From b9e587f5d57be3835eec5d69fa965a7a7eea8e2a Mon Sep 17 00:00:00 2001 From: Josh Gao Date: Wed, 12 Jun 2024 17:45:24 +0800 Subject: [PATCH] new post cvx-1.4 --- source/_posts/2024-06-08-josh-cvx-1-3.md | 9 +- source/_posts/2024-06-12-josh-cvx-1-4.md | 165 ++++++++++++++++++ ...-12-josh-cvx-1-4-170-MinimalAndMinimum.png | Bin 0 -> 44844 bytes ...06-12-josh-cvx-1-4-180-ThreeEllipsoids.png | Bin 0 -> 64413 bytes 4 files changed, 173 insertions(+), 1 deletion(-) create mode 100644 source/_posts/2024-06-12-josh-cvx-1-4.md create mode 100644 source/images/post/2024-06-12-josh-cvx-1-4/2024-06-12-josh-cvx-1-4-170-MinimalAndMinimum.png create mode 100644 source/images/post/2024-06-12-josh-cvx-1-4/2024-06-12-josh-cvx-1-4-180-ThreeEllipsoids.png 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 330cbca..04d609f 100644 --- a/source/_posts/2024-06-08-josh-cvx-1-3.md +++ b/source/_posts/2024-06-08-josh-cvx-1-3.md @@ -18,6 +18,13 @@ categories: tags: - 凸优化 - 凸集 + - 保凸运算 + - 交集 + - 仿射函数 + - 伸缩 + - 平移 + - 线性分式 + - 透视函数 abbrlink: a4a69a4f date: 2024-06-08 23:27:19 --- @@ -189,7 +196,7 @@ $$ > > > -> ![图 15.透视函数的小孔成象解释。深色的水平直线表示 $\bfR^3$ 中的平面 $x_3 = 0$,除了在原点处有一个小孔外,它是不透光的。平面之上的物体或光源呈现在浅色水平直线所示的象平面 $x_3 = -1$ 上。源位置向其象位置的映射对应于透视函数。](../images/post/2024-06-08-josh-cvx-1-3/2024-06-08-josh-cvx-1-3-150-PinHoleCameraInterpretationOfPerspectiveFuncation.png){width=500px} +> ![图 15. 透视函数的小孔成象解释。深色的水平直线表示 $\bfR^3$ 中的平面 $x_3 = 0$,除了在原点处有一个小孔外,它是不透光的。平面之上的物体或光源呈现在浅色水平直线所示的象平面 $x_3 = -1$ 上。源位置向其象位置的映射对应于透视函数。](../images/post/2024-06-08-josh-cvx-1-3/2024-06-08-josh-cvx-1-3-150-PinHoleCameraInterpretationOfPerspectiveFuncation.png){width=500px}   如果 $C \subseteq \dom P$ 是凸集,那么它的象 diff --git a/source/_posts/2024-06-12-josh-cvx-1-4.md b/source/_posts/2024-06-12-josh-cvx-1-4.md new file mode 100644 index 0000000..0678ed2 --- /dev/null +++ b/source/_posts/2024-06-12-josh-cvx-1-4.md @@ -0,0 +1,165 @@ +--- +title: Josh's Note — 凸优化
Part 1.4 凸集—广义不等式 +mathjax: true +comments: true +copyright: true +toc: + enable: true + number: false + wrap: true + expand_all: true + max_depth: 4 +categories: + - - Josh 的学习笔记 + - 凸优化 + - 凸集 + - - 数学 + - 凸优化 +tags: + - 凸优化 + - 凸集 + - 广义不等式 + - 正常锥 + - 最小元 + - 极小元 + - 最大元 + - 极大元 +abbrlink: aff3a091 +date: 2024-06-12 16:49:26 +--- + +# 1. 正常锥与广义不等式 + +$ +\newcommand{\bfR}{\mathbf{R}} +\newcommand{\bfS}{\mathbf{S}} +\newcommand{\E}{\mathcal{E}} +\newcommand{\TT}{\mathrm{T}} +\def\iint{\mathop{\bf int}} +$  称锥 $K \subseteq \bfR^n$ 即为**正常锥**(*proper cone*),如果它满足下列条件 + +- K 是凸的。 + +- K 是闭的。 + +- K 是**实**(*solid*)的,即具有非空内部。 + +- K 是**尖**(*pointed*)的,即不包含直线(或者等价地,$x\in K, -x \in K \Longrightarrow x = 0$)。 + +正常锥 $K$ 可以用来定义**广义不等式**(*generalized inequality*),即 $\bfR^n$ 上的偏序关系。这种偏序关系和 $\bfR$ 上的标准序有很多相同的性质。用正常锥 $K$ 可以定义 $\bfR^n$ 上的偏序关系如下 + +$$ +x \preceq_K y \Longleftrightarrow y - x \in K +$$ + +$y \prec_K x$ 也可以写为 $x \succeq_K y$。类似地,定义相应的严格偏序关系为 + +$$ +x \prec_K y \Longleftrightarrow y - x \in \iint K +$$ + +并且可以同样地定义 $x \succ_K y$。(为将广义不等式 $\preceq_K$ 与严格的广义不等式区分开,有时也称 $\preceq_K$ 为不严格的广义不等式)。 + + + +  当 $K = \bfR_+$ 时,偏序关系 $\preceq_K$ 就是通常意义上 $\bfR$ 中的序 $\leqslant$ 相应地,严格偏序关系 $\prec_K$ 与 $\bfR$ 上的严格序 $<$ 相同。因此,广义不等式包含了 $\bfR$ 上的(不严格和严格)不等式,它是广义不等式的一种特殊情况。 + +> **举例** **非负象限及分量不等式**。非负象限 $K = \bfR^n_+$ 是一个正常锥。相应的广义不等式 $\preceq_K$ 对应于向量间的分量不等式,即 $x \preceq_K y$ 等价于 $x_i \leqslant y_i, i = 1,\cdots,n$。相应地,其严格不等式对应于严格的分量不等式,即 $x \prec_K y$ 等价于 $x_i < y_i, i = 1,\cdots,n$。 +> +> 我们将常常使用对应于非负象限的不严格和严格的偏序关系,因此省略下标 $\bfR^n_+$。当 $\preceq$ 或 $\prec$ 出现在向量间的时候,该符号应被理解为分量不等式。 + +> **举例** **半正定锥和矩阵不等式**。半正定锥是 $S^n$ 空间中的正常锥,相应的广义不等式 $\preceq_K$ 就是通常的矩阵不等式,即 $X \preceq_K Y$ 等价于 $Y - X$ 为半正定矩阵。(在 $S^n$ 中)$S^n_+$ 的内部由正定矩阵组成,因此严格广义不等式也等同于通常的对称矩阵的严格不等式,即 $X \prec_K Y$ 等价于 $Y - X$ 为正定矩阵。 +> +> 这里,也是由于经常使用这种偏序关系,因此省略其下标,即对于对称矩阵,我们将广义不 +等式简写为 $X \preceq Y$ 或 $X \prec Y$, 它们表示关于半正定锥的广义不等式。 + +> **举例** **$[0, 1]$ 上非负的多项式锥**。$K$ 定义如下 +> +> $$\begin{equation} +> K = \{ c\in\bfR^n\ |\ c_1 + c_2 t + \cdots + c_n t^{n-1} \geqslant 0\ 对于\ t \in [0,1]\} +> \end{equation}$$ +> +> 即 $K$ 是 $[0, 1]$ 上最高 $n - 1$ 阶的非负多项式(系数)锥。可以看出 $K$ 是一个正常锥,其内部是 $[0, 1]$ 上为正的多项式的系数集合。 +> +> 两个向量 $c, d \in \bfR^n$ 满足 $c \preceq_K d$ 的充要条件是,对于所有 $t \in [0, 1]$ 有 +> +> $$ +> c_1 + c_2 t + \cdots + c_n t^{n-1} \leqslant d_1 + d_2 t + \cdots + d_n t^{n-1} +> $$ + +## 1.1 广义不等式的性质 + +  广义不等式 $\preceq_K$ 满足许多性质,例如 + +- **$\preceq_K$ 对于加法是保序的**:如果 $x \preceq_K y$ 并且 $u \preceq_K v$,那么 $x + u \preceq_K y + v$。 +- **$\preceq_K$ 具有传递性**:如果 $x \preceq_K y$ 并且 $y \preceq_K z$,那么 $x \preceq_K z$。 +- **$\preceq_K$ 对于非负数乘是保序的**:如果 $x \preceq_K y$ 并且 $\alpha \geqslant 0$,那么 $\alpha x \preceq_K \alpha y$。 +- **$\preceq_K$ 是自反的**:$x \preceq_K x$。 +- **$\preceq_K$ 是反对称的**:如果 $x \preceq_K y$ 并且 $y \preceq_K x$,那么 $x = y$。 +- **$\preceq_K$ 对于极限运算是保序的**:如果对于 $i = 1,2,\dots$ 均有 $x_i \preceq_K y_i$,当 $i \to \infty$ 时,有 $x_i \to x$ 和 $y_i \to y$,那么 $x \preceq_K y$。 + +相应的广义不等式 $\prec_K$ 也满足一些性质,例如 + +- 如果 $x \prec_K y$,那么 $x \preceq_K y$。 +- 如果 $x \prec_K y$ 并且 $u \prec_K v$, 那么 $x + u \prec_K y + v$。 +- 如果 $x \prec_K y$ 并且 $\alpha > 0$,那么 $\alpha x \prec_K \alpha y$。 +- $x \nprec_K x$。 +- 如果 $x \prec_K y$,那么对于足够小的 $u$ 和 $v$ 有 $x + u \prec_K y + v$。 + +这些性质可以从 $\preceq_K$ 和 $\prec_K$ 的定义以及正常锥的性质中直接得到。 + +# 2. 最小与极小元 + +  广义不等式的符号($\preceq_K$,$\prec_K$)似乎表明它们与 $\bfR$ 上的普通不等式($\leqslant$,$<$)有着相同的性质。虽然普通不等式的许多性质对于广义不等式确实成立,但很多重要的性质并不如此。最明显的区别在于,$\bfR$ 上的 $\leqslant$ 是一个**线性序**(*linear ordering*),即任意两点都是**可比的**(*comparable*),也就是说 $x\leqslant y$ 和 $y \leqslant x$ 二者必居其一。这个性质对于其他广义不等式并不成立。这导致了最小、最大这些概念在广义不等式环境下变得更加复杂。本节将对此进行简要的 +讨论。 + +  如果对于每个 $y \in S$,均有 $x \preceq_K y$,称 $x\in S$ 是 $S$(关于广义不等式 $\preceq_K$)的**最小元**(*minimum element*)。类似地,我们可以定义关于广义不等式的**最大元**(**maximum element**)。如果一个集合有最小(或最大)元,那么它们是唯一的。 + +  相对应的概念是**极小元**(*minimal element*)。如果 $y \in S$,$y \preceq_K x$ 可以推得 $y=x$,那么称 $x\in S$ 是 $S$ 上(关于广义不等式 $\preceq_K$)的**极小元**(*minimal element*)。同样地,可以定义**极大元**(*maximal element*)。一个集合可以有多个极小(或极大)元。 + +  用简单的集合符号,我们可以对最小元和极小元进行描述。元素 $x\in S$ 是 $S$ 中的一个最小元,当且仅当 + +$$ +S \subseteq x + K +$$ + +这里 $x+K$ 表示可以与 $x$ 相比并且大于或等于(根据 $\preceq_K$)$x$ 的所有元素。元素 $x\in S$ 是极小元,当且仅当 + +$$ +(x - K) \cap S = \{x\} +$$ + +这里 $x-K$ 表示可以与 $x$ 相比并且小于或等于(根据立$\preceq_K$)$x$ 的所有元素,它与 $S$ 的唯一共同点即是 $x$。 + +  $K=\bfR^n$ 导出的实际上就是 $\bfR$ 上一般的序。此时,极小和最小的概念是一致的,也符合集合最小元素的通常定义。 + +> **举例** 考虑锥 $\bfR^2_+$,它导出的是 $\bfR^2$ 上的关于分量的不等式。对此,可以给出一些关于极小元和最小元的简单的几何描述。不等式 $x \preceq y$ 的含义是 $y$ 在 $x$ 之上、之右。$x \in S$是集合 $S$ 的最小元,表明 $S$ 的其他所有点都在它之上、之右。而 $x$ 为集合 $S$ 的极小元,是指 $S$ 中没有任何一个点在 $x$ 之下、之左,其区别可见[图 17](#图17)。 +> +> +> +> ![图 17. **左**:集合出关于 $\bfR^2$ 上的分量不等式有最小元 $x_1$。集合 $x_1+K$ 由浅色阴影所示;$x_1$ 是 $S_1$ 的最小元,因为 $S_1 \subseteq x_1 + K$。**右**:点 $x_2$ 是 $S_2$ 的极小元。集合 $x_2-K$ 由浅色阴影所示。点 $x_2$ 是极小的,因为 $x_2-K$ 和 $S_2$ 只相交于 $x_2$](../images/post/2024-06-12-josh-cvx-1-4/2024-06-12-josh-cvx-1-4-170-MinimalAndMinimum.png){width=700px} + +> **举例** **对称矩阵集合中的最小元和极小元**。用 $A \in \bfS^n_{++}$ 表示一个圆心在原点的椭圆,即 +> +> $$ +> \E_A = \{ x\ |\ x^\TT A^{-1} x \leqslant 1 \} +> $$ +> +> 我们知道 $A \preceq B$ 等价于 $\E_A \subseteq \E_B$。 +> +> 给定 $v_1,\cdots,v_k \in \bfR^n$ 并定义 +> +> $$ +> S = \{ P \in \bfS ^n_{++}\ |\ v_i^\TT P^{-1} v_i \leqslant 1,\ i = 1,\dots k \} +> $$ +> +> 它对应于包含了点 $v_1,\cdots,v_k$ 的椭圆的集合。集合 $S$ 没有最小元:对于任意包含点$v_1,\cdots,v_k$ 的椭圆,总可以找到另一个包含这些点但不可比的椭圆。一个椭圆是极小的,如果它包含这些点但没有更小的椭圆也包含这些点。[图 18](#图18)显示了 $\bfR^2$ 上 $k = 2$ 时的一个例子。 +> +> +> +> ![图 18. $\bfR^2$ 中的三个椭球,它们均以原点(图中较低的点所示)为中心,并且包含较高处的点。椭球 $\E_1$ 不是极小的,因为存在包含这些点并且更小的椭球(例如,$\E_3$)。由于同样的原因,$\E_3$ 也不是极小的。椭球 $\E_2$ 是极小的,因为没有其他(以原点为中心)包含这些点并被 $\E_2$ 包含的椭球。](../images/post/2024-06-12-josh-cvx-1-4/2024-06-12-josh-cvx-1-4-180-ThreeEllipsoids.png){width=300px} + +# 参考文献 + +1. Stephen P. Boyd and Lieven Vandenberghe, *Convex optimization*. Cambridge, UK: Cambridge University Press, 2004. +2. Stephen P. 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