ECOSolveR

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

$$\mbox{Minimize } c'x \mbox{ such that } {\mathbf Ax} = {\mathbf b} \mbox{ and } {\mathbf G \mathbf x}\,\, \leq_{\mathbf K}\,\, {\mathbf h}$$ where the last inequality is generalized, that is, ${\mathbf h}-\mathbf{Gx}$ belongs to the cone ${\mathbf K}$.

ECOS supports the positive orthant ${\mathbf R}_+$, second-order cones ${\mathbf Q}_n$ defined as

$${\mathbf Q}_n = \bigl\{ (t,{\mathbf x}) | t >= \lVert{\mathbf x}\rVert_2 \bigr\}$$

with $t$ a scalar and ${\mathbf x} \in {\mathbf R}_{n-1}$, and the exponential cone ${\mathbf K}_e$ defined as

$$\mathbf{K}_e = \mbox{closure} \bigl\{ (x,y,z) | exp(x/z) <= y/z, z>0 \bigr\},$$

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

The cone ${\mathbf K}$ is therefore a direct product of the positive orthant, second-order, and exponential cones:

$${\mathbf K} = {\mathbf R}_+ \times {\mathbf Q}_{n_1} \times \cdots \times {\mathbf Q}_{n_N} \times {\mathbf K}_e \times \cdots \times {\mathbf K}_e.$$