CCOpt

A Julia package for solving Mathematical Programs with Complementarity Constraints (MPCCs). For details check out the implementation paper.

Installation

To install CCOpt, simply proceed to

pkg> add https://github.com/MadNLP/CCOpt.jl

Usage

CCOpt takes as input a nonlinear program formulated with NLPModels. Taking a nlp as input, a MPCC is defined using the package MPCCModels

using CCOpt
using MPCCModels
mpcc = MPCCModel(nlp, ind_x1, ind_x2)

with ind_x1 (resp. ind_x2) the indices of the variables appearing in the left-hand complementarity (resp. right-hand complementarity).

Formulating a MPCC with JuMP

CCOpt supports the modeler JuMP with the extension MathOptComplements. The following example shows how to formulate a MPCC with JuMP and solve it with CCOpt:

using JuMP
using MathOptComplements
using NLPModelsJuMP
using CCOpt

model = Model()
@variable(model, z[1:2] >= 0)
@objective(model, Min, z[1] + z[2])
@constraint(model, c1, z[2]^2 >= 1)
@constraint(model, comp, [z[1], z[2]] ∈ MOI.Complements(2))

MathOptComplements.Bridges.add_all_bridges(model)
set_optimizer(model, CCOpt.Optimizer)
JuMP.optimize!(model)

Solution methods

Relaxation method

Once specified, you can solve the MPCC problem implemented in mpcc using the relaxation method as

solver = CCOpt.RelaxationSolver(mpcc)
stats = CCOpt.solve_homotopy!(solver)

All the results (primal and dual solutions, objective, etc.) are stored in stats.

Penalty method

Alternatively, you can solve mpcc using the penalty method as

solver = CCOpt.PenaltySolver(mpcc)
stats = CCOpt.solve_homotopy!(solver)

Citation

If you use CCOpt.jl in your work, please cite:

@article{Pozharskiy2026,
  title={CCOpt: an Open-Source Solver for Large-Scale Mathematical Programs with Complementarity Constraints},
  author={Pozharskiy, Anton and Pacaud, Fran{\c{c}}ois and Diehl, Moritz and Nurkanovi{\'c}, Armin},
  journal={arXiv preprint arXiv:2604.18726},
  year={2026}
}