<aside>

Def of Cone

$C\in \R^n$, $\forall \lambda \ge 0$ / $\forall \lambda > 0$, we have $\lambda C\subset C$

besides convex set C ⇒ convex cone

Prop

cone K is convex ↔ $\mathcal K + \mathcal K \subset \mathcal K$

→ convex combination and affine in cone

← convex combination is two affine in the same cone

Def of Polar Cone 极锥

$$ \mathcal K^{\blue{\circ}}=\{s\in\R^n,\lang s,x\rang \le 0,\forall x\in \mathcal K\} $$

Remarks

Polar Cone is also a closed convex cone
Polar Cone and Primal Cone intersection is $\{0\}$
If $\mathcal K$ is subspace, $\mathcal K^\circ=\mathcal K^\perp$

conical hull of points → calculate its polar cone (algorithm)

Prop

$\mathcal K$ be a closed convex cone, then $y_x=P_{\mathcal K}x$ iff $y_x\in \mathcal K, x-y_x\in \mathcal K^\circ, \red{\lang x-y_x,y_x\rang }=0$

necessity 必要性 proj → dual

proj $y_x\in \mathcal K$ and internal production is non-positive

pick $y=0$ and $2y_x$, we get minus and positive of target equality LHS

reduce $y_x$and the internal production mean $\forall y \in \mathcal K$ in ^ is non-positive, which is the def of polar cone

sufficiency 充分性 proj ← dual

from $\mathcal K^\circ$ we get $\forall y$ then $y \in \mathcal K$ and sum to the internal production format as above.

</aside>

Example

$Ω_+=\{x∈\R^n: x_i\ge0, i=1,\cdots,n\}$

$$ P_{\mathcal \Omega_+}(x)=\begin{cases} \max (x_i,0), && x\notin \mathcal \Omega_+,\\[4pt] x, && x \in \mathcal\Omega_+ , \end{cases}= \bigl(\max\{x_i,0\}\bigr)_{i=1}^n. $$

Corollary

$P_{\mathcal K}x=0$ iff $x\in \mathcal K^\circ$

→ is def

← need $0\in \mathcal K$ then the dual equation $\lang x-0,0\rang=0$ mean 0 is projection since $x-0\in\mathcal K^\circ$
positive homogeneous $P_{\mathcal K}(\alpha x)=\alpha P_{\mathcal K}x,\alpha\ge0$ and $P_{\mathcal K}(-x)=-P_{-\mathcal K}x$

Theorem(JJ Moreau)

two point is projection in $\mathcal K$ and its polar ↔ these two point are ortho in $\mathcal K$ and its dual

Proof

we claim that

$$ \red{ P_{\mathcal K}x+P_{\mathcal K^\circ}x=x } $$

since we have $\mathcal K^\circ)^\circ=\mathcal K$ for a closed convex cone

<aside> 👐🏻

Conic Optimization / Conic Programming LP →CP

不等式约束替换为凸锥集（广义不等式）的 LP

$$ \begin{split}\begin{array}{ll} \operatorname{minimize} & c^Tx \\ \operatorname{subject\ to} & Ax=b,\\ & x\in K, \end{array}\end{split} $$

Linear Optimization $K=\mathbb R_+^n$
二次规划 $Q_r^n$
power cone $P_n^{\alpha,1-\alpha}$
exponential cone $K_{\exp}$
geometric programming → exponential cone
symmetric positive semidefinite → semidefinite programming $S_+^n$→ matrix value CP </aside>

<aside>

closed conic hull of its epigraph (conic extensions)

凸函数的上图可以用锥闭包（凸包）表示，只需增加一个透视 perspective 维度$(x,t\ge f(x))$ → $(x,s,t),s>0$

$$ cl\{(t,s,x)\in \mathbb R^{1\times 1\times n}\mid \frac ts\ge f(\frac xs),s\ge 0\} $$

单纯的凸集不能满足cone $kv\in \Omega,\forall v\in \Omega$

注意，“透视源”不一定是点，可以是线、面，只要作为极限存在并以边界点囊括入闭包即可。

Rockafellar, R.T.: Level sets and continuity of conjugate convex functions. Trans. Amer. Math. Soc. 123(1), 46–63 (1966). https://doi.org/10.1090/S0002-9947-1966-0192318-X

如指数锥常表示为第三象限和透视锥的并集。
指数函数生成指数锥

$$ \mathcal K_{\exp}\equiv cl([\mathcal K_{\exp}]{++}) =[\mathcal K{\exp}]{++} \cup[\mathcal K{\exp}]_{0} $$

其中$[\mathcal K_{\exp}]_{++}$是指数函数符合透视的集合（perspective interior）

$$ [\mathcal K_{\exp}]_{++} =\big\{ (t,s,x)\in \mathbb R^{1\times 1\times n}\mid \frac ts\ge \exp(\frac xs),s\ge 0 \big\} $$

作为闭包，需要考虑并包括边界点。这里就是取$s\to 0$后只有极限定义的点集（指数函数被无穷缩小，取极限后上图为第四象限全体， perspective boundary）

$$ [\mathcal K_{\exp}]_{0} =\big\{ s=0,t\ge 0,r\le 0 \big\} $$

<aside>

</aside>

3. 指数锥的投影问题