# COSMO: A conic operator splitting method for convex conic problems

arXiv: https://arxiv.org/abs/1901.10887 · HTML: https://mlsanity.com/paper/1901.10887/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Jan 2019.

## Built on by (most cited first, 8 of 8)

- [A Fast and Accurate Splitting Method for Optimal Transport: Analysis and Implementation](https://mlsanity.com/paper/2110.11738/) (2021)
- [Deep FlexQP: Accelerated Nonlinear Programming via Deep Unfolding](https://mlsanity.com/paper/2512.01565/) (2025)
- [Gauges and Accelerated Optimization over Smooth and/or Strongly Convex Sets](https://mlsanity.com/paper/2303.05037/) (2023)
- [Scalable Deep Unfolding of Conic Optimizers](https://mlsanity.com/paper/2606.13825/) (2026)
- [A Scalable Approach for Safe and Robust Learning via Lipschitz-Constrained Networks](https://mlsanity.com/paper/2506.23977/) (2025)
- [Stochastic Optimal Control via Measure Relaxations](https://mlsanity.com/paper/2508.00886/) (2025)
- [Finding Structure in Continual Learning](https://mlsanity.com/paper/2602.04555/) (2026)
- [Task Switching Without Forgetting via Proximal Decoupling](https://mlsanity.com/paper/2604.18857/) (2026)

## Abstract (arXiv)

This paper describes the Conic Operator Splitting Method (COSMO) solver, an operator splitting algorithm for convex optimisation problems with quadratic objective function and conic constraints. At each step the algorithm alternates between solving a quasi-definite linear system with a constant coefficient matrix and a projection onto convex sets. The low per-iteration computational cost makes the method particularly efficient for large problems, e.g. semidefinite programs that arise in portfolio optimisation, graph theory, and robust control. Moreover, the solver uses chordal decomposition techniques and a new clique merging algorithm to effectively exploit sparsity in large, structured semidefinite programs. A number of benchmarks against other state-of-the-art solvers for a variety of problems show the effectiveness of our approach. Our Julia implementation is open-source, designed to be extended and customised by the user, and is integrated into the Julia optimisation ecosystem.