Visit www.embotech.com/ECOS for detailed information on ECOS.

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

min  c'*x
s.t. A*x = b
     G*x <=_K h

where the last inequality is generalized, i.e. h - G*x belongs to the cone K. ECOS supports the positive orthant R_+ and second-order cones Q_n defined as

Q_n = { (t,x) | t >= || x ||_2 }

In the definition above, t is a scalar and x is in R_{n-1}. The cone K is therefore a direct product of the positive orthant and second-order cones:

K = R_+ x Q_n1 x ... x Q_nN

The latest version of ECOS is available via pip:

pip install ecos

This will download the relevant wheel for your machine.

If you are attempting to build the Python extension from source, then use

python setup.py install

The distribute.sh script is used to submit ECOS to the PyPi repository. Normal users can ignore it.

You will need Numpy and Scipy. For installation instructions, see their respective pages.

You may need sudo privileges for a global installation.

Windows users may experience some extreme pain when installing ECOS from source for Python 2.7. We suggest switching to Linux or Mac OSX.

If you must use (or insist on using) Windows, we suggest using the Miniconda distribution to minimize this pain.

If during the installation process, you see the error message Unable to find vcvarsall.bat, you will need to install Microsoft Visual Studio Express 2008, since Python 2.7 is built against the 2008 compiler.

If using a newer version of Python, you can use a newer version of Visual Studio. For instance, Python 3.3 is built against Visual Studio 2010.

##Calling ECOS from Python

After installing the ECOS interface, you must import the module with

import ecos

This module provides a single function ecos with one of the following calling sequences:

solution = ecos.solve(c,G,h,dims)
solution = ecos.solve(c,G,h,dims,A,b,**kwargs)

The arguments c, h, and b are Numpy arrays (i.e., matrices with a single column). The arguments G and A are Scipy sparse matrices in CSR format; if they are not of the proper format, ECOS will attempt to convert them. The argument dims is a dictionary with two fields, dims['l'] and dims['q']. These are the same fields as in the Matlab case. If the fields are omitted or empty, they default to 0. The argument kwargs can include the keywords

• feastol, abstol, reltol, feastol_inacc, abstol_innac, and reltol_inacc for tolerance values,
• max_iters for the maximum number of iterations,
• the Boolean verbose,
• bool_vars_idx, a list of ints which index the boolean variables,
• int_vars_idx, a list of ints which index the integer variables,
• mi_max_iters for maximum number of branch and bound iterations (mixed integer problems only),
• mi_abs_eps for the absolute tolerance between upper and lower bounds (mixed integer problems only), and
• mi_rel_eps for the relative tolerance, (U-L)/L, between upper and lower bounds (mixed integer problems only).

The arguments A, b, and kwargs are optional.

The returned object is a dictionary containing the fields solution['x'], solution['y'], solution['s'], solution['z'], and solution['info']. The first four are Numpy arrays containing the relevant solution. The last field contains a dictionary with the same fields as the info struct in the MATLAB interface.

CVXPY is a powerful Python modeling framework for convex optimization, similar to the MATLAB counterpart CVX. ECOS is one of the default solvers in CVXPY, so there is nothing special you have to do in order to use ECOS with CVXPY, besides specifying it as a solver. Here is a small example from the CVXPY tutorial:

# Solving a problem with different solvers.
x = Variable(2)
obj = Minimize(norm(x, 2) + norm(x, 1))
constraints = [x >= 2]
prob = Problem(obj, constraints)

# Solve with ECOS.
prob.solve(solver=ECOS)
print "optimal value with ECOS:", prob.value

ECOS is distributed under the GNU General Public License v3.0. Other licenses may be available upon request from embotech.

The solver is essentially based on Lieven Vandenberghe's CVXOPT ConeLP solver, although it differs in the particular way the linear systems are treated.

The following people have been, and are, involved in the development and maintenance of ECOS:

• Alexander Domahidi (principal developer)
• Eric Chu (Python interface, unit tests)
• Stephen Boyd (methods and maths)
• Michael Grant (CVX interface)
• Johan Löfberg (YALMIP interface)
• João Felipe Santos, Iain Dunning (Julia interface)
• Han Wang (ECOS branch and bound)

The main technical idea behind ECOS is described in a short paper. More details are given in Alexander Domahidi's PhD Thesis in Chapter 9.

If you find ECOS useful, you can cite it using the following BibTex entry:

@INPROCEEDINGS{bib:Domahidi2013ecos,
author={Domahidi, A. and Chu, E. and Boyd, S.},
booktitle={European Control Conference (ECC)},
title={{ECOS}: {A}n {SOCP} solver for embedded systems},
year={2013},
pages={3071-3076}
}