediately from results above: the image of $C$ under the affine mapping (2.12) is convex, and the image of the resulting set under the perspective function $P$, which yields $f(C)$, is convex. Similarly, if $C \subseteq \mathbf{R}^m$ is convex, then the inverse image $f^{-1}(C)$ is convex.

## 数学代写|凸优化作业代写Convex Optimization代考|Convex hull description of polyhedra

loperatorname{conv}}left{v_1, 1 dots, $\left.v_{-} k \mid r i g h t\right}=\backslash$ left ${t}$

Veft{ltheta_1 v_1+Icdots+ltheta_k v_k \mid Itheta_1+Icd

$C=\backslash$ 左 $\left{x|| x_{-} i \backslash m i d \mid l\right.$ leq $1, i=1, |$ dots, n\right } } \text { 。 }

$\pm e_i^T x \leq 1$ ，在哪里 $e_i$ 是个 $i$ 第单位向量。用凸包形式
(2.9) 来描述它至少需要 $2^n$ 要点:

## 数学代写|凸优化作业代写Convex Optimization代考|Linear-fractional functions

$$g(x)=\left[A c^T\right] x+[b d]$$

$f(x)=(A x+b) / \operatorname{left}\left(c^{\wedge} T x+d \backslash r i g h t\right)$, lquad loperatorname ${\Lambda$

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