# 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 JuMP, BilevelJuMP, Cbc
model = BilevelModel(Cbc.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(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.

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 tolance on the enforcing
constraint.

### QuadraticToBinary.jl

Note onQuadraticToBinary.jl is a package that converts quadratic terms in constraints and objective. To do so the pack acts like a solver on top of the real solver and most data is forwarded directly to the solver itself. For many solvers it is enough to use:

```
SOLVER = Xpress.Optimizer()
Q_SOLVER = QuadraticToBinary.Optimizer{Float64}(SOLVER)
BilevelModel(Q_SOLVER, mode = BilevelJuMP.ProductMode(1e-5))
```

However, this might lead to some solver not supporting certain functionality like Cbc. In this case we need to:

```
SOLVER = Cbc.Optimizer()
CACHED_SOLVER = MOI.Utilities.CachingOptimizer(
MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}()), SOLVER)
Q_SOLVER = QuadraticToBinary.Optimizer{Float64}(CACHED_SOLVER)
BilevelModel(()->Q_SOLVER, mode = BilevelJuMP.ProductMode(1e-5))
```

Note that we used `()->Q_SOLVER`

instead of just `Q_SOLVER`

because `BilevelModel`

requires as constructor and not an instance of an object.

## Advanced Features

### Lower level dual variables

Suppose you have a constraint `b`

in the lower level:

`@constraint(Lower(model), b, ...)`

It is possible to access the dual variable of `b`

to use it in the upper level:

`@variable(Upper(model), lambda, DualOf(b))`

### Conic lower level

BilevelJuMP allows users to write conic models in the lower level. However, solving this kind of problems is much harder and requires complex solution methods. Mosek's Conic MIP can be used with the aid of QuadraticToBinary.jl.

It is also possible to solve Second Order Cone constrained models with Ipopt. In this case we need to add a special, non-standard bridge, to Ipopt as follows:

```
IPO_OPT = Ipopt.Optimizer(print_level=0)
IPO = MOI.Bridges.Constraint.SOCtoNonConvexQuad{Float64}(IPO_OPT)
BilevelModel(()->IPO, mode = BilevelJuMP.ProductMode(1e-5))
```