From af188d3046433284b6193c37fe7b520500301588 Mon Sep 17 00:00:00 2001 From: Josh Gao Date: Sat, 8 Jun 2024 17:29:33 +0800 Subject: [PATCH] new post cvx-1.2 --- package.json | 1 - ...tion-1-1.md => 2022-03-29-josh-cvx-1-1.md} | 13 +- source/_posts/2024-06-08-josh-cvx-1-2.md | 377 ++++++++++++++++++ ...-1-1-010-TheLinePassingThroughX1AndX2.png} | Bin ...-020-SomeSimpleConvexAndNonconvexSets.png} | Bin ...h-cvx-1-1-030-TheConvexHullsOfTwoSets.png} | Bin ...22-03-29-josh-cvx-1-1-040-ThePieSlice.png} | Bin ...-03-29-josh-cvx-1-1-050-TheConicHulls.png} | Bin ...-06-08-josh-cvx-1-2-060-HyperplaneInR2.png | Bin 0 -> 12811 bytes ...vx-1-2-070-AHyperplaneAndTwoHalfspaces.png | Bin 0 -> 20114 bytes ...-josh-cvx-1-2-080-VectorsAndHalfspaces.png | Bin 0 -> 19647 bytes ...4-06-08-josh-cvx-1-2-090-EllipsoidInR2.png | Bin 0 -> 20907 bytes ...-cvx-1-2-100-BoundaryOfSecondOrderCone.png | Bin 0 -> 99703 bytes ...2024-06-08-josh-cvx-1-2-110-Polyhydron.png | Bin 0 -> 35497 bytes ...-120-BoundaryOfPostiveSemidefiniteCone.png | Bin 0 -> 136611 bytes 15 files changed, 384 insertions(+), 7 deletions(-) rename source/_posts/{2022-03-29-josh-convex-optimization-1-1.md => 2022-03-29-josh-cvx-1-1.md} (93%) create mode 100644 source/_posts/2024-06-08-josh-cvx-1-2.md rename source/images/post/{2022-03-29-josh-convex-optimization-1-1/2022-03-29-josh-convex-optimization-1-1-010-TheLinePassingThroughX1AndX2.png => 2022-03-29-josh-cvx-1-1/2022-03-29-josh-cvx-1-1-010-TheLinePassingThroughX1AndX2.png} (100%) rename source/images/post/{2022-03-29-josh-convex-optimization-1-1/2022-03-29-josh-convex-optimization-1-1-020-SomeSimpleConvexAndNonconvexSets.png => 2022-03-29-josh-cvx-1-1/2022-03-29-josh-cvx-1-1-020-SomeSimpleConvexAndNonconvexSets.png} (100%) rename source/images/post/{2022-03-29-josh-convex-optimization-1-1/2022-03-29-josh-convex-optimization-1-1-030-TheConvexHullsOfTwoSets.png => 2022-03-29-josh-cvx-1-1/2022-03-29-josh-cvx-1-1-030-TheConvexHullsOfTwoSets.png} (100%) rename source/images/post/{2022-03-29-josh-convex-optimization-1-1/2022-03-29-josh-convex-optimization-1-1-040-ThePieSlice.png => 2022-03-29-josh-cvx-1-1/2022-03-29-josh-cvx-1-1-040-ThePieSlice.png} (100%) rename source/images/post/{2022-03-29-josh-convex-optimization-1-1/2022-03-29-josh-convex-optimization-1-1-050-TheConicHulls.png => 2022-03-29-josh-cvx-1-1/2022-03-29-josh-cvx-1-1-050-TheConicHulls.png} (100%) create mode 100644 source/images/post/2024-06-08-josh-cvx-1-2/2024-06-08-josh-cvx-1-2-060-HyperplaneInR2.png create mode 100644 source/images/post/2024-06-08-josh-cvx-1-2/2024-06-08-josh-cvx-1-2-070-AHyperplaneAndTwoHalfspaces.png create mode 100644 source/images/post/2024-06-08-josh-cvx-1-2/2024-06-08-josh-cvx-1-2-080-VectorsAndHalfspaces.png create mode 100644 source/images/post/2024-06-08-josh-cvx-1-2/2024-06-08-josh-cvx-1-2-090-EllipsoidInR2.png create mode 100644 source/images/post/2024-06-08-josh-cvx-1-2/2024-06-08-josh-cvx-1-2-100-BoundaryOfSecondOrderCone.png create mode 100644 source/images/post/2024-06-08-josh-cvx-1-2/2024-06-08-josh-cvx-1-2-110-Polyhydron.png create mode 100644 source/images/post/2024-06-08-josh-cvx-1-2/2024-06-08-josh-cvx-1-2-120-BoundaryOfPostiveSemidefiniteCone.png diff --git a/package.json b/package.json index f6c1aaf..a013d01 100644 --- a/package.json +++ b/package.json @@ -25,7 +25,6 @@ "hexo-generator-tag": "^2.0.0", "hexo-helper-live2d": "^3.1.1", "hexo-neat": "^1.0.9", - "hexo-related-posts": "^1.5.2", "hexo-renderer-ejs": "^2.0.0", "hexo-renderer-pandoc": "^0.4.0", "hexo-renderer-stylus": "^3.0.1", diff --git a/source/_posts/2022-03-29-josh-convex-optimization-1-1.md b/source/_posts/2022-03-29-josh-cvx-1-1.md similarity index 93% rename from source/_posts/2022-03-29-josh-convex-optimization-1-1.md rename to source/_posts/2022-03-29-josh-cvx-1-1.md index 87df7e1..7a83ee5 100644 --- a/source/_posts/2022-03-29-josh-convex-optimization-1-1.md +++ b/source/_posts/2022-03-29-josh-cvx-1-1.md @@ -15,8 +15,9 @@ categories: - 凸优化 tags: - 凸优化 + - 凸集 - 仿射 - - 保凸运算 + - 锥 abbrlink: 43146ee9 date: 2022-03-29 11:07:04 --- @@ -51,7 +52,7 @@ $$ -![图 1. 通过 $x_1$ 和 $x_2$ 的直线可以参数化描述为 $\theta x_1 + (1-\theta)x_2$, 其中 $\theta$ 在 $\bfR$ 上变化。$x_1$ 和 $x_2$ 之间的线段由深色所示,对应处于 $0$ 和 $1$ 之间的 $\theta$。](../images/post/2022-03-29-josh-convex-optimization-1-1/2022-03-29-josh-convex-optimization-1-1-010-TheLinePassingThroughX1AndX2.png){width=600px} +![图 1. 通过 $x_1$ 和 $x_2$ 的直线可以参数化描述为 $\theta x_1 + (1-\theta)x_2$, 其中 $\theta$ 在 $\bfR$ 上变化。$x_1$ 和 $x_2$ 之间的线段由深色所示,对应处于 $0$ 和 $1$ 之间的 $\theta$。](../images/post/2022-03-29-josh-cvx-1-1/2022-03-29-josh-cvx-1-1-010-TheLinePassingThroughX1AndX2.png){width=600px} @@ -122,7 +123,7 @@ $$ -![图 2. 一些简单的凸和非凸集合。**左**:包含其边界的六边形是凸的。**中**:肾形集合不是凸的,因为图中所示集合中两点间的线段不为集合所包含。**右**:仅包含部分边界的正方形不是凸的。](../images/post/2022-03-29-josh-convex-optimization-1-1/2022-03-29-josh-convex-optimization-1-1-020-SomeSimpleConvexAndNonconvexSets.png){width=1000px} +![图 2. 一些简单的凸和非凸集合。**左**:包含其边界的六边形是凸的。**中**:肾形集合不是凸的,因为图中所示集合中两点间的线段不为集合所包含。**右**:仅包含部分边界的正方形不是凸的。](../images/post/2022-03-29-josh-cvx-1-1/2022-03-29-josh-cvx-1-1-020-SomeSimpleConvexAndNonconvexSets.png){width=1000px}   集合 $C$ 中所有点的凸组合的集合称为集合 $C$ 的凸包,记为 $\conv C$: @@ -134,7 +135,7 @@ $$ -![图 3. $\bfR^2$ 上两个集合的凸包。**左**:(如图所示的)十五个点的集合的凸包是一个五边形示)。**右**:[图 2](#图2) 中的肾形集合的凸包是阴影所示的集合。](../images/post/2022-03-29-josh-convex-optimization-1-1/2022-03-29-josh-convex-optimization-1-1-030-TheConvexHullsOfTwoSets.png){width=800px} +![图 3. $\bfR^2$ 上两个集合的凸包。**左**:(如图所示的)十五个点的集合的凸包是一个五边形示)。**右**:[图 2](#图2) 中的肾形集合的凸包是阴影所示的集合。](../images/post/2022-03-29-josh-cvx-1-1/2022-03-29-josh-cvx-1-1-030-TheConvexHullsOfTwoSets.png){width=800px}   凸组合的概念可以扩展到无穷级数、积分以及大多数形式的概率分布。假设 $\theta_1,\theta_2,\cdots$ 满足 @@ -168,7 +169,7 @@ $$ -![图 4. 扇形展示了所有具有形式 $\theta_1 x_1 + \theta_2 x_2$ 的点,其中 $\theta_1, \theta_2 \geqslant 0$。扇形的顶点 ($\theta_1 = \theta_2 = 0$) 在 $0$;其边界(对应于 $$\theta_1 = 0$ 或 $\theta_2 = 0$) 穿过点 $x_1$ 和 $x_2$。](../images/post/2022-03-29-josh-convex-optimization-1-1/2022-03-29-josh-convex-optimization-1-1-040-ThePieSlice.png){width=400px} +![图 4. 扇形展示了所有具有形式 $\theta_1 x_1 + \theta_2 x_2$ 的点,其中 $\theta_1, \theta_2 \geqslant 0$。扇形的顶点 ($\theta_1 = \theta_2 = 0$) 在 $0$;其边界(对应于 $$\theta_1 = 0$ 或 $\theta_2 = 0$) 穿过点 $x_1$ 和 $x_2$。](../images/post/2022-03-29-josh-cvx-1-1/2022-03-29-josh-cvx-1-1-040-ThePieSlice.png){width=400px}   具有 $\theta_1 x_1 + \cdots + \theta_k x_k, \theta_1,\cdots,\theta_k$ 形式的点称为 $x_1,\cdots,x_k$ 的**锥组合**(*conic combination*)(或**非负线性组合**(*nonnegative linear combination*))。如果 $x_i$ 均属于凸锥 $C$,那么,$x_i$ 的每一个锥组合也在 $C$ 中。反言之,集合 $C$ 是凸锥的充要条件是它包含其元素的所有锥组合。如同凸(或仿射)组合一样,锥组合的概念可以扩展到无穷级数和积分中。 @@ -182,7 +183,7 @@ $$ -![图 5. [图 3](#图3) 中两个集合的锥包(阴影所示)。](../images/post/2022-03-29-josh-convex-optimization-1-1/2022-03-29-josh-convex-optimization-1-1-050-TheConicHulls.png){width=600px} +![图 5. [图 3](#图3) 中两个集合的锥包(阴影所示)。](../images/post/2022-03-29-josh-cvx-1-1/2022-03-29-josh-cvx-1-1-050-TheConicHulls.png){width=600px} # 参考文献 diff --git a/source/_posts/2024-06-08-josh-cvx-1-2.md b/source/_posts/2024-06-08-josh-cvx-1-2.md new file mode 100644 index 0000000..8aa0605 --- /dev/null +++ b/source/_posts/2024-06-08-josh-cvx-1-2.md @@ -0,0 +1,377 @@ +--- +title: Josh's Note — 凸优化
Part 1.2 凸集—一些重要的凸集 +mathjax: true +comments: true +copyright: true +toc: + enable: true + number: false + wrap: true + expand_all: true + max_depth: 4 +categories: + - - Josh 的学习笔记 + - 凸优化 + - 凸集 +tags: + - 凸优化 + - 凸集 + - 射线 + - 超平面 + - 半空间 + - Euclid球 + - 椭球 + - 范数球 + - 范数锥 + - 二阶锥 + - 多面体 + - 单纯形 + - 半正定锥 +abbrlink: 9293b739 +date: 2024-06-08 14:57:52 +--- + +$$ +\newcommand{\bfR}{\mathbf{R}} +\newcommand{\bfS}{\mathbf{S}} +\newcommand{\TT}{\mathrm{T}} +\newcommand{\E}{\mathcal{E}} +\newcommand{\P}{\mathcal{P}} +\def\conv{\mathop{\bf conv}} +$$ + +  本文将描述一些重要的凸集。首先介绍一些简单的例子。 + +- 空集 $\emptyset$、任意一个点(即单点集(singleton))$\{x_0\}$ 、全空间 $\bfR^n$ 都是 $\bfR^n$ 的仿射(自然也是凸的)子集。 + +- 任意直线是仿射的。如果直线通过零点,则是子空间,因此,也是凸锥。 + +- 一条线段是凸的,但不是仿射的(除非退化为一个点)。 + +- 一条**射线**(*ray*),即具有形式 $\left\{x_0 + \theta v \middle| \theta \geqslant 0 \right\}, v \ne 0$ 的集合,是凸的,但不是仿射的。如果射线的基点 $x_0$ 是 $0$,则它是凸锥。 + +- 任意子空间是仿射的、凸锥(自然是凸的)。 + + + +# 1. 超平面与半空间 + +  **超平面**(*hyperplane*)是具有如下形式的集合 + +$$ +\left\{ x \middle| a^\TT x = b \right\} +$$ + +其中 $a \in \bfR^n, a \ne 0$ 且 $b \in \bfR$。解析地来看,超平面是关于 $x$ 的非平凡线性方程的解空间(因此是一个仿射集合)。几何上,超平面 $\left\{ x \middle| a^\TT x = b \right\}$ 可以理解为与给定向量 $a$ 的内积为常数的点的集合;也可以看成法线方向为 $a$ 的超平面,而常数 $b \in \bfR$ 决定了这个平面从原点的偏移。为更好地理解几何含义,可以将超平面表示成 + +$$ +\left\{ x \middle| a^\TT (x - x_0) = 0 \right\} +$$ + +其中 $x_0$ 是超平面上的任意一点(即任意满足 $a^\TT x_0 = b$ 的点)。进一步,可以表示为 + +$$ +\left\{ x \middle| a^\TT (x - x_0) = 0 \right\} = x_0 + a^\perp +$$ + +其中 $a^\perp$ 表示 $a$ 的正交补,即与 $a$ 正交的向量的集合: + +$$ +a^\perp = \left\{ v \middle| a^\TT v = 0 \right\} +$$ + +从中可以看出,超平面由偏移 $x_0$ 加上所有正交于(法)向量 $a$ 的向量构成,具体可见[图 6](#图6) 。 + + + +![图 6. $\bfR^2$ 中由法向量 $a$ 和超平面上一点 $x_0$ 确定的超平面。对于超平面上任意一点 $x$,$x - x_0$(如深色箭头所示)都垂直于 $a$。](../images/post/2024-06-08-josh-cvx-1-2/2024-06-08-josh-cvx-1-2-060-HyperplaneInR2.png){width=400px} + +  一个超平面将 $\bfR^n$ 划分为两个**半空间**(*halfspaces*)。(闭的)半空间是具有下列形式的集合, + +$$\begin{equation}\label{HalfSpace} + \left\{x \middle| a^\TT x \leqslant b \right\} +\end{equation} +$$ + +即(非平凡的)线性不等式的解空间,其中 $a\ne 0$。半空间是凸的,但不是仿射的,如[图 7](#图7) 所示。 + + + +![图 7. $\bfR^2$ 上由 $a^\TT x = b$ 定义的超平面决定了两个半空间。由 a^\TT x \geqslant b 决定的半空间(无阴影)是向 $a$ 扩展的半空间。由 a^\TT x \leqslant b 确定的半空间(阴影所示)向 $-a$ 方向扩展。向量 $a$ 是这个半空间向外的法向量。](../images/post/2024-06-08-josh-cvx-1-2/2024-06-08-josh-cvx-1-2-070-AHyperplaneAndTwoHalfspaces.png){width=400px} + +  半空间 $\eqref{HalfSpace}$ 也可以表示为 + +$$ +\left\{x \middle| a^\TT (x - x_0) \leqslant 0 \right\} +$$ + +其中 $x_0$ 是相应超平面上的任意一点,即 $x_0$ 满足 $a^\TT x_0 = b$。上式有一个简单的几何解释:半空间由 $x_0$ 加上任意与(向外的法)向量 $a$ 呈钝角(或直角)的向量组 +成,如[图 8](#图8) 所示。 + + + +![图 8. 阴影所示集合是由 $a^\TT (x - x_0) \leqslant 0$ 决定的半空间。向量 $x_1 - x_0$ 与 $a$ 的夹角为锐角,所以 $x_1$不在这个半空间中。向量 $x_2 - x_1$ 与 $a$ 的夹角为钝角,所以 $x_2$ 在半空间中。](../images/post/2024-06-08-josh-cvx-1-2/2024-06-08-josh-cvx-1-2-080-VectorsAndHalfspaces.png){width=400px} + +  半空间 $\eqref{HalfSpace}$ 的边界是超平面 $\left\{x \middle| a^\TT x = b \right\}$。集合 $\left\{x \middle| a^\TT x < b \right\}$ 是半空间 $\left\{x \middle| a^\TT x \leqslant b \right\}$ 的内部,称为**开半空间**(*open halfspace*)。 + +# 2. Euclid球和椭球 + +  $\bfR^n$ 中的空间 Euclid 球(*(Euclidean) ball*)(或简称为球)具有下面的形式: + +$$ +B(x_c,r) = \left\{x \middle| \left\| x - x_c \right\|_2 \leqslant r \right\} = \left\{ x \middle| (x-x_c)^\TT(x-x_c) \leqslant r^2 \right\} +$$ + +其中 $r > O$, $\| \cdot \|_2$ 表示 Euclid 范数,即 $\|u\|_2 = (u^\TT u)^{1/2}$。向量 $x_r$ 是**球心**(*center*),标量 $r$ 为半径。$B (x_c, r)$ 由距离球心 $x_r$ 距离不超过 $r$ 的所有点组成。 Euclid 球的另一个常见的表达式为: + +$$ +B(x_c,r) = \left\{ x_c + ru \middle| \|u\|_2 \leqslant 1 \right\} +$$ + +  Euclid 球是凸集,即如果 $\|x_1-x_2\|\leqslant r$,$\|x_2-x_c\|\leqslant r$,并且 $0 \leqslant \theta \leqslant 1$,那么 + +$$\begin{aligned} + \left\| \theta x_1 + (1- \theta)x_2 - x_c\right\|_2 &= \left\| \theta(x_1 - x_c) + (1-\theta)(x_2 - x_c)\right\|_2 + \\ + &\leqslant \theta \left\|x_1 - x_c\right\|_2 + (1-\theta) \left\| x_2 - x_c \right\|_2 + \\ + & \leqslant r +\end{aligned}$$ + +(此处利用了 $\| \cdot \|_2$ 的齐次性和三角不等式。) + +  一类相关的凸集是**椭球**(*ellipsoid*),它们具有如下的形式: + +$$\begin{equation}\label{ellipsoid-1} + \E = \left\{ x \middle| (x-x_c)^\TT P^{-1}(x-x_c) \leqslant 1 \right\} +\end{equation}$$ + +其中 $P = P^\TT \succ 0$,即 $P$ 是对称正定矩阵。向量 $x_c \in \bfR^n$ 为椭球的中心。矩阵 $P$ 决定了椭球从 $x_c$ 向各个方向扩展的幅度。$\E$ 的半轴长度由 $\sqrt{\lambda_i}$ 给出,这里 $\lambda_i$ 为 $P$ 的特征值。球可以看成 $P =r^2I$ 的椭球。[图 9](#图9) 给出了 $\bfR^2$ 上的一个椭球。 + + + +![图 9. 阴影所示为 $\bfR^2$ 中的一个椭球。中心 $x_c$ 由点显示,两个半轴由线段显示](../images/post/2024-06-08-josh-cvx-1-2/2024-06-08-josh-cvx-1-2-090-EllipsoidInR2.png){width=300px} + +  椭球另一个常用的表示形式是 + +$$\begin{equation}\label{ellipsoid-2} + \E = \left\{ x_c + Au \middle| \|u\|_2 \leqslant 1 \right\} +\end{equation}$$ + +其中 $A$ 是非奇异的方阵。在此类表示形式中,我们可以不失一般性地假设 $A$ 对称正定。取 $A=P^{1/2}$,这个表达式给出了由式 $\eqref{ellipsoid-1}$ 定义的椭球。当式 $\eqref{ellipsoid-2}$ 中的矩阵 $A$ 为对称半正定矩阵,但奇异时,集合 $\eqref{ellipsoid-2}$ 称为**退化的椭球**(*degenerate ellipsoid*),其仿射维数等于 $A$ 的秩。退化的椭球也是凸的。 + +# 3. 范数球和范数锥 + +  设 $\| \cdot \|$ 是 $\bfR^n$ 中的范数。由范数的一般性质可知,以 $r$ 为半径,$x_c$ 为球心的**范数球**(*norm ball*) $\left\{ x \middle| \|x-x_c\| \leqslant r \right\}$ 是凸的。关于范数 $\| \cdot \|$ 的**范数锥**(*norm cone*)是集合 + +$$ +C = \left\{ (x,t) | \|t\| \leqslant t \right\} \subseteq \bfR^{n+1} +$$ + +顾名思义,它是一个凸锥。 + +> **举例** **二阶锥**(*second-order cone*)是由 Euclid 范数定义的范数锥,即 +> +> $$\begin{aligned} +> C &= \left\{ (x,t) \in \bfR^{n+1} \middle| \|x\|_2 \leqslant t\right\} +> \\ +> &= \left\{ \left[\begin{array}{c} +> x \\ t +> \end{array} \right] \middle| \left[\begin{array}{c} +> x \\ t +> \end{array} \right]^\TT\left[\begin{array}{c} +> I & 0 \\ 0 & -1 +> \end{array} \right] \left[\begin{array}{c} +> x \\ t +> \end{array} \right] \leqslant 0,\ t\geqslant 0 \right\} +> \end{aligned}$$ +> +> 二阶锥的其他名字也常常被使用。它由二次不等式定义,因此也被称为**二次锥**(*quadratic cone*)。同时,也称其为 **Lorentz 锥**(*Lorentz cone*)或**冰激凌锥**(*ice-cream cone*)。[图 10](#图10) 显示了 $\bfR^3$ 上一个的二阶锥。 +> +> +> +> ![图 10. $\bfR^3$ 中二阶锥 $\left\{(x_1,x_2,t)\middle| (x_1^2 + x_2^2)^{1/2} \leqslant t\right\}$ 的边界](../images/post/2024-06-08-josh-cvx-1-2/2024-06-08-josh-cvx-1-2-100-BoundaryOfSecondOrderCone.png){width=500px} + +# 4. 多面体 + +  **多面体**(*polyhedron*)被定义为有限个线性等式和不等式的解集: + +$$\begin{equation}\label{polyhedron-1} + \P = \left\{ x \middle| a_j^\TT x \leqslant b_j,\ j = 1,\cdots,m,\ c_j^\TT x = d_j,\ j = 1,\cdots,p\right\} +\end{equation}$$ + +因此,多面体是有限个半空间和超平面的交集。仿射集合(例如子空间、超平面、直线)、射线、线段和半空间都是多面体。显而易见,多面体是凸集。有界的多面体有时也称为**多胞形**(*polytope*),但也有一些作者反过来使用这两个概念(即用多胞形表示具有 $eqref{polyhedron}$ 形式的集合,而当其有界时称为多面体)。[图 11](#图11) 显示了一个由五个半空间的交集定义的多面体。 + + + +![图 11. 多面体 $\P$(阴影所示)是外法向量为 $a_1,\cdots,a_5$ 的五个半空间的交集。](../images/post/2024-06-08-josh-cvx-1-2/2024-06-08-josh-cvx-1-2-110-Polyhydron.png){width=400px} + +  可以方便地使用紧凑表达式 + +$$\begin{equation}\label{polyhedron-2} + \P = \left\{ x \middle| Ax \preceq b,\ Cx = d \right\} +\end{equation}$$ + +来表示 $\eqref{polyhedron-1}$,其中 + +$$ +A = \left[\begin{array}{c} + a_1^\TT \\ \vdots \\ a_m^\TT +\end{array}\right],\qquad C = \left[\begin{array}{c} + c_1^\TT \\ \vdots \\ c_p^\TT +\end{array}\right] +$$ + +此处的 $\preceq$ 代表 $\bfR^m$ 上的**向量不等式**(*vector inequality*)或**分量不等式**(*componentwise inequality*):$u \preceq v$ 表示 $u_i \leqslant v_i, i = 1,\cdots, m$。 + +> **举例** **非负象限**(*nonnegative orthant*)是具有非负分量的点的集合,即 +> +> $$ +> \bfR_+^n = \left\{ x\in\bfR^n \middle| x_i \geqslant 0,\ i = 1,\cdots,n \right\} = \left\{ x\in \bfR^n \middle| x \succeq 0 \right\} +> $$ +> +> (此处 $\bfR_+$ 表示非负实数的集合,即 $\bfR_+ = \left\{ x\in\bfR \middle| x \geqslant 0 \right\}$。)非负象限既是多面体也是锥(因此称为**多面体锥**(*polyhedral cone*))。 + +## 4.1 单纯形 + +  **单纯形**(*simplex*)是一类重要的多面体。设 $k+1$ 个点 $v_0,\cdots,v_k \in \bfR^n$ 仿射独立,即 $v_1 - v_0, \cdots, v_k-v_0$ 线性独立,那么,这些点决定了一个单纯形: + +$$\begin{equation}\label{simplex} + C = \conv{\{v_0,\cdots,v_k\}} = \left\{ \theta_0v_0 + \cdots + \theta_k v_k \middle| \theta \succeq 0,\ \mathbf{1}^\TT\theta = 1 \right\} +\end{equation}$$ + +其中 $\mathbf{1}$ 表示所有分量均为 $1$ 的向量。这个单纯形的仿射维数为 $k$,因此也称为 $\bfR^n$ 空间的 $k$ 维单纯形。 + +> **举例** **一些常见的单纯形。** 1 维单纯形是一条线段;2 维单纯形是一个三角形(包含其内部); 3 维单纯形是一个四面体。 +> +> **单位单纯形**(*unit simplex*)是由零向量和单位向量 $\mathbf{0}, e_1,\cdots, e_n \in \bfR^n$ 决定的 $n$ 维单纯形。它可以表示为满足下列条件的向量的集合, +> +> $$ +> x \succeq 0,\qquad \mathbf{1}^\TT x\leqslant 1 +> $$ +> +> **概率单纯形**(*probability simplex*)是由单位向量 $e_1,\cdots, e_n \in \bfR^n$ 决定的 $n — 1$ 维单纯形。它是满足下列条件的向量的集合, +> +> $$ +> x \succeq 0,\qquad \mathbf{1}^\TT x= 1 +> $$ +> +> 概率单纯形中的向量对应于含有 $n$ 个元素的集合的概率分布,$x_i$ 可理解为第 $i$ 个元素的概率。 + +为用多面体来描述单纯形 $\eqref{simplex}$,我们采用以下步骤将其变换为 $\eqref{polyhedron-2}$ 的形式。由定义可知,$x\in C$ 的充要条件是,对于某些 $\theta\succeq 0, \mathbf{1}^\TT\theta=1$, 有 $x = \theta_0 v_0 + \theta_1 v_1 + \cdots + \theta_k v_k$。等价地,如果定义 $y = (\theta_1,\cdots,\theta_k)$ 和 + +$$ +B = \left[ \begin{array}{c} + v_1-v_0 & \cdots & v_k-v_0 +\end{array} \right] \in \bfR ^{n\times k} +$$ + +我们知道 $x\in C$ 的充要条件是 + +$$\begin{equation}\label{IFFCondition} + x = v_0 + By +\end{equation} +$$ + +对于 $y \succeq 0$,$\mathbf{1}^\TT y \leqslant 1$ 成立。注意到 $v_0,\cdots,v_k$ 仿射独立意味着矩阵 $B$ 的秩为 $k$。因此,存在非奇异矩阵 $A=(A_1,A_2)\in\bfR^{n\times n}$ 使得 + +$$ +AB = \left[ \begin{array}{c} + A_1 \\ A_2 +\end{array} \right] B = \left[ \begin{array}{c} + I \\ 0 +\end{array} \right] +$$ + +用 $A$ 左乘 $\eqref{IFFCondition}$,得到 + +$$ +A_1 x = A_1 v_0 + y,\qquad A_2 x = A_2 v_0 +$$ + +从中可以看出,$x\in C$ 当且仅当 $A_2 x = A_2 v_0$ 并且向量 $y = A_1 v_0 - A_1 x$ 满足 $y\succeq 0$ 和 $\mathbf{1}^\TT y \leqslant 1$。换言之得到了 $x\in C$ 的充要条件, + +$$ +A_2 x = A_2 v_0, \qquad A_1 x \succeq A_1 v_0, \qquad \mathbf{1}^\TT A_1 x \leqslant 1 + \mathbf{1}^\TT A_1 v_0 +$$ + +这些是 $x$ 的线性等式和不等式,因此,描述了一个多面体。 + +## 4.2 多面体的凸包描述 + +  有限集合 $\{v_1,\cdots,v_k\}$ 的凸包是 + +$$ +\conv\{v_1,\cdots,v_k\} = \left\{ \theta_1 v_1 + \cdots + \theta_k v_k \middle| \theta \succeq 0, \ \mathbf{1}^\TT \theta = 1 \right\} +$$ + +它是一个有界的多面体,但是(除非是例如单纯形这样的特殊情况)无法简单地用形如 $\eqref{polyhedron-1}$ 的式子,即用线性等式和不等式的集合将其表示。 + +  凸包表达式的一个扩展表示是, + +$$\begin{equation}\label{ExtendedHull} + \left\{ \theta_1 v_1 + \cdots + \theta_k v_k \middle| \theta_1 + \cdots + \theta_m = 1,\ \theta_i\geqslant 0, \ i=1,\cdots,k \right\} +\end{equation}$$ + +其中 $m\leqslant k$。此处我们考虑 $v_i$ 的非负线性组合,但是仅仅要求前 $m$ 个系数之和为一。此外,我们可以将 $\eqref{ExtendedHull}$ 解释为点 $v_1,\cdots,v_m$ 的凸包加上点 $v_{m+1},\cdots,v_k$ 的锥包。集合 $\eqref{ExtendedHull}$ 定义了一个多面体,反之亦然,每个多面体都可以表示为此类形式(虽然这一点此处并未证明)。 + +  “如何表示多面体”这一问题的求解是十分巧妙的,并且有一些非常实用的结果。一个简单的例子是定义在 $\ell_\infty$-范数空间的 $\bfR^n$ 上的单位球, + +$$ +C = \left\{x \middle| |x_i| \leqslant 1, \ i = 1,\cdots n\right\} +$$ + +$C$ 可以由 $2n$ 个线性不等式 $\pm e_i^\TT x \leqslant 1$ 表示为 $\eqref{polyhedron-1}$ 的形式,其中 $e_i$ 表示第 $i$ 维的单位向量。为了将其描述为形如 $\eqref{ExtendedHull}$ 的凸包,需要至少 $2^n$ 个点: + +$$ +C = \conv\{v_1,\cdots,v_{2^n}\} +$$ + +其中 $v_1,\cdots,v_{2^n}$ 是以 $1$ 和 $-1$ 为分量的全部向量,共 $2^n$ 个。可见,当 $n$ 很大时,这两种描述方式的规模相差极大。 + +# 5. 半正定锥 + +  用 $\bfS^n$ 表示对称 $n\times n$ 矩阵的集合,即 + +$$ +\bfS^n = \left\{ X \in \bfR ^{n\times n} \middle| X = X^\TT \right\} +$$ + +这是一个维数为 $n(n + 1)/2$ 的向量空间。用 $\bfS^n_+$ 表示对称半正定矩阵的集合: + +$$ +\bfS^n_+ = \left\{ X \in \bfS^n \middle| X \succeq 0 \right\} +$$ + +用 $\bfS^n_{++}$ 表示对称正定矩阵的集合: + +$$ +\bfS^n_{++} = \left\{ X \in \bfS^n \middle| X \succ 0 \right\} +$$ + +(这些符号与 $\bfR_+$ 相对应:$\bfR_+$ 表示非负实数,而 $\bfR_{++}$ 表示正实数。) + +  集合 $\bfS^n_+$ 是一个凸锥:如果 $\theta_1,\theta_2 \geqslant 0$ 并且 $A,B\in\bfS^n_+$,那么 $\theta_1 A + \theta_2 B\in\bfS^n_+$。从半正定矩阵的定义可以直接得到:对于任意 $x \in \bfR^n$,如果 $A\succeq 0$,$B\succeq 0$ 并且 $\theta_1,\theta_2 \geqslant 0$,那么有 + +$$ +x^\TT (\theta_1 A + \theta_2 B)x = \theta_1 x^\TT A x + \theta_2 x^\TT Bx \geqslant 0 +$$ + +> **举例** $\bfS^2$ 上的**半正定锥**(*Positive semidefinite cone*)。我们有 +> +> $$ +> X = \left[\begin{array}{c} + x & y \\ y & z +\end{array}\right] \in \bfS^2_+ \quad \Longleftrightarrow \quad x\geqslant 0, \quad z \geqslant 0, \quad xz\geqslant y^2 +> $$ +> +> [图 12](#图12) 给出了这个锥的边界,按 $(x,y,z)$ 表示在 $\bfR^3$ 中。 +> +> +> +> ![图 12. $\bfS^2$ 中半正定锥的边界。](../images/post/2024-06-08-josh-cvx-1-2/2024-06-08-josh-cvx-1-2-120-BoundaryOfPostiveSemidefiniteCone.png){width=500px} + +# 参考文献 + +1. Stephen P. Boyd and Lieven Vandenberghe, *Convex optimization*. Cambridge, UK: Cambridge University Press, 2004. +2. Stephen P. Boyd and Lieven Vandenberghe, *凸优化*. 北京: 清华大学出版社, 2013. diff --git a/source/images/post/2022-03-29-josh-convex-optimization-1-1/2022-03-29-josh-convex-optimization-1-1-010-TheLinePassingThroughX1AndX2.png b/source/images/post/2022-03-29-josh-cvx-1-1/2022-03-29-josh-cvx-1-1-010-TheLinePassingThroughX1AndX2.png similarity index 100% rename from source/images/post/2022-03-29-josh-convex-optimization-1-1/2022-03-29-josh-convex-optimization-1-1-010-TheLinePassingThroughX1AndX2.png rename to source/images/post/2022-03-29-josh-cvx-1-1/2022-03-29-josh-cvx-1-1-010-TheLinePassingThroughX1AndX2.png diff --git a/source/images/post/2022-03-29-josh-convex-optimization-1-1/2022-03-29-josh-convex-optimization-1-1-020-SomeSimpleConvexAndNonconvexSets.png b/source/images/post/2022-03-29-josh-cvx-1-1/2022-03-29-josh-cvx-1-1-020-SomeSimpleConvexAndNonconvexSets.png similarity index 100% rename from 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