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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>

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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）

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

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Friberg - 2023 - Projection onto the exponential cone a univariate root-finding problem.pdf

指数锥优势
1. expressive abilities
  
  指数函数上图，对数函数下图，exponentials指数多项式，logarithms，entropy funxtions熵，product logarithms， soft-max， soft-plus
2. numerically stable 3-self-concordant barrier functions
  
  兼容对偶、facial reduction、内点特性
3. 非symmetric cone，仍然能够快速收敛 </aside>
3. 指数锥的投影问题

Project onto P/D Exponential cone ←

Hien - 2015 - Differential properties of Euclidean projection onto power cone.pdf

Cederberg and Boyd - 2025 - Projections onto Spectral Matrix Cones.pdf

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Duality

feasible set 可行集

$$ {\cal F}_p=\{x\in \R^n \mid Ax=b\} \cap K $$
优化问题的值的下确界infimum

$$ p^\star = \inf\{c^Tx~:~ x\in{\cal F}_p\}, $$

Dual cone

$$ K^* = \{ y\in \real^n~:~ y^Tx\geq 0\ \forall x\in K\}. $$

如何直观理解凸优化理论中【对偶锥】的概念 https://www.zhihu.com/question/264853229/answer/286677771

self-dual

$(\real_+^n)^* = \real_+^n.$ $(Q^n)^=Q^n,(Q_r^n)^,(S_+^n)^*$

Infeasibility certificates

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Farkas lemma (see Sec. 2.3 (Infeasibility in linear optimization))

定义$A(K) = \{Ax~:~x\in K\}.$ 可行性↔ $b\in A(K)$ 不可行且$b\not\in\mathrm{cl}(A(K))$，存在过远点的超平面划分b和锥，对偶锥可以找到$y$

$$ b^Ty > 0, \quad (Ax)^Ty \leq 0\ \forall x\in K. $$

可行
limit-feasible $\|Ax_n-b\|\to 0$ → ill-posed
不可行，但存在$y$满足刚才的不等式组 → infeasibility certificates </aside>

Lagrangian

$$ L(x, y, s) = c^Tx + y^T(b-Ax) - s^T x. $$

可行点$x^\in{\cal F}_p$ 对偶锥$(y^,s^)\in\real^m\times K^$

满足$x\in K$等价于$\exists s\in S^*,s^Tx\ge 0$

$$ L(x^, y^, s^)=c^Tx^+(y^)^T\cdot 0 - (s^)^Tx^\leq c^T x^. $$

对偶目标函数是拉格朗日函数的极小值

$$ \begin{split}g(y, s) = \min_x L(x,y,s) = \min_x x^T( c - A^T y - s ) + b^T y = \left \{ \begin{array}{ll} b^T y, & c - A^T y - s = 0,\\ -\infty, & \text{otherwise}. \end{array} \right.\end{split} $$