锥扩展 (epigraphical reformulation)

https://www.cvxgrp.org/scs/api/cones.html

t/s\ge -\sum _i^n\blue{\log}(x_i/s) \right\}

=\left\{ (t,s,\vec x)\in \R \times \R_{++}\times\R_{++}^n :

t/s\ge -\sum i^n\log(x_i/s) \right\} \bigcup (\R+\times\{0\}\times\R_+^n)$ for assosiated matrix: $K_{log-det}\equiv cl\left\{ (t,s,\mathbf X)\in \R \times \R_{++}\times \green{ \mathbb S_{++}^n} :

t/s\ge -\blue{\log}\left(\blue{\det }\frac{\mathbf X}{s}\right) \right\}$ | systematic approach based on Newton’s method.https://arxiv.org/abs/2511.01089v1https://arxiv.org/abs/2511.01089v1 | | | PSD - Spectral Matrix Cone: trace-inverse cone | in spectral space $K_{inv}\equiv cl\left\{ (t,s,\vec x)\in \R\times \R_{++}\times\R_{++}^n: t/s\ge\frac {\sum i^n1/x_i}v=v\sum_i \blue{1/x_i} \right\}= \left\{ (t,s,\vec x)\in \R\times \R{++}\times\R_{++}^n: t/s\ge\frac {\sum i^n1/x_i}v=v\sum_i {1/x_i} \right\}\bigcup (\R+\times\{0\}\times\R_+^n)$ for associated spectral matrix cone $K_{log-det}\equiv cl\left\{ (t,s,\mathbf X)\in \R \times \R_{++}\times \green{ \mathbb S_{++}^n} :

t/s\ge s\blue{ \operatorname{Tr} \left( \mathbf X^{-1}

\right)} \right\}$ | systematic approach based on Newton’s method. https://arxiv.org/abs/2511.01089v1 | | | PSD - Spectral Matrix Cone: entropy cone | in spectral space $K_{\text{vEnt}} \equiv cl \{ (t, v, x) \in \reals \times \reals_{++}\times \reals^{n}{+} \: | \: \sum{i=1}^n \blue{x_i \log (x_i/v)} \leq t\}

\left\{ (t, v, x) \in \reals \times \reals_{++} \times \reals^{n}{+} \: | \: \sum{i=1}^n x_i \log (x_i/v) \leq t\right\} \bigcup \: (\reals_+ \times \reals_+ \times \{0\}^n)$ for associated spectral matrix cone $K_{\text{mEnt}} \equiv cl \{ (t, v, X) \in \reals \times \reals_{++} \times \green{\mathbb S^{n}{++}} \: | \: \blue{\operatorname {Tr}(X \log (X/v))} \leq t \}$ | systematic approach based on Newton’s method. https://arxiv.org/abs/2511.01089v1 | | | PSD - Spectral Matrix Cone: root-determinant cone | in spectral space $K{\text{geomean}}  \triangleq \{(t, x) \in \reals \times \reals^n_{+} \: | \: - \prod_{i=1}^n x_i^{1/n} \leq t\}.$ for associated spectral matrix cone $K_{\det} \triangleq \{(t, X) \in \reals \times \mathbb S^n_{+} \: | \: -( \det(X))^{1/n} \leq t \}.$ | systematic approach based on Newton’s method. https://arxiv.org/abs/2511.01089v1 | | | PSD - Spectral Matrix Cone: dual cone | | systematic approach based on Newton’s method. https://arxiv.org/abs/2511.01089v1 | |

1. 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。

2. 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>

   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

3. epigraphical reformulation

   SCS

   Project onto P/D Exponential cone