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CVXPY 提供凸性检测，原理是 Disciplined Convex Programming ，网页入口：
https://dcp.stanford.edu/home | DCP analyzer:
将函数用cvxpy自带的 Functions 表示，会返回凸性分析结果。
将函数用cvxpy自带的 Functions 表示，会返回凸性分析结果。
[ ] 凸函数与凸集什么关系？为什么说凸集的定义更普适？ </aside>
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$\forall x,y\in S$ and $\alpha \in [0,1]$, we have their convex combination / line segment is also belongs to $S$
$$ \alpha x+(1-\alpha)y\in S $$
凸集定义为 存在加法运算和对应标量空间的集合的子集，满足集合中任意元素的凸组合仍在集合内。
A set is convex: $x\in S,y\in S$→ segment $xy\subset S$.
segment $xy:\alpha x+\beta y,\alpha,\beta\ge0,\alpha+\beta =1$. to Convex Combination to barycentric coordinates.
$$ \boxed{ \alpha x+\beta y\in \Omega}\\x,y\in \Omega,\alpha +\beta=1,\alpha,\beta \in \R $$
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a specific convex set with the line rather line segment
$$ \alpha x+(1-\alpha )y\in S,\ \alpha \in \R $$
凸组合是两变量的线性组合，其线性因子和为1且0~1。
affine combination $\sum_i \alpha_i=1\to \sum_i \alpha_ix_i$, $A(M\in \R^n)=\sum_i^r \alpha_im_i,r\in N_+\}$
锥组合类似，线性因子非负
Theorem
An affine set is subspace of $\R^n$
iff $0\in S$
S is parallel to $U$ if $\exists x$
$$ S=x+U. $$
⇒ affine space = its any element + unique subspace with U//S = x + $\ker S$
and Affine dimension: $\dim S=\dim U$
Affine hyperplane: $H_{S,r}=\{x\in \R^n:\lang s,x\rang =r\}$ s 法向量，r是截距
$H_{S,0}=\{s\}^\perp$ 垂直超平面
Half Spaces: $\{x\in \R^n:\lang s, r\rang\le r\}$is closed and $\{x\in \R^n:\lang s, r\rang>r\}$is open affine set
Affine Hull: $Aff\ M=\bigcap_{M\subset S} S_{affine}$ smallest affine hull of subset
property
Affine independent
$x_0,x_1,\cdots,x_p$ are affine independent iff $x_1-x_0,\cdots,x_p-x_0$ are linear independent.
prop these are quivalent
$x_0,x_1,\cdots,x_p$ are affine independent
$(x_0,1),(x_1,1),\cdots,(x_p,1)$ are linear independent
below admits a unique solution
$$ \begin{aligned} \sum_{i-0}^p\alpha_ix_i=0,\\ \sum_{i=0}^p \alpha_i=0 \end{aligned} $$
(1) ⇒ (2) $\sum_{i-0}^p\alpha_ix_i-\sum_{i=0}^px_0=$affine independent
(2) <⇒ (3)
(3) ⇒ (1) let $\alpha_0=-\sum_{i=1}^P \alpha_i$, we get linear combination
$$ \Delta n=\{ \alpha\in \R^n:\sum{i=1}^n\alpha_i=1,\alpha_i\ge 0 \}\subset \R^n\\= \{\alpha\in \R^n:e^T\alpha =1,e_i^T\alpha\ge 0\} $$
$e_i$ is orthonormal basis;
n half-space within a hyperplane
intersection of a family of convex sets is convex set
笛卡尔积$\prod C_i$ 也是 convex set
明可夫斯基和 $\lambda A+\mu B$ is also convex set
$int\ C$ and $cl\ C$ are convex set
$cl\ C$ is intersection of $C$ and some small unit ball $cl \ C =\prod_{\epsilon >0} (C+\epsilon B)$
convex combination of two neighborhood set are also subset to the same set
$$ U_1\subset S,U_2\subset S\to convex(U_1,U_2)\subset S $$
Affine mapping / function
affine combination of image is the same as the affine combination of function
$$ F(\lambda x+(1-\lambda)y)=\lambda F(x)+(1-\lambda )F(y) $$
homework：寻找更多性质 $F(S)$ $F(aff\ M)$
$conv\ M=\prod_{M\subset C}C$
set of all convex combination of any two pint in M: $conv M=\{\sum_{i=1}^k\lambda_ix_i:\lambda _i\in \Delta_n,x_i\in M\}$
Caratheodary Theorem
prop:
A set $M\in \R^n$ is compact (bounded & closed ), then $conv\ M$ is also compact
corollary
bounded set ⇒ $conv \ M$ is bounded
bounded M → $cl\ M$ → $conv(cl\ M)$ →$conv(M)\subset conv(cl\ M)$
closedness of M ≠> closedness of $conv\ M$找一个极限点不属于 convex hull的反例
closed convex hull
$$ \bar {conv} \ M=\prod_{M\subset C, \text{C is closed and convex}}C $$
$conv \ M\subset \bar{conv}( M)$ hou者条件更宽松，因为cl 包括的更多
prop
closed convex hull is closure of convex hull $\bar{conv}\ M=cl \ (conv \ M)$
relative open set → relative topology → boundary
Python库:SciPy.spatial.ConvexHull
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Some formal definitions and approaches.
凸函数的上图可通过透视法扩展为凸锥（包），凸优化 Convex Optimization 进而规约为锥规划 Conic Optimization
一阶优化算法中的投影法 3. 指数锥的投影问题
内点法 Clarabel