1 Exponential Cone Programming
对于约束，
\[ \left\lbrace \begin{aligned} & x \in \mathbb{R}, z \geq y e^{x / y}, && y > 0 \\ & x \leq 0, z \geq 0, && y = 0 \end{aligned} \right. \]
将该曲面上方部分所形成的空间，定义为指数锥（Exponential Cone）
\[ \mathcal{K}_{exp} := \left\lbrace (x, y, z) \mid y > 0, z \geq y \exp\left(\frac{x}{y}\right) \right\rbrace \cup \left\lbrace (x, 0, z) \mid x \leq 0, z \geq 0 \right\rbrace \]这是一个凸的可行域，底层一般采用内点法求解EP。对于我们做上层应用，可以调用Mosek或COPT来求解。
2 KL-Divergence DRO
考虑如下DRO问题，
\[ (\text{KL-DRO}) \ \min_{\pmb{x} \in \mathcal{X}} \ \pmb{c}^\top \pmb{x} + \sup_{\pmb{p} \in \mathcal{P}} \sum_{i=1}^{N} p_i \mathcal{Q}(\pmb{x}, \hat{\pmb{\xi}}_i) \]其中，给定历史数据集\(\mathcal{N}:=\left\lbrace\hat{\pmb{\xi}}_1, \cdots, \hat{\pmb{\xi}}_N\right\rbrace\)，记经验分布为
\[ \hat{\mathbb{P}} = \sum_{i=1}^N \hat{p}_i \delta_{\xi_i} \]则KL-divergence模糊集定义为
\[ \mathcal{P} := \left\lbrace \mathbf{p} \in \mathbb{R}^N_+ : \sum_{i=1}^N p_i = 1,\; \sum_{i=1}^N p_i \log\left(\frac{p_i}{\hat{p}_i}\right) \le \rho \right\rbrace \]对于给定的\(\pmb{x}\), 内部最坏情形期望可写为
\[ \begin{aligned} \sup_{\mathbf{p}}\quad & \sum_{i=1}^N p_i Q(\pmb{x}, \hat{\pmb{\xi}}_i) \\ \text{s.t.}\quad & \sum_{i=1}^N p_i = 1 \\ & \sum_{i=1}^N p_i \log\left(\frac{p_i}{\hat{p}_i}\right) \le \rho \\ & p_i \ge 0, && \forall i \in [N] \end{aligned} \]通过拉格朗日对偶，其等价于
\[ \min_{\eta > 0} \left\lbrace \eta \rho + \eta \log \left( \sum_{i=1}^N \hat{p}_i \exp\left(\frac{\mathcal{Q}(\pmb{x}, \hat{\pmb{\xi}}_i)}{\eta}\right) \right) \right\rbrace \]因此，KL-DRO等价于
\begin{equation} \begin{aligned} \min_{\pmb{x}, \eta, u}\quad & \pmb{c}^\top \pmb{x} + \eta \rho + u \\ \text{s.t.}\quad & u \geq \eta \log \left( \sum_{i=1}^N \hat{p}_i \exp\left(\frac{\mathcal{Q}(\pmb{x}, \hat{\pmb{\xi}}_i)}{\eta}\right) \right) \\ & \pmb{x} \in \mathcal{X}, \; \eta > 0 \end{aligned} \label{eq:cons-1} \end{equation}进一步地，由于\(\eta>0\)，约束\(\eqref{eq:cons-1}\)等价于
\[ \frac{u}{\eta} \geq \log \left( \sum_{i=1}^N \hat{p}_i \exp\left(\frac{\mathcal{Q}(\pmb{x}, \hat{\pmb{\xi}}_i)}{\eta}\right) \right) \]对两边取指数，可得
\[ 1 \geq \sum_{i=1}^N \hat{p}_i \exp\left(\frac{\mathcal{Q}(\pmb{x}, \hat{\pmb{\xi}}_i)-u}{\eta}\right) \]进一步，两边同时乘以\(\eta\)，得到
\[ \sum_{i=1}^N \hat{p}_i \eta \exp\left(\frac{\mathcal{Q}(\pmb{x}, \hat{\pmb{\xi}}_i)-u}{\eta}\right) \leq \eta \]为表示上述约束，引入辅助变量\(t_i\)和\(s_i\)，其中\(t_i\)用于刻画 recourse function 的 epigraph，即
\begin{equation} \label{eq:cons-1-1} t_i \geq \mathcal{Q}(\pmb{x}, \hat{\pmb{\xi}}_i), \quad \forall i \in [N] \end{equation}而\(s_i\) 用于刻画指数项，即
\begin{equation} \label{eq:cons-1-2} s_i \geq \eta \exp \left(\frac{t_i - u}{\eta}\right), \quad \forall i \in [N] \end{equation}因此，约束\(\eqref{eq:cons-1}\) 等价于
\begin{equation} \left\lbrace \begin{aligned} & \eta \geq \sum_{i=1}^{N}\hat{p}_i s_i \\ & \text{Constraints \eqref{eq:cons-1-1} and \eqref{eq:cons-1-2}} \end{aligned} \right. \end{equation}对于约束\(\eqref{eq:cons-1-2}\)，根据指数锥的定义，可以写成如下指数锥约束
\[ (t_i - u, \eta, s_i) \in \mathcal{K}_{exp}, \quad \forall i \in [N] \]综上，KL-DRO最终可表述为如下ECP问题，
\begin{equation} \begin{aligned} \min_{\pmb{x}, \eta, u}\quad & \pmb{c}^\top \pmb{x} + \eta \rho + u \\ \text{s.t.}\quad & \eta \geq \sum_{i=1}^{N}\hat{p}_i s_i \\ & t_i \geq \mathcal{Q}(\pmb{x}, \hat{\pmb{\xi}}_i), && \forall i \in [N] \\ & (t_i - u, \eta, s_i) \in \mathcal{K}_{exp}, && \forall i \in [N] \\ & \pmb{x} \in \mathcal{X}, \eta > 0 \end{aligned} \end{equation}