BilevelJuMP.jl

A bilevel optimization extension of the JuMP package.

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Introduction

BilevelJuMP is a package for modeling and solving bilevel optimization problems in Julia. As an extension of the JuMP package, BilevelJuMP allows users to employ the usual JuMP syntax with minor modifications to describe the problem and query solutions.

BilevelJuMP is built on top of MathOptInterface and makes strong use of its features to reformulate the problem as a single level problem and solve it with available MIP, NLP, and other solvers.

The currently available methods are based on re-writing the problem using the KKT conditions of the lower level. For that we make strong use of Dualization.jl

Example

using BilevelJuMP, SCIP

model = BilevelModel(SCIP.Optimizer, mode = BilevelJuMP.SOS1Mode())

@variable(Lower(model), x)
@variable(Upper(model), y)

@objective(Upper(model), Min, 3x + y)
@constraints(Upper(model), begin
    x <= 5
    y <= 8
    y >= 0
end)

@objective(Lower(model), Min, -x)
@constraints(Lower(model), begin
     x +  y <= 8
    4x +  y >= 8
    2x +  y <= 13
    2x - 7y <= 0
end)

optimize!(model)

objective_value(model) # = 3 * (3.5 * 8/15) + 8/15 # = 6.13...
value(x) # = 3.5 * 8/15 # = 1.86...
value(y) # = 8/15 # = 0.53...

The option BilevelJuMP.SOS1Mode() indicates that the solution method used will be a KKT reformulation emplying SOS1 to model complementarity constraints and solve the problem with MIP solvers (Cbc, Xpress, Gurobi, CPLEX, SCIP).

Alternatively, the option BilevelJuMP.IndicatorMode() is almost equivalent to the previous. The main difference is that it relies on Indicator constraints instead. This kind of constraints is available in some MIP solvers.

A third and classic option it the BilevelJuMP.FortunyAmatMcCarlMode(), which relies on the Fortuny-Amat and McCarl big-M method that requires a MIP solver with very basic functionality, i.e., just binary variables are needed. The main drawback of this method is that one must provide bounds for all primal and dual variables. However, if the bounds are good, this method can be more efficient than the previous. Bound hints to compute the big-Ms can be passed with the methods: set_primal_(upper\lower)_bound_hint(variable, bound), for primals; and set_dual_(upper\lower)_bound_hint(constraint, bound) for duals. We can also call FortunyAmatMcCarlMode(primal_big_M = vp, dual_big_M = vd), where vp and vd are, repspectively, the big M fallback values for primal and dual variables, these are used when some variables have no given bounds, otherwise the given bounds are used instead.

Another option is BilevelJuMP.ProductMode() that reformulates the complementarity constraints as products so that the problem can be solved by NLP (Ipopt, KNITRO) solvers or even MIP solvers with the aid of binary expansions (see QuadraticToBinary.jl). Note that binary expansions require variables to have upper and lower bounds. Also, note that the Gurobi solver supports products, but requires setting the "NonConvex" options.

Finally, one can use BilevelJuMP.MixedMode(default = mode) where mode is one of the other modes described above. With this method it is possible to set complementarity reformulations per constraint with BilevelJuMP.set_mode(ctr, mode).

An alternative to complementarity constraint reformulation is the Strong Duality reformulation which add the constraint enforcing primal dual equality. The option is BilevelJuMP.StrongDualityMode(eps) where eps is the tolerance on the enforcing constraint.