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

## TL;DR

COSMO introduces an efficient operator splitting solver for large convex conic problems, leveraging sparsity and decomposition techniques, with promising benchmarks and an open-source Julia implementation.

## Contribution

The paper presents COSMO, a novel conic operator splitting method that efficiently solves large structured convex conic problems using sparsity exploitation and decomposition.

## Key findings

- Outperforms existing solvers on benchmark problems
- Efficient for large-scale semidefinite programs
- Open-source Julia implementation available

## Abstract

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.