Embedded Conic Solver in R. This is an R wrapper around the ecos project on GitHub which describes ECOS as below.

ECOS is a numerical software for solving convex second-order cone programs (SOCPs) of type

\[
\mbox{Minimize } c'x \mbox{ such that } Ax = b \mbox{ and } Gx <=_K h
\] where the last inequality is generalized, that is, \(h-Gx\) belongs to the cone \(K\).

ECOS supports the positive orthant \({\mathbf R}_+\), second-order cones \(Q_n\) defined as

\[
Q_n = \bigl\{ (t,{\mathbf x}) | t >= \lVert{\mathbf x}\rVert_2 \bigr\}
\]

with \(t\) a scalar and \({\mathbf x} \in {\mathbf R}_{n-1}\), and the exponential cone \(K_e\) defined as

\[
K_e = \mbox{closure} \bigl\{ (x,y,z) | exp(x/z) <= y/z, z>0 \bigr\},
\]

where \((x,y,z) \in {\mathbf R}^3\).

The cone \(K\) is therefore a direct product of the positive orthant, second-order, and exponential cones:

\[
\begin{array}{l}
K = R_+ \times Q_{n_1} \times \cdots \times Q_{n_N} \times K_e \times \cdots \times K_e.
\end{array}
\]

##
Further Details

Note that the ECOS C language sources are included here. Changes to the original source are clearly delineated for easy reference.