A fast, reliable, and open-source convex cone solver.

SCS (Splitting Conic Solver) is a numerical optimization package for solving large-scale convex quadratic cone problems. The code is freely available on GitHub. It solves primal-dual problems of the form

$$\begin{array}{r}\begin{array}{lcr}\begin{array}{ll}\text{minimize}& (1/2)x^{\mathrm{\top }}Px+c^{\mathrm{\top }}x\\ \text{subject to}& Ax+s=b\\ & s\in \mathcal{K}\end{array}& & \begin{array}{ll}\text{maximize}& -(1/2)x^{\mathrm{\top }}Px-b^{\mathrm{\top }}y\\ \text{subject to}& Px+A^{\mathrm{\top }}y+c=0\\ & y\in {\mathcal{K}}^{\ast }\end{array}\end{array}\end{array}$$

over variables

$x\in {\mathbf{R}}^n$	primal variable
$y\in {\mathbf{R}}^m$	dual variable
$s\in {\mathbf{R}}^m$	slack variable

with data

$A\in {\mathbf{R}}^{m\times n}$	sparse data matrix, see Data matrices
$P\in {\mathbf{S}}_{+}^n$	sparse, symmetric positive semidefinite matrix
$c\in {\mathbf{R}}^n$	dense primal cost vector
$b\in {\mathbf{R}}^m$	dense dual cost vector
$\mathcal{K}\subseteq {\mathbf{R}}^m$	nonempty, closed, convex cone, see Cones
${\mathcal{K}}^{\ast }\subseteq {\mathbf{R}}^m$	dual cone to $\mathcal{K}$

At termination SCS will either return points $(x^{\star },y^{\star },s^{\star })$ that satisfies the optimality conditions to the desired accuracy, or a certificate of primal or dual infeasibility to the designated infeasibility accuracy.

Features

• Efficient: Designed to scale to large problems.

• Flexible: Supports quadratic objectives and a large range of cones.

• Free and open source: Distributed under the permissive MIT license.

• Detects infeasibility: Robustly and reliably detects infeasible problems.

• Interfaces: Bindings for many languages, including C, Python, Julia, R, MATLAB, Ruby, and JavaScript via WebAssembly.

• Warm starts: Easily warm-started, and the matrix factorization can be cached.

• Matrix-free: Optionally use an indirect linear system solver, or a GPU version.

• Supported: A supported solver in CVX, CVXPY, YALMIP, Convex.jl and JuMP.

• Accelerated: Includes acceleration that can improve convergence to high accuracy.

• Battle-tested: The first ADMM-based solver available, and in wide usage.

Performance

SCS is a fast and reliable optimization library. For instance, it is one of the most performant solvers as determined by the third-party QP solvers benchmark on the challenging Maros-Meszaros QP test suite. This is despite SCS being a general quadratic conic solver and not specifically tailored for QPs.

SCS is faster and more reliable than most other solvers.

Development

SCS is a community project, built from the contributions of many researchers and engineers. The primary maintainer is Brendan O’Donoghue. We appreciate all contributions. To get involved, see our contributing guide.