#  Interior Point Conic Optimization for Julia

Clarabel.jl is a Julia implementation of an interior point numerical solver for convex optimization problems using a novel homogeneous embedding. Clarabel.jl solves the following problem:

$$$\begin{array}{r} \text{minimize} & \frac{1}{2}x^T P x + q^T x\\\\[2ex] \text{subject to} & Ax + s = b \\\\[1ex] & s \in \mathcal{K} \end{array}$$$

with decision variables $x \in \mathbb{R}^n$, $s \in \mathbb{R}^m$ and data matrices $P=P^\top \succeq 0$, $q \in \mathbb{R}^n$, $A \in \mathbb{R}^{m \times n}$, and $b \in \mathbb{R}^m$. The convex set $\mathcal{K}$ is a composition of convex cones.

Clarabel is also available in a Rust / Python implementation. See here.

## Features

• Versatile: Clarabel.jl solves linear programs (LPs), quadratic programs (QPs), second-order cone programs (SOCPs) and semidefinite programs (SDPs). It also solves problems with exponential and power cone constraints.
• Quadratic objectives: Unlike interior point solvers based on the standard homogeneous self-dual embedding (HSDE), Clarabel.jl handles quadratic objectives without requiring any epigraphical reformulation of the objective. It can therefore be significantly faster than other HSDE-based solvers for problems with quadratic objective functions.
• Infeasibility detection: Infeasible problems are detected using a homogeneous embedding technique.
• JuMP / Convex.jl support: We provide an interface to MathOptInterface (MOI), which allows you to describe your problem in JuMP and Convex.jl.
• Arbitrary precision types: You can solve problems with any floating point precision, for example, Float32 or Julia's BigFloat type, using either the native interface, or via MathOptInterface / Convex.jl.
• Clarabel.jl can be added via the Julia package manager (type ]): pkg> add Clarabel