Convex.jl

Convex.jl is a Julia package for Disciplined Convex Programming (DCP).

Convex.jl can solve linear programs, mixed-integer linear programs, and DCP-compliant convex programs using a variety of solvers, including Mosek, Gurobi, ECOS, SCS, and GLPK, through MathOptInterface.

Convex.jl also supports optimization with complex variables and coefficients.

For usage questions, please contact us via Discourse.

Installation

import Pkg
Pkg.add("Convex")

Quick Example

# Let us first make the Convex.jl module available
using Convex, SCS

# Generate random problem data
m = 4;  n = 5
A = randn(m, n); b = randn(m, 1)

# Create a (column vector) variable of size n x 1.
x = Variable(n)

# The problem is to minimize ||Ax - b||^2 subject to x >= 0
# This can be done by: minimize(objective, constraints)
problem = minimize(sumsquares(A * x - b), [x >= 0])

# Solve the problem by calling solve!
solve!(problem, SCS.Optimizer)

# Check the status of the problem
problem.status

# Get the optimal value
problem.optval

Using with JuMP

The master branch of this package (not yet released) contains an experimental JuMP solver. This solver reformulates a nonlinear JuMP model into a conic program using DCP. Note that it currently supports only a limited subset of scalar nonlinear programs, such as those involving log and exp.

julia> model = Model(() -> Convex.Optimizer(Clarabel.Optimizer))
A JuMP Model
Feasibility problem with:
Variables: 0
Model mode: AUTOMATIC
CachingOptimizer state: EMPTY_OPTIMIZER
Solver name: Convex with Clarabel

julia> @variable(model, x >= 1);

julia> @variable(model, t);

julia> @constraint(model, t >= exp(x))
t - exp(x) ≥ 0

julia> @objective(model, Min, t);

julia> optimize!(model)
-------------------------------------------------------------
           Clarabel.jl v0.5.1  -  Clever Acronym
                   (c) Paul Goulart
                University of Oxford, 2022
-------------------------------------------------------------

problem:
  variables     = 3
  constraints   = 5
  nnz(P)        = 0
  nnz(A)        = 5
  cones (total) = 2
    : Nonnegative = 1,  numel = 2
    : Exponential = 1,  numel = 3

settings:
  linear algebra: direct / qdldl, precision: Float64
  max iter = 200, time limit = Inf,  max step = 0.990
  tol_feas = 1.0e-08, tol_gap_abs = 1.0e-08, tol_gap_rel = 1.0e-08,
  static reg : on, ϵ1 = 1.0e-08, ϵ2 = 4.9e-32
  dynamic reg: on, ϵ = 1.0e-13, δ = 2.0e-07
  iter refine: on, reltol = 1.0e-13, abstol = 1.0e-12,
               max iter = 10, stop ratio = 5.0
  equilibrate: on, min_scale = 1.0e-04, max_scale = 1.0e+04
               max iter = 10

iter    pcost        dcost       gap       pres      dres      k/t        μ       step
---------------------------------------------------------------------------------------------
  0   0.0000e+00   4.4359e-01  4.44e-01  8.68e-01  8.16e-02  1.00e+00  1.00e+00   ------
  1   2.2037e+00   2.6563e+00  2.05e-01  7.34e-02  6.03e-03  5.44e-01  1.01e-01  9.33e-01
  2   2.5276e+00   2.6331e+00  4.17e-02  1.43e-02  1.26e-03  1.27e-01  2.26e-02  7.84e-01
  3   2.6758e+00   2.7129e+00  1.39e-02  4.09e-03  3.42e-04  4.35e-02  6.00e-03  7.84e-01
  4   2.7167e+00   2.7178e+00  3.90e-04  1.18e-04  9.82e-06  1.25e-03  1.72e-04  9.80e-01
  5   2.7182e+00   2.7183e+00  9.60e-06  3.39e-06  2.82e-07  3.15e-05  4.95e-06  9.80e-01
  6   2.7183e+00   2.7183e+00  1.92e-07  6.74e-08  5.62e-09  6.29e-07  9.84e-08  9.80e-01
  7   2.7183e+00   2.7183e+00  4.70e-09  1.94e-09  1.61e-10  1.59e-08  2.83e-09  9.80e-01
---------------------------------------------------------------------------------------------
Terminated with status = solved
solve time =  941μs

julia> value(x), value(t)
(0.9999999919393833, 2.7182818073461403)

More Examples

A number of examples can be found here. The basic usage notebook gives a simple tutorial on problems that can be solved using Convex.jl.

Citing this package

If you use Convex.jl for published work, we encourage you to cite the software using the following BibTeX citation:

@article{convexjl,
 title = {Convex Optimization in {J}ulia},
 author = {Udell, Madeleine and Mohan, Karanveer and Zeng, David and Hong, Jenny and Diamond, Steven and Boyd, Stephen},
 year = {2014},
 journal = {SC14 Workshop on High Performance Technical Computing in Dynamic Languages},
 archivePrefix = "arXiv",
 eprint = {1410.4821},
 primaryClass = "math-oc",
}