MPCC

Documentation for MPCC.

MPCC: Optimization Models with Complementarity Constraints

Set of tools to model nonlinear continuous optimization model

\[ \min f(x) \text{ s.t. } l ≤ c(x) ≤ u,\]

extended with complementarity constraints:

\[ G(x) ≥ 0, H(x) ≥ 0, G(x) ∘ H(x) ≤ 0.\]

The resulting model is called an MPCC.

This package also handles vanishing constraints (MPVC)

\[ H(x) ≥ 0, G(x) ∘ H(x) ≤ 0,\]

kink constraints

\[ H(x) ≥ 0, G(x) ∘ H(x) = 0,\]

and switching constraints

\[ G(x) ∘ H(x) = 0.\]

Denote ∘ the componentwise product of two vectors.

This package extends the NLPModel API defined in NLPModels.jl. The role of NLPModels.jl is to define an API that users and solvers can rely on.

The package contains the basic tools to use the Stopping framework, and exports an MPCCStopping.

Installation

] add MPCC

Example

The simplest way to define an MPCCModel uses automatic differentiation:

using MPCC
f(x) = sum(x) # objective function
x0 = ones(6)
G(x) = [x[1]; x[3]]
H(x) = [x[2]; x[4]]
lccg, lcch = zeros(2), zeros(2)
lvar, uvar = fill(-10.0, size(x0)), fill(10.0, size(x0))
admpcc = ADMPCCModel(G, H, lccg, lcch, f, x0, lvar, uvar)

The resulting is an instance of an AbstractMPCCModel with all capabilities of an AbstractNLPModel such as

x = rand(6)
grad(admpcc, x) # returns the gradient of the objective function
neval_grad(admpcc) # returns the number of evaluation of grad
reset!(admpcc) # reset the internal counter

but also exports functions to deal with complementarity constraints

x = rand(6)
(consG(admpcc, x), consH(admpcc, x))

but also jac_G_structure, jac_H_structure, jac_G_coord, jac_H_coord, jGprod, jHprod, jGtprod, jHtprod, hGprod, hHprod.

It also possible to convert the problem as a classical nonlinear optimization model treating the complementarity constraint as a nonlinear inequalities

\[ H(x) ≥ 0, G(x) ≥ 0, G(x) ∘ H(x) ≤ 0,\]

using NLMPCC as follows

nlp = NLMPCC(admpcc)

so that

cons(nlp, x)

returns a vector consisting of [c(x), G(x), H(x), G(x) ∘ H(x)].

Contributors

Tangi Migot 📆 🚧 📖 ⚠️ 💻