# 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.

- 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.

- 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.