## 数学代写|凸优化作业代写Convex Optimization代考|ELEN90026

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|Hyperplanes and halfspaces

A hyperplane is a set of the form
$$\left{x \mid a^T x=b\right},$$
where $a \in \mathbf{R}^n, a \neq 0$, and $b \in \mathbf{R}$. Analytically it is the solution set of a nontrivial linear equation among the components of $x$ (and hence an affine set). Geometrically, the hyperplane $\left{x \mid a^T x=b\right}$ can be interpreted as the set of points with a constant inner product to a given vector $a$, or as a hyperplane with normal vector $a$; the constant $b \in \mathbf{R}$ determines the offset of the hyperplane from the origin. This geometric interpretation can be understood by expressing the hyperplane in the form
$$\left{x \mid a^T\left(x-x_0\right)=0\right},$$
where $x_0$ is any point in the hyperplane (i.e., any point that satisfies $a^T x_0=b$ ). This representation can in turn be expressed as
$$\left{x \mid a^T\left(x-x_0\right)=0\right}=x_0+a^{\perp},$$
where $a^{\perp}$ denotes the orthogonal complement of $a$, i.e., the set of all vectors orthogonal to it:
$$a^{\perp}=\left{v \mid a^T v=0\right} .$$
This shows that the hyperplane consists of an offset $x_0$, plus all vectors orthogonal to the (normal) vector $a$. These geometric interpretations are illustrated in figure $2.6$.

A hyperplane divides $\mathbf{R}^n$ into two halfspaces. A (closed) halfspace is a set of the form
$$\left{x \mid a^T x \leq b\right},$$
where $a \neq 0$, i.e., the solution set of one (nontrivial) linear inequality. Halfspaces are convex, but not affine. This is illustrated in figure $2.7$.

## 数学代写|凸优化作业代写Convex Optimization代考|Euclidean balls and ellipsoids

A (Euclidean) ball (or just ball) in $\mathbf{R}^n$ has the form
$$B\left(x_c, r\right)=\left{x \mid\left|x-x_c\right|_2 \leq r\right}=\left{x \mid\left(x-x_c\right)^T\left(x-x_c\right) \leq r^2\right},$$
where $r>0$, and $|\cdot|_2$ denotes the Euclidean norm, i.e., $|u|_2=\left(u^T u\right)^{1 / 2}$. The vector $x_c$ is the center of the ball and the scalar $r$ is its radius; $B\left(x_c, r\right)$ consists of all points within a distance $r$ of the center $x_c$. Another common representation for the Euclidean ball is
$$B\left(x_c, r\right)=\left{x_c+r u \mid|u|_2 \leq 1\right} .$$

A Euclidean ball is a convex set: if $\left|x_1-x_c\right|_2 \leq r,\left|x_2-x_c\right|_2 \leq r$, and $0 \leq \theta \leq 1$, then
\begin{aligned} \left|\theta x_1+(1-\theta) x_2-x_c\right|_2 & =\left|\theta\left(x_1-x_c\right)+(1-\theta)\left(x_2-x_c\right)\right|_2 \ & \leq \theta\left|x_1-x_c\right|_2+(1-\theta)\left|x_2-x_c\right|_2 \ & \leq r . \end{aligned}
(Here we use the homogeneity property and triangle inequality for $|\cdot|_2$; see $\S$ A.1.2.) A related family of convex sets is the ellipsoids, which have the form
$$\mathcal{E}=\left{x \mid\left(x-x_c\right)^T P^{-1}\left(x-x_c\right) \leq 1\right},$$
where $P=P^T \succ 0$, i.e., $P$ is symmetric and positive definite. The vector $x_c \in \mathbf{R}^n$ is the center of the ellipsoid. The matrix $P$ determines how far the ellipsoid extends in every direction from $x_c$; the lengths of the semi-axes of $\mathcal{E}$ are given by $\sqrt{\lambda_i}$, where $\lambda_i$ are the eigenvalues of $P$. A ball is an ellipsoid with $P=r^2 I$. Figure $2.9$ shows an ellipsoid in $\mathbf{R}^2$.
Another common representation of an ellipsoid is
$$\mathcal{E}=\left{x_c+A u \mid|u|_2 \leq 1\right},$$
where $A$ is square and nonsingular. In this representation we can assume without loss of generality that $A$ is symmetric and positive definite. By taking $A=P^{1 / 2}$, this representation gives the ellipsoid defined in (2.3). When the matrix $A$ in (2.4) is symmetric positive semidefinite but singular, the set in (2.4) is called a degenerate ellipsoid; its affine dimension is equal to the rank of $A$. Degenerate ellipsoids are also convex.

## 数学代写|凸优化作业代写Convex Optimization代考|Hyperplanes and halfspaces

$\backslash$ eft ${x \backslash$ mid $a \wedge T x=b \backslash r i g h t}$

$\backslash$ eft $\left{x \backslash\right.$ mid $a^{\wedge} T \backslash l$ eft $(x-x$ 아ight $)=0$ \right } } ,

$a^{\wedge}{\backslash$ perp $}=\backslash l$ eft $\left{v \backslash\right.$ mid $a^{\wedge} T v=0 \backslash$ right $}$ 。

$\backslash$ eft $\left{x \backslash\right.$ mid $a^{\wedge} T x \backslash$ leq b\right } } ,

## 数学代写|凸优化作业代写Convex Optimization代考|Euclidean balls and ellipsoids

$\left|x_1-x_c\right|_2 \leq r,\left|x_2-x_c\right|_2 \leq r ， \quad$ 和 $0 \leq \theta \leq 1 ＼mathrm{~ ， ~}$ 然后
$$\left|\theta x_1+(1-\theta) x_2-x_c\right|_2=\mid \theta\left(x_1-x_c\right)+(1-\theta)$$
(这里我们使用同质性和三角不等式来表示 $|\cdot|_2$; 看
A.1.2.) 一个相关的凸集族是椭圆体，它有以下形式

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