I realize that a second-order cone is a cone, and thus an SOCP is a type of conic problem. However, to me it doesn't seem so apparent, looking at their equations. Could someone explain how one could convert an SOCP to conic program (and show the math involved)?
As an example, if there's an SOCP of the form:
\begin{aligned} & \text{minimize} & & \mathbf{f}^\intercal\mathbf{x} \\ & \text{subject to} & & {\|\mathbf{A}_i \mathbf{x}+\mathbf{b}_i \|}_2 \leq \mathbf{c}_{i}^{T} \mathbf{x}+d_i \quad \quad i=1,\ldots,m\\ & & & \mathbf{Fx = g} \end{aligned}
where $\mathbf{x} \in \mathbb R ^n$ is the optimization variable, $\mathbf{A}_i \in \mathbb R^{n_i \times n}$, $F \in \mathbb R ^{p \times n}$. How can it has to be converted to the form:
\begin{aligned} & \text{minimize} & & \mathbf{c}^\intercal\mathbf{x} \\ & \text{subject to} & & \mathbf{Mx+p} {\preceq}_K 0\\ & & & \mathbf{Qx = r}. \end{aligned}
where $K$, which will be the cone of the $m$ second-order cones combined?
Both forms have been picked up from Boyd & Vandenberghe's Convex Optimization book.
Let $\mathcal{Q}^{n_i}\subseteq\mathbb{R}^{n_i}\times\mathbb{R}$ be the second-order cone of dimension $n_i+1$. Then using your notation, we have $c=f$, $Q=F$, $r=g$, and $$M=-\begin{bmatrix} A_1 \\ c_1^T \\ A_2 \\ c_2^T \\ \vdots \\ A_m \\ c_m^T \end{bmatrix} \quad p=-\begin{bmatrix} b_1 \\ d_1 \\ b_2 \\ d_2 \\ \vdots \\ b_m \\ d_m \end{bmatrix} \quad K = \mathcal{Q}^{n_1} \times \mathcal{Q}^{n_2} \times \dots \times \mathcal{Q}^{n_m}.$$ That's it! Once you understand this, then you will see that really there is no need to be so dogmatic about your single-cone form. There's no reason you can't do this: \begin{array}{ll} \text{minimize} & c^T x \\ \text{subject to} & M^{(i)} x + p^{(i)} \succeq_{K^{(i)}} 0, ~i=1,2,\dots, m \\ & Q x = r \end{array} Using $\succeq$ instead of $\preceq$ avoids those pesky negative signs above.