Indicates and object that is part of the dual of the lower level problem, and is shared with the upper level.

Indicates and object that is part of the lower level problem, but is shared with the upper level.

Indicates and object that is part of the lower level problem, but is not shared with the upper level.

Activates the indicator constraint on the primal constraint if the auxiliaty binary is one and activates the indicator constraint on the dual variable if the auxiliary binary is one.

Indicates and object that is part of the upper level problem, but is shared with the lower level.

Indicates and object that is part of the upper level problem, but is not shared with the lower level.

Activates the indicator constraint on the primal constraint if the auxiliaty binary is zero and activates the indicator constraint on the dual variable if the auxiliary binary is one.

Activates the indicator constraint on the primal constraint if the auxiliaty binary is zero and activates the indicator constraint on the dual variable if the auxiliary binary is zero.

BigMMode(; with_slack = false, primal_big_M = Inf, dual_big_M = Inf)

Used to solve a bilevel problem with the MPEC reformulation using Fortuny-Amat and McCarl's big-M method to convert complementarity constraints to a mixed integer formulation.

• with_slack indicates whether to use slack variables to convert the complementarity constraints to a mixed integer formulation. If false, the reformulation of a constraint like expr <= 0 is expr <= big_M * (1 - binary) and var <= big_M * binary, where var is the associated dual variable. If true, the reformulation is expr == slack, slack <= big_M * (1 - binary) and var <= big_M * binary.

• primal_big_M is a big-M used to primal variables that have no bounds so we can compute the big-M for the primal constraint.

• dual_big_M is a big-M used to dual variables that have no bounds so we can compute the big-M for the dual constraint.

Also known as FortunyAmatMcCarlMode (which can be used interchangeably).

BilevelVariableRef

Alias for GenericAffExpr{Float64,BilevelVariableRef}.

BilevelModel()

Create an empty BilevelModel with default settings, no solver and no solve mode.

Example

julia> model = BilevelModel()

BilevelModel(solver::Function; mode = BilevelJuMP.SOS1Mode(), add_bridges::Bool = true)

Create a BilevelModel with the given solver and solve mode.

• solver: is a functions that takes no arguments and returns a JuMP solver object.
• mode: is a solve mode object that defines how the model is solved.
• add_bridges: if true (default) then bridges are added to the model. If false then bridges are not added and the model is not modified.

Example

julia> model = BilevelModel(
    HiGHS.Optimizer,
    mode = BilevelJuMP.FortunyAmatMcCarlMode(primal_big_M = 1e6, dual_big_M = 1e6))

which is equivalent to

julia> model = BilevelModel(
    ()->HiGHS.Optimizer(),
    mode = BilevelJuMP.FortunyAmatMcCarlMode(primal_big_M = 1e6, dual_big_M = 1e6))

and equivalent to

julia> model = BilevelModel()

julia> BilevelJuMP.set_solver(model, HiGHS.Optimizer)

julia> BilevelJuMP.set_mode(model, BilevelJuMP.FortunyAmatMcCarlMode(primal_big_M = 1e6, dual_big_M = 1e6))

BilevelQuadExpr

Alias for GenericQuadExpr{Float64,BilevelVariableRef}.

BilevelVariableRef

Holds a reference to a variable in a bilevel model.

ComplementMode(; with_slack = false)

Used to solve a bilevel problem with the MPEC reformulation using actual complementarity constraints. A limited number of solvers support this mode. One example is Knitro.

• with_slack indicates whether to use slack variables to reformulate the complementarity constraints. Given a pair expr and var, the reformulation is expr == slack and var ⟂ slack instead of expr ⟂ slack.

DualOf(constraint::ConstraintRef)

Get the dual variable associated with a constraint. This is only valid for constraints in the upper level of a bilevel model.

Examples

julia> m = BilevelModel();

julia> @variable(Lower(m), x >= 0);

julia> @constraint(Lower(m), c, x <= 1);

julia> @variable(Upper(m), y, DualOf(c));

FortunyAmatMcCarlMode

See BigMMode for more details.

IndicatorMode(method::IndicatorSetting = BilevelJuMP.ONE_ONE)

Used to solve a bilevel problem with the MPEC reformulation using indicator constaints to convert complementarity constraints to a mixed integer formulation.

• method indicates how the indicator constraints are activated for primal cosntraints and dual variables. See IndicatorSetting for more details.

IndicatorSetting

The type of indicator function to use in the IndicatorMode mode.

Level

The level of a variable in a bilevel problem.

MixedMode(; default = SOS1Mode())

A mode that allows to mix different modes for different constraints and variables.

• default is the default mode to use for all constraints and variables that are not explicitly mapped to a mode.

ProductMode(epsilon = 0.0; with_slack = false, aggregation_group = nothing)

Used to solve a bilevel problem with the MPEC reformulation using products to convert complementarity constraints into non-convex quadratic constraints.

• with_slack indicates whether to use slack variables to reformulate the complementarity constraints. Given a pair expr and var, the reformulation is expr == slack and var * slack == 0 instead of expr * slack == 0.

• aggregation_group indicates whether to aggregate the products into a single quadratic constraint. If aggregation_group is nothing, then each product is converted into a quadratic constraint. If aggregation_group is a positive integer, then products with the same aggregation_group are aggregated into a single quadratic constraint.

SOS1Mode()

Used to solve a bilevel problem with the MPEC reformulation using SOS1 constraints to convert complementarity constraints into mixed-integer constraints.

StrongDualityMode(eps = 0.0, inequality = true)

A mode that adds a strong duality constraint of the lower level problem instead of reformulating the complementarity constraints.

• eps: The tolerance for the strong duality constraint. Defaults to 0.0.

• inequality: If true the strong duality constraint is added as two inequality constraints. If false the strong duality constraint is added as an equality constraint. Defaults to true.

Lower(model::BilevelModel)

Create a reference to the lower level of a bilevel model.

Example

julia> model = BilevelModel();

julia> @variable(Lower(model), x >= 0)

LowerOnly(model::BilevelModel)

Create a special reference to the lower level of a bilevel model. Variables created with this reference will not be shared with the upper level.

Upper(model::BilevelModel)

Create a reference to the upper level of a bilevel model.

Example

julia> model = BilevelModel();

julia> @variable(Upper(model), x >= 0)

UpperOnly(model::BilevelModel)

Create a special reference to the upper level of a bilevel model. Variables created with this reference will not be shared with the lower level.

build_time(model::BilevelModel)

Return the time it took to build the model.

get_copy_names(model::BilevelModel)

Return the value of the copy_names attribute of the solver.

get_dual_lower_bound_hint(cref)

Get the lower bound to the dual variable of the constraint cref that was set with set_dual_lower_bound_hint.

get_dual_upper_bound_hint(cref)

Get the upper bound to the dual variable of the constraint cref that was set with set_dual_upper_bound_hint.

get_mode(vi::BilevelVariableRef)

Get the mode of the bounds of a variable. This is used in MixedMode reformulations.

get_mode(ci::BilevelConstraintRef)

Get the mode of a constraint. This is used in MixedMode reformulations.

get_pass_start(model::BilevelModel)

Checks if passing start values (both primal and dual) to the solver is activated.

get_primal_lower_bound_hint(cref)

Get the lower bound to the primal variable of the constraint cref that was set with set_primal_lower_bound_hint.

get_primal_upper_bound_hint(cref)

Get the upper bound to the primal variable of the constraint cref that was set with set_primal_upper_bound_hint.

lower_objective_value(model::BilevelModel; result::Int = 1)

Return the value of the objective function of the lower level problem.

set_copy_names(model::BilevelModel)

Set the copy_names attribute of the solver to true.

set_dual_lower_bound_hint(cref, value)

Set a lower bound to the dual variable of the constraint cref to value. This bound will not be dualized. The dual lower bound hint is used to help the solution method.

Solution modes can be benefitted from this hint:

• BigMMode will use this information to compute a tighter bound for the dual variable.

• Other modes will be stabilized by the existence of the bounds on variables that would otherwise no be bounded.

• Bounds that are not dualized are also useful for binary expansions of products of variables that can be done with QuadraticToBinary.jl.

set_dual_upper_bound_hint(cref, value)

Set a upper bound to the dual variable of the constraint cref to value. This bound will not be dualized. The dual upper bound hint is used to help the solution method.

Solution modes can be benefitted from this hint:

• BigMMode will use this information to compute a tighter bound for the dual variable.

• Other modes will be stabilized by the existence of the bounds on variables that would otherwise no be bounded.

• Bounds that are not dualized are also useful for binary expansions of products of variables that can be done with QuadraticToBinary.jl.

set_mode(bm::BilevelModel, mode::AbstractBilevelSolverMode)

Set the mode of a bilevel model.

set_mode(vi::BilevelVariableRef, mode::AbstractBilevelSolverMode)

Set the mode of the bounds of a variable. This is used in MixedMode reformulations.

set_mode(ci::BilevelVariableRef, mode::AbstractBilevelSolverMode)

Set the mode of a constraint. This is used in MixedMode reformulations.

set_pass_start(model::BilevelModel)

Activate passing start values (both primal and dual) to the solver.

set_primal_lower_bound_hint(vref, value)

Set a lower bound to the prima variable vref to value. This bound will not be dualized. The lower bound hint is used to help the solution method.

Solution modes can be benefitted from this hint:

• BigMMode will use this information to compute a tighter bound for the primal constraint variable.

• Other modes will be stabilized by the existence of the bounds on variables that would otherwise no be bounded.

• Bounds that are not dualized are also useful for binary expansions of products of variables that can be done with QuadraticToBinary.jl.

set_primal_upper_bound_hint(vref, value)

Set a upper bound to the prima variable vref to value. This bound will not be dualized. The upper bound hint is used to help the solution method.

Solution modes can be benefitted from this hint:

• BigMMode will use this information to compute a tighter bound for the primal constraint variable.

• Other modes will be stabilized by the existence of the bounds on variables that would otherwise no be bounded.

• Bounds that are not dualized are also useful for binary expansions of products of variables that can be done with QuadraticToBinary.jl.

solve_with_MibS(model::BilevelModel, mibs_call; kwargs...)

Inputs

• model::BilevelModel: the model to optimize
• mibs_call: shoul be MibS_jll.mibs remember to import MibS_jll before.
• verbose_results::Bool = false: controls the verbosity of the solver output.

If verbose_results=false, nothing is printed. Set to true to display the MibS output.

• verbose_file::Bool = false: Writes MibS input files to screen.
• keep_files::Bool = false: Saves MibS input files to pwd().

Outputs

This function returns a NamedTuple with fields:

• status::Bool: true if the problem is feasible and has an optimal solution. false otherwise.
• objective::Float64: objective value (cost) of the upper problem
• nonzero_upper::Dict{Int, Float64}: it returns Dict{index => value}, in which the index refers to the index of upper variables with non zero values and the index starts from 0. Here, the order of the variables is based on their order of appearance in the MPS file.
• nonzero_lower::Dict{Int, Float64}: it has the same structure as nonzero_upper, but it represents the index of non-zero variables in the lower problem.
• all_upper::Dict{String, Float64}: it returns Dict{name => value} which contains all upper variables values (zero and non-zero). For recalling the variables, you need to use the same name as you used to define the variables, e.g., for @variable(Upper(model), y, Int), we need to use all_upper["y"] to get the value of the variable y.
• all_lower::Dict{String, Float64}: it has the same structure as the all_upper but is defined for lower variables.
• all_var::Dict{MOI.VariableIndex, Float64}: it contains information on all variables (upper and lower) in the format of MOI.VariableIndex and their output values.

Split variable because actual owner of the variable should be the one holding the bounds.

unset_copy_names(model::BilevelModel)

Set the copy_names attribute of the solver to false.

unset_mode(vi::BilevelVariableRef)

Unset the mode of the bounds of a variable. This will use the default mode for the bounds. This is used in MixedMode reformulations.

unset_mode(ci::BilevelConstraintRef)

Unset the mode of a constraint. This will use the default mode for the constraint. This is used in MixedMode reformulations.

unset_pass_start(model::BilevelModel)

Deactivate passing start values (both primal and dual) to the solver.