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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:

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

Further Details

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