COSMO: An ADMM-based solver for convex conic optimisation problems (LP, QP, SOCP, SDP, ExpCP, PowCP). Implemented in Julia



This is a Julia implementation of the Conic operator splitting method (COSMO) solver. It can solve large convex conic optimization problems of the following form:

with decision variables x ϵ R^n, s ϵ R^m and data matrices P=P'>=0, q ϵ R^n, A ϵ R^(m×n), and b ϵ R^m. The convex set K is a composition of convex sets and cones.

For more information take a look at the COSMO.jl Documentation (stable | dev).


  • Versatile: COSMO solves linear programs, quadratic programs, second-order cone programs, semidefinite programs and problems involving exponential and power cones
  • Quad SDPs: Positive semidefinite programs with quadratic objective functions are natively supported
  • Infeasibility detection: Infeasible problems are detected without a homogeneous self-dual embedding of the problem
  • JuMP / Convex.jl support: We provide an interface to MathOptInterface (MOI), which allows you to describe your problem in JuMP and Convex.jl.
  • Warm starting: COSMO supports warm starting of the decision variables
  • Custom sets and linear solver: Customize COSMO's components by defining your own convex constraint sets and by choosing from a number of direct and indirect linear system solvers, e.g. QDLDL, Pardiso, Conjugate Gradient and MINRES
  • Arbitrary precision types: You can solve problems with any floating point precision.
  • Open Source: Our code is free to use and distributed under the Apache 2.0 Licence
  • Chordal decomposition: COSMO tries to decompose large structured PSD constraints using chordal decomposition techniques. This often results in a significant speedup compared to the original problem.
  • Smart clique merging: After an initial decomposition of a structured SDP, COSMO recombines overlapping cliques/blocks to speed up the algorithm.


  • COSMO can be added via the Julia package manager (type ]): pkg> add COSMO


If you find COSMO useful in your project, we kindly request that you cite the following paper:

  author        = {Michael Garstka and Mark Cannon and Paul Goulart},
  title         = {{COSMO}: A conic operator splitting method for large convex problems},
  booktitle     = {European Control Conference},
  year          = {2019},
  location      = {Naples, Italy},
  doi            = {10.23919/ECC.2019.8796161},
  eprint        = {1901.10887},
  url           = {https://arxiv.org/abs/1901.10887},
  archiveprefix = {arXiv},
  keywords      = {Mathematics - Optimization and Control},
  primaryclass  = {math.OC},

A preprint can be downloaded here.


  • Contributions are always welcome. Our style guide can be found here.
  • Current issues, tasks and future ideas are listed in Issues. Please report any issues or bugs that you encounter.
  • As an open source project we are also interested in any projects and applications that use COSMO. Please let us know!

Licence 🔍

This project is licensed under the Apache License - see the LICENSE.md file for details.

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