BilevelOptimization

This package is a Julia toolbox based on JuMP.jl for solving bilevel optimization problems. These are encountered in various applications, including power grids, security games, market equilibria or chemical reaction optimization.

The generic problem can be written:

min_{x} F(x,y)
such that
  G * x + H * y ⩽ q
  y ∈ arg min_y {d^T y + x^T F * y
                 such that
                   A * x + B * y ⩽ b
                   y ⩾ 0
  }
  x_j integer ∀ j ∈ Jx

x represents the upper-level decision variable and y the lower-level one. y is thus the solution to a parametric optimization sub-problem, depending on the value of x. The required data describing this problem are the feasibility domains of the upper and lower level and the coefficients of the objective functions. All these are regrouped within the BilevelLP type of this package.

The formulation is made as general as possible for the problem to remain approachable with plain Mixed-Integer Solvers (CBC, GLPK, SCIP, Gurobi, CPLEX). For a simple linear-linear problem, the user can set Jx = ∅ and F as a zero matrix of appropriate dimension. The problem can be made as complex as wanted at the upper level, as long as JuMP and the solver used support the constraints and objective.

The main function is build_bilevel_lp, which will build the JuMP model or modify it for the bilevel problem.

The signature:
build_blp_model(bp::BilevelOptimization.BilevelLP, solver; comp_method) builds the model from scratch. It will return a Tuple: (m, x, y, λ, s) with:

• m the JuMP model
• x the upper-level variable vector
• y the lower-level variable vector
• λ the dual of the lower-level problem
• s the lower-level slack variable vector

The function can also be called with a model already built:
build_blp_model(m::JuMP.Model, bp::BilevelOptimization.BilevelLP, x, y; comp_method) In which case it will add the lower-level optimality constraints, it returns the same tuple.

If the user is not willing to describe the whole problem using a BilevelLP, the following signature can be used: build_blp_model(m::JuMP.Model, B::M, d, s; comp_method)

With B the lower-level constraint matrix, d the lower-level objective and s the lower-level slack variable. Only the KKT conditions are added to the model in that case (not the lower-level feasibility constraints).

The package can be installed using Julia Pkg tool:

julia> ]
(v1.0) pkg> add BilevelOptimization

You will also need an optimization solver up and running with JuMP.

Tests can be performed using Pkg:

julia> ]
(v1.0) pkg> test BilevelOptimization

From the Julia REPL, type ? to show the help prompt, then type the identifier you want the documentation for.

julia> using BilevelOptimization

help?> BilevelOptimization.BilevelLP
  A bilevel linear optimization problem of the form:

  min cx^T * x + cy^T * y
  s.t. G x + H y <= q
       x_j ∈ [xl_j,xu_j]
       x_j ∈ ℤ ∀ j ∈ Jx
       y ∈ arg min {
          d^T * y + x^T * F * y
          s.t. A x + B y <= b
               y_j ∈ [yl_j,yu_j]
          }

  Note that integer variables are allowed at the upper level. 

The "hard" part of the reduction of a bilevel problem is the set of complementarity constraints of the form:

λ ⋅ (b - Ax - By) = 0

These constraints cannot be handled directly, different methods have been developed in the literature and implemented in this package. The standard way is to give a different algorithm in build_blp_model:

build_blp_model(args..., comp_method::ComplementarityMethod = my_method)

Special-ordered Sets of type 1 or SOS1 are used by default for complementarity constraints. The option to pass is:

build_blp_model(args..., comp_method = SOS1Complementarity())

The most common technique for these constraints is the linearization of the constraint with a formulation developed in Fortuny-Amat and McCarl, 1981, using so-called big-M constraints.

build_blp_model(args..., comp_method = BoundComplementarity(MD, MP))

MD, MP are primal and dual bounds, both can be either a scalar for one bound per variable type or an abstract vector for one bound per variable.

Users can create a custom method for the complementarity constraint, by creating a type T <: ComplementarityMethod (sub-typing is optional but helps for clarity). They also need to implement a method:

add_complementarity_constraint(m, cm::T, s, λ)

Within which the complementarity constraints are added to the JuMP model m.

As a special application of the above model, the module BilevelFlowProblems offers the following problem:

• The upper-level, acting as a leader of the Stackelberg game, chooses taxes to set on some arcs of a directed graph.
• The lower-level, acting as the follower, makes a minimum-cost flow with a given minimum amount from the source to the sink.
• Each arc has an invariant base cost and a tax level decided upon by the leader.

This has been investigated in the literature as the "toll-setting problem". The required data include:

• the initial cost of each arc for all i,j
• which edges can be taxed by the leader for all i,j
• the tax options (at which level can each edge be taxed) for all i,j,k
• flow capacities of each edge
• the minimum flow the follower has to pass from source to sink

using BilevelOptimization.BilevelFlowProblems

init_cost = [
	0. 1. 1. 4.
	0. 0. 0. 1.
	0. 0. 0. 1.
	0. 0. 0. 0.
]
taxable_edges = [
	false true true false
	false false false true
	false false false true
	false false false false
]
tax_options = zeros(4,4,5)
for i in 1:4, j in 1:4
	if taxable_edges[i,j]
    	tax_options[i,j,:] .= (0.0, 0.5, 1.0, 1.5, 2.0)
    end
end

capacities = [
	0. 3. 2. 3.
    0. 0. 0. 2.
    0. 0. 0. 1.
    0. 0. 0. 0.
]

minflow = 3.

BilevelFlowProblem(init_cost,taxable_edges,capacities,tax_options, minflow)

(m, r, y, f, λ) = build_blp_model(bfp, optimizer_with_attributes(Cbc.Optimizer, "LogLevel" => 0))
st = JuMP.optimize!(m)

# objective_value(m) ≈ 6.
#     for j in 1:size(r)[2]
#        for i in 1:size(r)[1]
#            @test JuMP.value(r[i,j]) ≈ sum(JuMP.value.(y[i,j,:]) .* bfp.tax_options[i,j,:]) * JuMP.value(f[i,j])
#        end
#    end

Problems with the package and its usage can be explained through Github issues, ideally with a minimal working example showing the problem. Pull requests (PR) are welcome.

Please read detailed information in CONTRIBUTING.md.

• BilevelJuMP.jl is more flexible on the form of the problem input, and has been designed for MathOptInterface from the beginning, the two packages may be merged at some point
• Complementarity.jl solving a generic class including bilevel problems using non-linear techniques
• MibS for problems where the lower-level also includes integer variables. KKT conditions can therefore not be used and other branching and cutting plane techniques are leveraged.
• YALMIP includes a bilevel solver and offers roughly the same features (and a bit more) as BilevelOptimization.jl

See CITATION.bib, prefer citing the paper published in the Journal of Open-Source Software.