Complementarity.jl

This package provides modeling language for (1) mixed complementarity problems (MCP) and (2) mathematical programs with equilibrium problems (MPEC).

NOTE @complmentarity for MCP and @complements for MPEC.

Mixed Complementarity Problems (MCP)

NOTE: Differences between PATHSolver.jl and Complementarity.jl:

PATHSolver.jl provides a wrapper for the C API of the PATH solver.
PATHSolver.jl also enables JuMP for solving MCP, but limited to linear problems.
Complementarity.jl provides a JuMP extension for solving MCP, both linear and nonlinear, using the C API wrapper in PATHSolver.jl.

MCP Documentation

This package provides a modeling and computational interface for solving Mixed Complementarity Problems (MCP): modeling by JuMP.jl and computing by PATHSolver.jl and NLsolve.jl. See the documentation.

F(x) ⟂ lb ≤ x ≤ ub

A very simple example:

(x+2) x = 0,  x ≥ 0,   x+2 ≥ 0

using Complementarity, JuMP
m = MCPModel()
@variable(m, x >= 0)
@mapping(m, F, x+2)
@complementarity(m, F, x)
status = solveMCP(m)
@show result_value(x)

Mathematical Programs with Equilibrium Constraints (MPEC)

NOTE: For solving MPEC, JuMP.jl v0.21 has started supporting complementarity constraints. At this moment, GAMS.jl and KNITRO support complementarity constraints.

MPEC Documentation

For solving mathematical programs with equilibrium constraints (MPEC), this package provides an extension to JuMP.jl by providing a macro that accepts complementarity conditions as constraints. Then it reformulates the complementarity conditions as a set of equality and inequality constraints so that a nonlinear optimization solver such as Ipopt.jl can solve the problem. See the documentation.

min  f(x)
s.t. g(x) ≤ 0
     F(x) ⟂ lb ≤ x ≤ ub

A very simple example:

min  x^3
s.t. (x+2) x = 0,  x ≥ 0,   x+2 ≥ 0

using JuMP, Ipopt, Complementarity
m = Model(Ipopt.Optimizer)
@variable(m, x>=0)
@NLobjective(m, Min, x^3)
@complements(m, 0 <= x+2,   x >= 0)
solve(m)
@show getvalue(x)

Installation

Pkg.add("Complementarity")

This will also install a few other packages.