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.