From abbc0da71b0e3060b5329565b5eaf4d90d4df7c2 Mon Sep 17 00:00:00 2001 From: Josh Gao Date: Tue, 11 Jun 2024 20:10:20 +0800 Subject: [PATCH] update cvx post --- package-lock.json | 100 ------------------ package.json | 1 - source/_data/variables.styl | 2 +- source/_posts/2024-06-08-josh-cvx-1-2.md | 8 +- source/_posts/2024-06-08-josh-cvx-1-3.md | 88 ++++++++++++++- ...raInterpretationOfPerspectiveFuncation.png | Bin 0 -> 25146 bytes 6 files changed, 90 insertions(+), 109 deletions(-) create mode 100644 source/images/post/2024-06-08-josh-cvx-1-3/2024-06-08-josh-cvx-1-3-150-PinHoleCameraInterpretationOfPerspectiveFuncation.png diff --git a/package-lock.json b/package-lock.json index 994c820..4aa478d 100644 --- a/package-lock.json +++ b/package-lock.json @@ -21,7 +21,6 @@ "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", @@ -975,15 +974,6 @@ "node": ">=0.4.0" } }, - "node_modules/afinn-165": { - "version": "1.0.4", - "resolved": "https://registry.npmjs.org/afinn-165/-/afinn-165-1.0.4.tgz", - "integrity": "sha512-7+Wlx3BImrK0HiG6y3lU4xX7SpBPSSu8T9iguPMlaueRFxjbYwAQrp9lqZUuFikqKbd/en8lVREILvP2J80uJA==", - "funding": { - "type": "github", - "url": "https://github.com/sponsors/wooorm" - } - }, "node_modules/agent-base": { "version": "6.0.2", "resolved": "https://registry.npmmirror.com/agent-base/-/agent-base-6.0.2.tgz", @@ -1143,17 +1133,6 @@ "node": ">= 8" } }, - "node_modules/apparatus": { - "version": "0.0.10", - "resolved": "https://registry.npmjs.org/apparatus/-/apparatus-0.0.10.tgz", - "integrity": "sha512-KLy/ugo33KZA7nugtQ7O0E1c8kQ52N3IvD/XgIh4w/Nr28ypfkwDfA67F1ev4N1m5D+BOk1+b2dEJDfpj/VvZg==", - "dependencies": { - "sylvester": ">= 0.0.8" - }, - "engines": { - "node": ">=0.2.6" - } - }, "node_modules/aproba": { "version": "1.2.0", "resolved": "https://registry.npmmirror.com/aproba/-/aproba-1.2.0.tgz", @@ -5281,18 +5260,6 @@ "node": ">=14" } }, - "node_modules/hexo-related-posts": { - "version": "1.5.2", - "resolved": "https://registry.npmjs.org/hexo-related-posts/-/hexo-related-posts-1.5.2.tgz", - "integrity": "sha512-IXGTMp6r1AT6rBS4vsWwcnVQ6tPmMA4cju5ldcske6zs7ff9SdqQ6t05UHBJWjyK45NtNu5V5IsqG4o+lzjJ1w==", - "dependencies": { - "natural": "^2.1.5" - }, - "peerDependencies": { - "chalk": "^4.1.0", - "hexo": "^4.2.0 || ^5.2.0 || ^6.3.0 || ^7.1.1" - } - }, "node_modules/hexo-renderer-ejs": { "version": "2.0.0", "resolved": "https://registry.npmmirror.com/hexo-renderer-ejs/-/hexo-renderer-ejs-2.0.0.tgz", @@ -7088,28 +7055,6 @@ "jju": "^1.1.0" } }, - "node_modules/json-stable-stringify": { - "version": "1.1.1", - "resolved": "https://registry.npmjs.org/json-stable-stringify/-/json-stable-stringify-1.1.1.tgz", - "integrity": "sha512-SU/971Kt5qVQfJpyDveVhQ/vya+5hvrjClFOcr8c0Fq5aODJjMwutrOfCU+eCnVD5gpx1Q3fEqkyom77zH1iIg==", - "dependencies": { - "call-bind": "^1.0.5", - "isarray": "^2.0.5", - "jsonify": "^0.0.1", - "object-keys": "^1.1.1" - }, - "engines": { - "node": ">= 0.4" - }, - "funding": { - "url": "https://github.com/sponsors/ljharb" - } - }, - "node_modules/json-stable-stringify/node_modules/isarray": { - "version": "2.0.5", - "resolved": "https://registry.npmjs.org/isarray/-/isarray-2.0.5.tgz", - "integrity": "sha512-xHjhDr3cNBK0BzdUJSPXZntQUx/mwMS5Rw4A7lPJ90XGAO6ISP/ePDNuo0vhqOZU+UD5JoodwCAAoZQd3FeAKw==" - }, "node_modules/json5": { "version": "2.2.3", "resolved": "https://registry.npmmirror.com/json5/-/json5-2.2.3.tgz", @@ -7140,14 +7085,6 @@ "node": ">= 10.0.0" } }, - "node_modules/jsonify": { - "version": "0.0.1", - "resolved": "https://registry.npmjs.org/jsonify/-/jsonify-0.0.1.tgz", - "integrity": "sha512-2/Ki0GcmuqSrgFyelQq9M05y7PS0mEwuIzrf3f1fPqkVDVRvZrPZtVSMHxdgo8Aq0sxAOb/cr2aqqA3LeWHVPg==", - "funding": { - "url": "https://github.com/sponsors/ljharb" - } - }, "node_modules/jsonparse": { "version": "1.3.1", "resolved": "https://registry.npmmirror.com/jsonparse/-/jsonparse-1.3.1.tgz", @@ -8447,22 +8384,6 @@ "node": ">=0.10.0" } }, - "node_modules/natural": { - "version": "2.4.5", - "resolved": "https://registry.npmjs.org/natural/-/natural-2.4.5.tgz", - "integrity": "sha512-fxQou4ubo6Rzf7XL0jjwtC2vHm6Sdicu7ztcE0k1ODZs+d1UzcKF1Hu2IwE2y7OgzU3Sa9o8obw7rnyNrjsd1A==", - "dependencies": { - "afinn-165": "^1.0.2", - "apparatus": "^0.0.10", - "json-stable-stringify": "^1.0.1", - "sylvester": "^0.0.12", - "underscore": "^1.9.1", - "wordnet-db": "^3.1.11" - }, - "engines": { - "node": ">=0.4.10" - } - }, "node_modules/ncp": { "version": "2.0.0", "resolved": "https://registry.npmmirror.com/ncp/-/ncp-2.0.0.tgz", @@ -11072,14 +10993,6 @@ "node": ">= 0.4" } }, - "node_modules/sylvester": { - "version": "0.0.12", - "resolved": "https://registry.npmjs.org/sylvester/-/sylvester-0.0.12.tgz", - "integrity": "sha512-SzRP5LQ6Ts2G5NyAa/jg16s8e3R7rfdFjizy1zeoecYWw+nGL+YA1xZvW/+iJmidBGSdLkuvdwTYEyJEb+EiUw==", - "engines": { - "node": ">=0.2.6" - } - }, "node_modules/symbol-tree": { "version": "3.2.4", "resolved": "https://registry.npmmirror.com/symbol-tree/-/symbol-tree-3.2.4.tgz", @@ -11447,11 +11360,6 @@ "url": "https://github.com/sponsors/ljharb" } }, - "node_modules/underscore": { - "version": "1.13.6", - "resolved": "https://registry.npmjs.org/underscore/-/underscore-1.13.6.tgz", - "integrity": "sha512-+A5Sja4HP1M08MaXya7p5LvjuM7K6q/2EaC0+iovj/wOcMsTzMvDFbasi/oSapiwOlt252IqsKqPjCl7huKS0A==" - }, "node_modules/undici": { "version": "5.28.4", "resolved": "https://registry.npmjs.org/undici/-/undici-5.28.4.tgz", @@ -11994,14 +11902,6 @@ "node": ">=0.10.0" } }, - "node_modules/wordnet-db": { - "version": "3.1.14", - "resolved": "https://registry.npmjs.org/wordnet-db/-/wordnet-db-3.1.14.tgz", - "integrity": "sha512-zVyFsvE+mq9MCmwXUWHIcpfbrHHClZWZiVOzKSxNJruIcFn2RbY55zkhiAMMxM8zCVSmtNiViq8FsAZSFpMYag==", - "engines": { - "node": ">=0.6.0" - } - }, "node_modules/wrap-ansi": { "version": "7.0.0", "resolved": "https://registry.npmjs.org/wrap-ansi/-/wrap-ansi-7.0.0.tgz", 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/_data/variables.styl b/source/_data/variables.styl index ffcee67..0b7681c 100644 --- a/source/_data/variables.styl +++ b/source/_data/variables.styl @@ -5,4 +5,4 @@ $border-radius = 10px; // $font-family-chinese = "PingFang SC", "Microsoft YaHei"; // Font size of articles. -$font-size-base = 18px \ No newline at end of file +// $font-size-base = 18px \ No newline at end of file diff --git a/source/_posts/2024-06-08-josh-cvx-1-2.md b/source/_posts/2024-06-08-josh-cvx-1-2.md index 2e87acb..5e961c7 100644 --- a/source/_posts/2024-06-08-josh-cvx-1-2.md +++ b/source/_posts/2024-06-08-josh-cvx-1-2.md @@ -113,13 +113,13 @@ $$ # 2. Euclid球和椭球 -  $\bfR^n$ 中的空间 **Euclid 球**（*(Euclidean) ball*）（或简称为球）具有下面的形式： +  $\bfR^n$ 中的空间 **Euclid 球**（*Euclidean ball*）（或简称为球）具有下面的形式： $$ 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\} $$ -其中 $r > 0$, $\| \cdot \|_2$ 表示 Euclid 范数，即 $\|u\|_2 = (u^\TT u)^{1/2}$。向量 $x_r$ 是**球心**（*center*），标量 $r$ 为**半径**（*radius*）。$B (x_c, r)$ 由距离球心 $x_r$ 距离不超过 $r$ 的所有点组成。 Euclid 球的另一个常见的表达式为： +其中 $r > 0$，$\| \cdot \|_2$ 表示 Euclid 范数，即 $\|u\|_2 = (u^\TT u)^{1/2}$。向量 $x_r$ 是**球心**（*center*），标量 $r$ 为**半径**（*radius*）。$B (x_c, r)$ 由距离球心 $x_r$ 距离不超过 $r$ 的所有点组成。 Euclid 球的另一个常见的表达式为： $$ B(x_c,r) = \left\{ x_c + ru\ \middle|\ \|u\|_2 \leqslant 1 \right\} @@ -143,7 +143,7 @@ $$\begin{equation}\label{ellipsoid-1} \E = \left\{ x\ \middle|\ (x-x_c)^\TT P^{-1}(x-x_c) \leqslant 1 \right\} \end{equation}$$ -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。 +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。 @@ -155,7 +155,7 @@ $$\begin{equation}\label{ellipsoid-2} \E = \left\{ x_c + Au\ \middle|\ \|u\|_2 \leqslant 1 \right\} \end{equation}$$ -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。 +其中 $A$ 是非奇异的方阵。在此类表示形式中，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 diff --git a/source/_posts/2024-06-08-josh-cvx-1-3.md b/source/_posts/2024-06-08-josh-cvx-1-3.md index 14a6f62..c4c423c 100644 --- a/source/_posts/2024-06-08-josh-cvx-1-3.md +++ b/source/_posts/2024-06-08-josh-cvx-1-3.md @@ -27,13 +27,15 @@ $ \newcommand{\bfS}{\mathbf{S}} \newcommand{\TT}{\mathrm{T}} \newcommand{\H}{\mathcal{H}} +\newcommand{\E}{\mathcal{E}} +\def\dom{\mathop{\bf dom}} $  本文将描述一些保凸运算，利用保凸运算可以使用凸集构造出其他凸集。这些运算与[*Part 1.2 凸集—一些重要的凸集*](https://josh-gao.top/posts/9293b739.html)中描述的凸集的简单例子一起构成了凸集的演算，可以用来确定或构建集合的凸性。 # 1. 交集   交集运算是保凸的：如果 $S_1$ 和 $S_2$ 是凸集，那么 $S_1 \cap S_2$ 也是凸集。这个性质可以扩展到无穷个集合的交：如果对于任意 $\alpha\in A$，$S_\alpha$ 都是凸的，那么 $\displaystyle \bigcap_{\alpha \in A} S_\alpha$ 也是凸集。（子空间、仿射集合和凸锥对与任意交运算也是封闭的。）举一个简单的例子，多面体是半空间和超平面（它们都是凸集）的交集，因而是凸的。 -> **举例** 半正定锥 $\bfS^n_+$ 可以表示为， +> **举例** 半正定锥 $\bfS^n_+$ 可以表示为 > > $$ > \bigcap_{z\ne 0} \left\{ X \in \bfS^n\ \middle|\ z^\TT Xz \geqslant 0 \right\} @@ -79,7 +81,7 @@ $$ # 2. 仿射函数 -  函数 $f: \bfR^n \to \bfR^m$ 是**仿射**（*affine*）的，如果它是一个线性函数和一个常数的和，即具有 $f(x) = Ax+ b$ 的形式，其中 $A \in \bfR^{m\times n}$，$b \in \bfR^m$。假设 $S \subseteq \bfR^n$ 是凸的，并且 $f: \bfR^n \to \bfR^m$ 是仿射函数。那么，$S$ 在 f$ 下的象 +  函数 $f: \bfR^n \to \bfR^m$ 是**仿射**（*affine*）的，如果它是一个线性函数和一个常数的和，即具有 $f(x) = Ax+ b$ 的形式，其中 $A \in \bfR^{m\times n}$，$b \in \bfR^m$。假设 $S \subseteq \bfR^n$ 是凸的，并且 $f: \bfR^n \to \bfR^m$ 是仿射函数。那么，$S$ 在 $f$ 下的象 $$ f(S) = \left\{ f(x)\ \middle|\ x\in S \right\} @@ -93,7 +95,7 @@ $$ 是凸的。 -  两个简单的例子是**伸缩**（*scaling*）和**平移**（*translation*）。如果 $S \subseteq \bfR^n$ 是凸集，$\alpha \in \bfR^n$ 并且 $a \in \bfR^n$，那么，集合 $\alpha S$ 和 $S+a$ 是凸的，其中 +  两个简单的例子是**伸缩**（*scaling*）和**平移**（*translation*）。如果 $S \subseteq \bfR^n$ 是凸集，$\alpha \in \bfR$ 并且 $a \in \bfR^n$，那么，集合 $\alpha S$ 和 $S+a$ 是凸的，其中 $$ \alpha S = \left\{ \alpha x\ \middle|\ x \in S \right\},\qquad S + a = \left\{ x + a\ \middle|\ x \in S \right\} @@ -129,6 +131,86 @@ $$ 其中 $x\in\bfR^n$，$y_i \in \bfR^m$。$m=0$ 时，部分和给出了 $S_1$ 和 $S_2$的交集；$n=0$ 时，部分和等于集合之和。凸集的部分和是凸集。 +> **举例** **多面体**。$\{ x\ |\ Ax \preceq b, Cx = d \}$ 可以表示为非负象限和原点的 Cartesian 乘积在仿射函数 $f(x) = (b -Ax, d - Cx)$ 下的原象： +> +> $$ +> \{x\ |\ Ax\preceq b, Cx = d\} = \{ x\ |\ f(x) \in \bfR^m_+ \times \{0\}\} +> $$ + +> **举例** **线性矩阵不等式的解**。条件 +> +> $$\begin{equation} +> A(x) = x_1 A_1 + \cdots + x_nA_n \preceq B +> \end{equation}$$ +> +> 称为关于 $x$ 的**线性矩阵不等式**（*Linear Matrix Inequality, LMI*），其中 $B,A_i \in \bfS^m$（注意它与有序线性不等式 +> +> $$ +> a^\TT x = x_1a_1+\cdots x_na_n \leqslant b +> $$ +> +> 的相似性，其中 $b,a_i\in \bfR$。） +> +> 线性矩阵不等式的解 $\{ x\ |\ A(x) \preceq B\}$ 是凸集。事实上，它是半正定锥在由 $f(x) = B-A(x)$ 给定的仿射映射 $f: \bfR^n \to \bfS^m$ 下的原象。 + +> **举例** **双曲锥**。集合 +> +> $$ +> \{ x\ |\ x^\TT Px \leqslant (c^\TT x) ^2, c^\TT x \geqslant 0 \} +> $$ +> +> 是凸集，其中 $P\in\bfS^n_+$，$c\in\bfR$ 这是因为它是二阶锥 +> +> $$ +> \{ (z,t)\ |\ z^\TT z \leqslant t ^2 , t \geqslant 0 \} +> $$ +> +> 在仿射函数 $f(x) = (P^{1/2}x,c^\TT x)$ 下的原象。 + +> **举例** **椭球**。椭球 +> +> $$ +> \E = \{ x\ |\ (x-x_c)^\TT P^{-1} (x-x_c) \leqslant 1 \} +> $$ +> +> 是单位 Euclid 球 $\{ u\ |\ \|u\|_2 \leqslant 1 \}$ 在仿射映射 $f(u) = P^{1/2}u + x_c$ 下的象， 其中 $P \in \bfS^n_{++}$。（同时也是单位球在仿射映射 $g(x) = P^{-1/2}(x — x_c)$ 下的原象。） + +# 3. 线性分式及透视函数 + +  本节将讨论一类称为**线性分式**（*linear-fractional*）的函数，它比仿射函数更普遍，并且仍然保凸。 + +## 3.1 透视函数 + +  定义 $P:\bfR^{n+1}\to\bfR^n$ 为**透视函数**（*perspective function*），其定义域为 $\dom P = \bfR^n \times \bfR_{++}$。（此处的 $\bfR_{++}$ 表示正实数集合，即 $\bfR_{++} = \{ x\in\bfR\ |\ x > 0 \}$。）透视函数对向量进行伸缩（scale），或称为规范化（normalize），使得最后一维分量为1 并舍弃之。 + +> **注释** 我们用小孔成象来解释透视函数。($\bfR^3$ 中的）小孔照相机由一个不透明的水平面 $x_3 = 0$ 和一个在原点的小孔组成，光线通过这个小孔在 $x_3 = -1$ 呈现出一个水平图像。在相机上方 $x(x_3 > 0)$ 处的一个物体，在相平面的点 $=(x_1/x_3,x_2/x_3,1)$ 处形成一个图像。忽略象点的最后一维分量（因为它恒等于 $-1$），$x$ 处的点的象在象平面上呈现于 $y=-(x_1/x_3,x_2/x_3)$ 处。[图 15](#图15)显示了这个过程。 +> +> +> +> ![图 15.透视函数的小孔成象解释。深色的水平直线表示 $\bfR^3$ 中的平面 $x_3 = 0$，除了在原点处有一个小孔外，它是不透光的。平面之上的物体或光源呈现在浅色水平直线所示的象平面 $x_3 = -1$ 上。源位置向其象位置的映射对应于透视函数。](../images/post/2024-06-08-josh-cvx-1-3/2024-06-08-josh-cvx-1-3-150-PinHoleCameraInterpretationOfPerspectiveFuncation.png){width=500px} + +  如果 $C \subseteq \dom P$ 是凸集，那么它的象 + +$$ +P(C) = \{ P(x)\ |\ x\in C \} +$$ + +也是凸集。这个结论很直观：通过小孔观察一个凸的物体，可以得到凸的象。为解释这个事实，我们将说明在透视函数作用下，线段将被映射成线段。（也可以这样理解，通过小孔，一条线段的象是一条线段。）假设 $x = (\tilde{x},x_{n+1}), y = (\tilde{y}, y_{n+1}) \in \bfR^{n+1}$，并且 $x_{n+1}>0, y_{n+1}>0$ 。那么，对于 $0 \leqslant \theta \leqslant 1$ + +$$ +P(\theta x + (1-\theta)y) = \frac{\theta\tilde{x}+ (1-\theta)\tilde{y}}{\theta x_{n+1} + (1-\theta)y_{n+1}} = \mu P(x) + (1-\mu) P(x) +$$ + +其中 + +$$ +\mu = \frac{\theta x_{n+1}}{\theta x_{n+1} + (1-\theta) y_{n+1}}\in[0,1] +$$ + +$\theta$ 和 $\mu$ 之间的关系是单调的：当 $\theta$ 在 $0$、$1$ 间变化时（形成线段 $[x,y]$），$μ$ 也在 $0$、$1$ 间变化（形成线段 $[P(x),P(y)]$。这说明 $P([x,y]) = [P(x),P(y)]$。 + +  现在假设 $C$ 是凸的，并且有 $C \subseteq \dom P$（即对于所有 $x\in C$，$x_{n+1} > 0$) 及 $x,y\in C$。为显示 $P(C)$ 的凸性，需要说明线段 $[P(x), P(y)]$ 在 $P(C)$ 中。这条线段是线段 $[x,y]$ 在 $P$ 的象，因而属于 $P(C)$。 + # 参考文献 1. Stephen P. Boyd and Lieven Vandenberghe, *Convex optimization*. 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