ConicIP.jl

ConicIP.jl is a pure-Julia conic interior-point solver for optimization problems with linear, second-order cone, and (experimental) semidefinite constraints.

Why ConicIP?

Pure Julia — no binary dependencies or external solver installations
Custom KKT solver callbacks — exploit problem structure for speed
Extensible — plug in your own factorization at each interior-point iteration
Nesterov-Todd scaling — symmetric primal-dual scaling for good numerical behavior
Infeasibility detection — returns certificates for infeasible/unbounded problems

Problem Formulation

ConicIP solves problems of the form:

minimize    ½ yᵀQy - cᵀy
subject to  Ay ≥_K b
            Gy  = d

where Q ≽ 0 and K is a Cartesian product of cones:

Cone	Spec	Description
Nonnegative orthant	`("R", n)`	Linear inequalities
Second-order cone	`("Q", n)`	Norm constraints
Semidefinite (experimental)	`("S", k)`	Matrix positivity

Quick Example

using ConicIP, SparseArrays, LinearAlgebra, Random
Random.seed!(42)

# Box-constrained QP: minimize ½ y'Qy - c'y subject to 0 ≤ y ≤ 1
n = 5
Q = sparse(Diagonal(rand(n) .+ 0.1))
c = randn(n)

# Constraints: [I; -I] y ≥ [0; -1]
A = sparse([I(n); -I(n)])
b = [zeros(n); -ones(n)]
cone_dims = [("R", 2n)]

sol = conicIP(Q, c, A, b, cone_dims; verbose=false)
sol.status

:Optimal

round.(sol.y, digits=4)

5-element Vector{Float64}:
 0.0
 0.0
 1.0
 1.0
 0.0

Two Ways to Use ConicIP

Direct API — full control, supports quadratic objectives:

sol = conicIP(Q, c, A, b, cone_dims; verbose=false)

JuMP/MOI — algebraic modeling, linear objectives only:

using JuMP, ConicIP
model = Model(ConicIP.Optimizer)
@variable(model, x[1:n] >= 0)
@objective(model, Min, sum(x))
optimize!(model)

See the JuMP Integration guide for details.

Choosing a Solver

ConicIP is a good fit when you need:

A pure-Julia solver with no binary dependencies
Custom KKT solver callbacks to exploit problem structure
A solver for moderate-size LP/QP/SOCP problems

For large-scale production use, consider: