ConicIP.jl: A Pure Julia Conic QP Solver

ConicIP (Conic Interior Point) is a pure-Julia interior-point solver for optimizing quadratic objectives with linear equality constraints, and polyhedral, second-order cone, and (experimental) semidefinite cone constraints. Because ConicIP is written in Julia, it allows abstract input and custom KKT solver callbacks for exploiting problem structure.

Installation

ConicIP.jl is not yet registered in the Julia General registry. Install directly from GitHub:

using Pkg
Pkg.add(url="https://github.com/MPF-Optimization-Laboratory/ConicIP.jl")

Requires Julia 1.10 or later.

Basic Usage

ConicIP has the interface

sol = conicIP( Q , c , A , b , 𝐾 , G , d )

For the problem

minimize    ½yᵀQy - cᵀy
s.t         Ay ≧𝐾 b,  𝐾 = 𝐾₁  × ⋯ × 𝐾ⱼ
            Gy  = d

𝐾 is a list of tuples of the form (Cone Type ∈ {"R", "Q", "S"}, Cone Dimension) specifying the cone 𝐾ᵢ. For example, the cone 𝐾 = 𝑅² × 𝑄³ × 𝑅² has the following specification:

𝐾 = [ ("R",2) , ("Q",3),  ("R",2) ]

ConicIP returns sol, a structure containing the status (sol.status), primal variables (sol.y), dual variables (sol.v, sol.w), objective values (sol.pobj, sol.dobj), and convergence residuals (sol.prFeas, sol.duFeas, sol.muFeas).

To solve the problem

minimize    ½yᵀQy - cᵀy
such that   y ≧ 0

for example, use ConicIP as follows

using ConicIP
using SparseArrays, LinearAlgebra

n = 1000

Q = sprandn(n, n, 0.1)
Q = Q'*Q
c = ones(n)
A = sparse(1.0I, n, n)
b = zeros(n)
𝐾 = [("R", n)]

sol = conicIP(Q, c, A, b, 𝐾, verbose=true);

Usage with JuMP

ConicIP implements a MathOptInterface wrapper, so it can be used as a solver in JuMP. Note: only linear objectives are supported through JuMP; use the direct conicIP API for quadratic programs.

using JuMP
using ConicIP

model = Model(ConicIP.Optimizer)
@variable(model, x[1:10] >= 0)
@constraint(model, sum(x) == 1.0)
@objective(model, Min, sum(x))
optimize!(model)
value.(x) # should be ≈ [0.1, 0.1, …, 0.1]

Documentation

For full documentation including tutorials, how-to guides, and API reference, see the documentation.