### Set Initial Starting Conditions

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/examples/FOBP_example3

Configures starting values for primal and dual variables to assist the solver.

```julia
JuMP.set_start_value.(x, 0)
JuMP.set_start_value.(ya, 1)
JuMP.set_start_value.(yb, 0)
JuMP.set_start_value(z, 1)
for i in 1:I
    JuMP.set_dual_start_value.(b1, 0)
    JuMP.set_dual_start_value.(b2, 0)
    JuMP.set_dual_start_value.(b3, -1)
end
```

--------------------------------

### Install BilevelJuMP

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable

Use the Julia package manager to add the BilevelJuMP package.

```julia
julia> import Pkg

julia> Pkg.add("BilevelJuMP")
```

--------------------------------

### Initialize QuadraticToBinary with HiGHS

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/tutorials/quad_to_bin

Standard setup for using QuadraticToBinary with the HiGHS solver in a BilevelModel.

```julia
using BilevelJuMP, QuadraticToBinary, HiGHS

SOLVER = HiGHS.Optimizer()
Q_SOLVER = QuadraticToBinary.Optimizer{Float64}(SOLVER, lb = -10, ub = 10)
model = BilevelModel(()->Q_SOLVER, mode = BilevelJuMP.ProductMode(1e-6))

@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)
@assert abs(objective_value(model) - (3 * (3.5 * 8/15) + 8/15)) < 1e-1 # src
```

--------------------------------

### set_pass_start(model::BilevelModel)

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/reference

Manages passing start values to the solver.

```APIDOC
## set_pass_start(model::BilevelModel)
## get_pass_start(model::BilevelModel)
## unset_pass_start(model::BilevelModel)

### Description
Functions to activate, check, or deactivate passing start values (primal and dual) to the solver.
```

--------------------------------

### Setting JuMP Start Values

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/tutorials/conic_lower

Alternative syntax for defining initial values for variables and duals in JuMP.

```julia
set_start_value(x, 6)
set_dual_start_value(con2, 0)
```

```julia
0
```

--------------------------------

### Manage pass start values attribute

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/reference

Controls whether primal and dual start values are passed to the solver.

```julia
set_pass_start(model::BilevelModel)
```

```julia
get_pass_start(model::BilevelModel)
```

```julia
unset_pass_start(model::BilevelModel)
```

--------------------------------

### IPOPT Solver Output Log

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/tutorials/conic_lower

Example of an iterative solver log showing objective values, infeasibility metrics, and final convergence statistics.

```text
  22r 7.1782285e+00 9.24e-01 1.00e+03   0.1 0.00e+00    -  0.00e+00 4.09e-07R  3
  23r 7.2536425e+00 9.25e-01 1.21e+03   0.1 1.19e+03    -  9.06e-03 1.17e-03f  1
  24  7.2536348e+00 9.25e-01 1.10e+05  -1.0 9.53e-01  -0.1 9.22e-01 1.70e-05h  1
  25r 7.2536348e+00 9.25e-01 1.00e+03  -0.0 0.00e+00    -  0.00e+00 2.59e-07R  6
  26r 7.5343972e+00 9.34e-01 1.22e+03  -0.0 9.54e+01    -  8.79e-03 3.97e-03f  1
  27r 7.5327092e+00 9.46e-01 9.90e+02  -0.0 1.70e+00   2.0 1.33e-01 1.52e-02f  1
  28r 7.5181606e+00 9.80e-01 9.71e+02  -0.0 7.28e-01   2.4 2.49e-03 1.46e-01f  1
  29r 7.5135600e+00 9.88e-01 9.55e+02  -0.0 7.04e-01   2.9 6.85e-02 2.06e-01f  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  30r 7.5135662e+00 9.88e-01 2.66e+03  -0.0 8.57e-02   3.3 1.00e+00 1.86e-02h  1
  31r 7.5227783e+00 9.94e-01 9.97e+00  -0.0 1.58e-02   2.8 1.00e+00 1.00e+00f  1
  32r 7.5237814e+00 1.00e+00 1.29e+00  -1.4 6.12e-03   2.3 1.00e+00 1.00e+00f  1
  33r 7.5248565e+00 1.00e+00 1.37e-01  -2.1 1.95e-03   1.8 1.00e+00 1.00e+00f  1
  34r 7.5265535e+00 1.00e+00 7.02e+01  -3.2 3.47e-03    -  1.00e+00 7.46e-01f  1
  35r 9.2295707e+00 1.00e+00 6.44e+02  -3.2 2.52e+00    -  4.06e-01 9.10e-01f  1
  36r 9.1585683e+00 1.00e+00 2.89e+02  -3.2 2.61e-01   1.4 6.81e-01 1.00e+00h  1
  37r 9.1584002e+00 1.00e+00 8.13e+01  -3.2 1.15e-03   1.8 5.41e-01 8.53e-01h  1
  38r 9.1611165e+00 1.00e+00 4.80e+01  -3.2 3.20e-03   1.3 1.57e-01 1.00e+00f  1
  39r 9.1870694e+00 9.90e-01 5.54e+01  -3.2 3.68e-02   1.7 3.29e-02 8.23e-01f  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  40r 9.1893066e+00 9.72e-01 4.56e+01  -3.2 1.82e-02   3.1 6.00e-01 1.00e+00f  1
  41r 9.1979046e+00 9.52e-01 2.81e+01  -3.2 6.66e-02   2.6 3.62e-01 3.05e-01f  1
  42r 9.3297133e+00 8.04e-01 7.97e+01  -3.2 1.08e+00   2.1 5.48e-02 1.62e-01f  1
  43  9.3307037e+00 8.04e-01 6.64e+03  -1.0 2.93e+01  -0.6 6.97e-02 4.51e-05f  1
  44  9.9071010e+00 7.70e-01 1.39e+04  -1.0 1.82e+01  -1.1 1.23e-01 4.22e-02f  2
  45  1.1984910e+01 5.61e-01 2.05e+04  -1.0 1.22e+01  -0.6 3.78e-01 2.72e-01H  1
  46  1.2009065e+01 3.55e-01 9.73e+04  -1.0 5.61e-01   0.7 1.00e+00 3.66e-01h  1
  47  1.2007891e+01 2.63e-01 2.24e+05  -1.0 3.55e-01   1.1 1.00e+00 2.59e-01h  1
  48  1.2015766e+01 1.20e-01 2.26e+05  -1.0 2.63e-01   0.6 1.00e+00 5.43e-01h  1
  49  1.2021975e+01 5.78e-02 3.75e+05  -1.0 1.20e-01   1.1 1.00e+00 5.20e-01h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  50  1.2032201e+01 2.04e-02 4.94e+05  -1.0 7.12e-02   0.6 1.00e+00 6.46e-01h  1
  51  1.2053842e+01 6.82e-03 8.67e+05  -1.0 7.96e-02   0.1 1.00e+00 6.66e-01h  1
  52  1.2068410e+01 8.41e-04 4.74e+05  -1.0 3.85e-02   0.5 1.00e+00 8.77e-01h  1
  53  1.2084847e+01 0.00e+00 5.33e+00  -1.0 3.41e-02   0.1 1.00e+00 1.00e+00f  1
  54  1.2023010e+01 2.22e-16 1.51e+05  -5.7 1.26e-01  -0.4 9.84e-01 1.00e+00f  1
  55  1.1999979e+01 0.00e+00 2.91e+02  -5.7 5.58e-02    -  9.98e-01 8.16e-01h  1
  56  1.1999973e+01 0.00e+00 4.25e-06  -5.7 3.02e-04    -  1.00e+00 1.00e+00h  1
  57  1.1999970e+01 0.00e+00 5.91e+00  -8.6 5.01e-06    -  9.68e-01 1.00e+00h  1
  58  1.1999970e+01 0.00e+00 2.61e-10  -8.6 1.18e-07    -  1.00e+00 1.00e+00h  1

Number of Iterations....: 58

                                   (scaled)                 (unscaled)
Objective...............:   1.1999969855039765e+01    1.1999969855039765e+01
Dual infeasibility......:   2.6066093158514153e-10    2.6066093158514153e-10
Constraint violation....:   0.0000000000000000e+00    0.0000000000000000e+00
Variable bound violation:   0.0000000000000000e+00    0.0000000000000000e+00
Complementarity.........:   2.5705110524677697e-09    2.5705110524677697e-09
Overall NLP error.......:   2.5705110524677697e-09    2.5705110524677697e-09


Number of objective function evaluations             = 77
Number of objective gradient evaluations             = 43
Number of equality constraint evaluations            = 77
Number of inequality constraint evaluations          = 77
Number of equality constraint Jacobian evaluations   = 62
Number of inequality constraint Jacobian evaluations = 62
Number of Lagrangian Hessian evaluations             = 58
Total seconds in IPOPT                               = 0.039

EXIT: Optimal Solution Found.
```

--------------------------------

### Configure SCIP for QuadraticToBinary

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/tutorials/quad_to_bin

Setup required for solvers like SCIP that require a CachingOptimizer wrapper.

```julia
using SCIP
```

```julia
SOLVER = SCIP.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))
```

--------------------------------

### Install HiGHS solver

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/tutorials/getting_started

Add the HiGHS solver package to enable solving the optimization models.

```julia
import Pkg
Pkg.add("HiGHS")
```

--------------------------------

### Get model build time

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/reference

Returns the duration taken to build the bilevel model.

```julia
build_time(model::BilevelModel)
```

--------------------------------

### Define Variables

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/examples/FOBP_example2

Defines variables for the upper and lower levels with initial start values.

```julia
@variable(Upper(model), y, start = 8 / 15)

@variable(Lower(model), x, start = 3.5 * 8 / 15)
```

--------------------------------

### Get bilevel component mode

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/reference

Retrieves the current mode for a constraint or variable bounds.

```julia
get_mode(ci::BilevelConstraintRef)
```

```julia
get_mode(vi::BilevelVariableRef)
```

--------------------------------

### Get dual upper bound hint

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/reference

Retrieves the dual upper bound hint previously set for a constraint.

```julia
get_dual_upper_bound_hint(cref)
```

--------------------------------

### Get dual lower bound hint

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/reference

Retrieves the dual lower bound hint previously set for a constraint.

```julia
get_dual_lower_bound_hint(cref)
```

--------------------------------

### Get primal upper bound hint

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/reference

Retrieves the primal upper bound hint previously set for a constraint.

```julia
get_primal_upper_bound_hint(cref)
```

--------------------------------

### Get primal lower bound hint

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/reference

Retrieves the primal lower bound hint previously set for a constraint.

```julia
get_primal_lower_bound_hint(cref)
```

--------------------------------

### Initialize Bilevel Model

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/examples/FOBP_example4

Sets up the BilevelModel using the Ipopt solver and ProductMode.

```julia
using BilevelJuMP
using Ipopt

model = BilevelModel(Ipopt.Optimizer; mode = BilevelJuMP.ProductMode(1e-9))
```

--------------------------------

### Initialize Bilevel Model

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/examples/SOCBLP_example1

Sets up the BilevelModel with the required SOCtoNonConvexQuad bridge and Ipopt optimizer.

```julia
using BilevelJuMP
using Ipopt

model = BilevelModel(
    () -> MOI.Bridges.Constraint.SOCtoNonConvexQuad{Float64}(Ipopt.Optimizer());
    mode = BilevelJuMP.ProductMode(1e-9),
)
```

--------------------------------

### Configure BilevelModel step-by-step

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/tutorials/getting_started

Initializes an empty model and configures the optimizer and mode sequentially.

```julia
julia> model = BilevelModel()

julia> set_optimizer(model, HiGHS.Optimizer)

julia> BilevelJuMP.set_mode(model,
           BilevelJuMP.FortunyAmatMcCarlMode(primal_big_M = 100, dual_big_M = 100))
```

--------------------------------

### get_mode(ci::BilevelConstraintRef)

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/reference

Gets the mode of a constraint or variable bounds.

```APIDOC
## get_mode(ci::BilevelConstraintRef)
## get_mode(vi::BilevelVariableRef)

### Description
Get the mode of a constraint or the bounds of a variable. Used in MixedMode reformulations.
```

--------------------------------

### Configure and Optimize with ProductMode

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/tutorials/modes

Sets the optimizer to Ipopt and the bilevel mode to ProductMode before running the optimization.

```julia
set_optimizer(model, Ipopt.Optimizer)

BilevelJuMP.set_mode(model, BilevelJuMP.ProductMode())

optimize!(model)

objective_value(model)

@assert abs(objective_value(model) - (3 * (7/2 * 8/15) + 8/15)) < 1e-1 # src
```

--------------------------------

### Initialize Bilevel Model

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/examples/MibS_example1

Imports necessary packages and initializes a new BilevelModel.

```julia
using BilevelJuMP
using Test
using MibS_jll

model = BilevelModel()
```

--------------------------------

### Load required packages

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/tutorials/lower_duals

Imports the necessary libraries for bilevel modeling and optimization.

```julia
using BilevelJuMP
using Ipopt
using QuadraticToBinary
using HiGHS
```

--------------------------------

### Configure Solver and Solution Method

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/tutorials/lower_duals

Set the optimizer to use the QuadraticToBinary wrapper with HiGHS and apply the Fortuny-Amat and McCarl solution mode.

```julia
set_optimizer(model,
    ()->QuadraticToBinary.Optimizer{Float64}(HiGHS.Optimizer()))
BilevelJuMP.set_mode(model,
    BilevelJuMP.FortunyAmatMcCarlMode(dual_big_M = 100))
optimize!(model)
```

--------------------------------

### Get dual variable of a constraint

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/reference

Retrieves the dual variable associated with a constraint in the upper level.

```julia
julia> m = BilevelModel();

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

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

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

--------------------------------

### Create a BilevelModel with solver and mode

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/reference

Initializes a BilevelModel with a specific solver and solve mode.

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

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

```julia
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))
```

--------------------------------

### Get lower level objective value

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/reference

Retrieves the objective function value of the lower level problem.

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

--------------------------------

### Create an empty BilevelModel

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/reference

Initializes a BilevelModel with default settings.

```julia
julia> model = BilevelModel()
```

--------------------------------

### Display Model

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/tutorials/getting_started

Print the structure of the bilevel model to verify the levels and constraints.

```julia
julia> print(model)
```

--------------------------------

### Configure and Optimize with StrongDualityMode

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/tutorials/modes

Sets the optimizer to Ipopt and the bilevel mode to StrongDualityMode before running the optimization.

```julia
set_optimizer(model, Ipopt.Optimizer)

BilevelJuMP.set_mode(model, BilevelJuMP.StrongDualityMode())

optimize!(model)

objective_value(model)

@assert abs(objective_value(model) - (3 * (7/2 * 8/15) + 8/15)) < 1e-1 # src
```

--------------------------------

### Define and solve a bilevel optimization problem

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/tutorials/getting_started

Initializes a bilevel model using the Fortuny-Amat-McCarl reformulation, defines hierarchical levels, and retrieves the optimal solution.

```julia
julia> using BilevelJuMP

julia> using HiGHS

julia> model = BilevelModel(
           HiGHS.Optimizer,
           mode = BilevelJuMP.FortunyAmatMcCarlMode(primal_big_M = 100, dual_big_M = 100))
An Abstract JuMP Model
Feasibility problem with:
Variables: 0
Upper Constraints: 0
Lower Constraints: 0
Bilevel Model
Solution method: BilevelJuMP.BigMMode{Float64}(false, 100.0, 100.0, BilevelJuMP.ComplementBoundCache(Dict{MathOptInterface.VariableIndex, BilevelJuMP.BilevelVariableInfo}(), Dict{MathOptInterface.VariableIndex, BilevelJuMP.BilevelVariableInfo}(), Dict{MathOptInterface.ConstraintIndex, BilevelJuMP.BilevelConstraintInfo}(), Dict{MathOptInterface.VariableIndex, BilevelJuMP.BilevelVariableInfo}()))
Solver name: HiGHS

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

julia> @variable(Upper(model), y)
y

julia> @objective(Upper(model), Min, 3x + y)
3 x + y

julia> @constraint(Upper(model), u1, x <= 5)
u1 : x ≤ 5

julia> @constraint(Upper(model), u2, y <= 8)
u2 : y ≤ 8

julia> @constraint(Upper(model), u3, y >= 0)
u3 : y ≥ 0

julia> @objective(Lower(model), Min, -x)
-x

julia> @constraint(Lower(model), l1, x +  y <= 8)
l1 : x + y ≤ 8

julia> @constraint(Lower(model), l2, 4x +  y >= 8)
l2 : 4 x + y ≥ 8

julia> @constraint(Lower(model), l3, 2x +  y <= 13)
l3 : 2 x + y ≤ 13

julia> @constraint(Lower(model), l4, 2x - 7y <= 0)
l4 : 2 x - 7 y ≤ 0

julia> print(model)
Upper level:
Min 3 x + y
Subject to
 u3 : y ≥ 0
 u1 : x ≤ 5
 u2 : y ≤ 8

Lower level:
Min -x
Subject to
 l2 : 4 x + y ≥ 8
 l1 : x + y ≤ 8
 l3 : 2 x + y ≤ 13
 l4 : 2 x - 7 y ≤ 0

julia> optimize!(model)
┌ Warning: primal_var_dual_quad_slack field is deprecated, use primal_var_in_quad_obj_to_dual_slack_var instead
└ @ Dualization ~/.julia/packages/Dualization/ihzlf/src/structures.jl:268
┌ Warning: primal_parameter field is deprecated, use primal_parameter_to_dual_parameter instead
└ @ Dualization ~/.julia/packages/Dualization/ihzlf/src/structures.jl:265
Running HiGHS 1.5.1 [date: 1970-01-01, git hash: 93f1876e4]
Copyright (c) 2023 HiGHS under MIT licence terms
WARNING: Row      0 has         infeasibility of           1 from [lower, value, upper] = [             -1;               0;              -1]
WARNING: Row      2 has         infeasibility of           8 from [lower, value, upper] = [              8;               0;             inf]
Solution has               num          max          sum
Col     infeasibilities      0            0            0
Integer infeasibilities      0            0            0
Row     infeasibilities      4           13           30
Row     residuals            0            0            0
Attempting to find feasible solution of continuous variables for user-supplied values of discrete variables
Presolving model
13 rows, 6 cols, 24 nonzeros
Problem status detected on presolve: Infeasible
Model   status      : Infeasible
Objective value     :  0.0000000000e+00
HiGHS run time      :          0.00
Presolving model
13 rows, 10 cols, 32 nonzeros
13 rows, 10 cols, 32 nonzeros

Solving MIP model with:
   13 rows
   10 cols (4 binary, 0 integer, 0 implied int., 6 continuous)
   32 nonzeros

        Nodes      |    B&B Tree     |            Objective Bounds              |  Dynamic Constraints |       Work      
     Proc. InQueue |  Leaves   Expl. | BestBound       BestSol              Gap |   Cuts   InLp Confl. | LpIters     Time

         0       0         0   0.00%   0               inf                  inf        0      0      0         0     0.0s
 R       0       0         0   0.00%   6.133333333     6.133333333        0.00%        0      0      0         5     0.0s

Solving report
  Status            Optimal
  Primal bound      6.13333333333
  Dual bound        6.13333333333
  Gap               0% (tolerance: 0.01%)
  Solution status   feasible
                    6.13333333333 (objective)
                    0 (bound viol.)
                    0 (int. viol.)
                    0 (row viol.)
  Timing            0.00 (total)
                    0.00 (presolve)
                    0.00 (postsolve)
  Nodes             1
  LP iterations     6 (total)
                    0 (strong br.)
                    0 (separation)
                    0 (heuristics)

julia> termination_status(model)
OPTIMAL::TerminationStatusCode = 1

julia> primal_status(model)
FEASIBLE_POINT::ResultStatusCode = 1

julia> dual_status(Lower(model))
FEASIBLE_POINT::ResultStatusCode = 1

julia> dual_status(Upper(model))
NO_SOLUTION::ResultStatusCode = 0

julia> objective_value(model)
6.133333333333333

julia> objective_value(Lower(model))
-1.8666666666666667

julia> objective_value(Upper(model))
6.133333333333333

julia> value(x)
1.8666666666666667

julia> value(y)
0.5333333333333332

julia> dual(l1)
1-element Vector{Float64}:
 0.0

julia> dual(l2)
1-element Vector{Float64}:
 49.75
```

--------------------------------

### Load BilevelJuMP and Solver

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/tutorials/getting_started

Load the BilevelJuMP package and a compatible solver like HiGHS.

```julia
julia> using BilevelJuMP
```

```julia
julia> using HiGHS
```

--------------------------------

### Verify Solution

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/examples/MibS_example2

Run tests to verify the solution status and objective value.

```julia
@test solution.status == true
@test solution.objective ≈ -8.0
@test solution.nonzero_upper == Dict(0 => 6.0)
@test solution.nonzero_lower == Dict(0 => 5.0)
@test solution.all_upper["x"] == 6.0
@test solution.all_upper["z"] == 0
@test solution.all_lower["y"] == 5.0
```

--------------------------------

### Solve and Verify

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/examples/SOCBLP_example1

Executes the optimization and retrieves the results.

```julia
optimize!(model)

primal_status(model)

termination_status(model)

objective_value(model)

value.(y)
```

--------------------------------

### Configure and Solve Bilevel Model

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/tutorials/lower_duals

Sets the reformulation mode to Strong Duality and optimizes the model using the Ipopt solver.

```julia
BilevelJuMP.set_mode(model, BilevelJuMP.StrongDualityMode())
set_optimizer(model, Ipopt.Optimizer)
optimize!(model)
```

--------------------------------

### Solve Bilevel Model with Ipopt and SOC Bridge

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/tutorials/conic_lower

Configures the model to use the ProductMode and the SOCtoNonConvexQuad bridge to handle second-order cones with Ipopt.

```julia
using Ipopt
BilevelJuMP.set_mode(model, BilevelJuMP.ProductMode(1e-5))
set_optimizer(model,
    ()->MOI.Bridges.Constraint.SOCtoNonConvexQuad{Float64}(Ipopt.Optimizer()))
optimize!(model)
```

--------------------------------

### Initialize BilevelModel with optimizer attributes

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/tutorials/getting_started

Passes solver attributes directly during the construction of the BilevelModel using optimizer_with_attributes.

```julia
julia> model = BilevelModel(
           optimizer_with_attributes(HiGHS.Optimizer, "output_flag" => false))
An Abstract JuMP Model
Feasibility problem with:
Variables: 0
Upper Constraints: 0
Lower Constraints: 0
Bilevel Model
Solution method: BilevelJuMP.SOS1Mode{Float64}()
Solver name: HiGHS
```

--------------------------------

### Configure BilevelJuMP with SOS1Mode and SCIP

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/tutorials/modes

Sets the optimizer to SCIP and configures the model to use SOS1Mode for bilevel optimization.

```julia
set_optimizer(model, SCIP.Optimizer)

BilevelJuMP.set_mode(model, BilevelJuMP.SOS1Mode())

optimize!(model)

objective_value(model)

@assert abs(objective_value(model) - (3 * (3.5 * 8/15) + 8/15)) < 1e-1 # src
```

--------------------------------

### Verify Solution

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/examples/MibS_example1

Performs automated tests to verify the solver output.

```julia
@test solution.status == true
@test solution.objective ≈ -53
@test solution.nonzero_upper == Dict(0 => 6.0)
@test solution.nonzero_lower == Dict(0 => 5.0)
@test solution.all_upper["x"] == 6.0
@test solution.all_lower["y"] == 5.0
```

--------------------------------

### Product Reformulation Mode

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/reference

Converts complementarity constraints into non-convex quadratic constraints using products.

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

--------------------------------

### BilevelModel

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/reference

Constructs a new BilevelModel instance, optionally specifying a solver and solve mode.

```APIDOC
## BilevelModel()

### Description
Creates an empty BilevelModel with default settings, no solver, and no solve mode.

### Signature
`BilevelModel()`

## BilevelModel(solver::Function; mode, add_bridges)

### Description
Creates a BilevelModel with the specified solver and solve mode.

### Parameters
- **solver** (Function) - A function that takes no arguments and returns a JuMP solver object.
- **mode** (Any) - A solve mode object defining how the model is solved (default: BilevelJuMP.SOS1Mode()).
- **add_bridges** (Bool) - If true, bridges are added to the model (default: true).
```

--------------------------------

### Configure BilevelJuMP with SCIP IndicatorMode

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/tutorials/modes

Sets the optimizer to SCIP and configures the model to use IndicatorMode for solving bilevel problems.

```julia
set_optimizer(model, SCIP.Optimizer)

BilevelJuMP.set_mode(model, BilevelJuMP.IndicatorMode())

optimize!(model)

objective_value(model)

@assert abs(objective_value(model) - (3 * (3.5 * 8/15) + 8/15)) < 1e-1 # src
```

--------------------------------

### Cite the BilevelJuMP paper

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/background

BibTeX entry for referencing the BilevelJuMP.jl modeling and solving framework.

```bibtex
@article{diasgarcia2022bileveljump,
    title={{BilevelJuMP. jl}: {M}odeling and solving bilevel optimization in {J}ulia},
    author={{Dias Garcia}, Joaquim and Bodin, Guilherme and Street, Alexandre},
    journal={arXiv preprint arXiv:2205.02307},
    year={2022}
}
```

--------------------------------

### build_time(model::BilevelModel)

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/reference

Returns the time taken to build the model.

```APIDOC
## build_time(model::BilevelModel)

### Description
Return the time it took to build the model.

### Parameters
- **model** (BilevelModel) - Required - The bilevel model instance.
```

--------------------------------

### Define a bilevel model

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/tutorials/modes

Construct a bilevel model and define variables, objectives, and constraints for both upper and lower levels.

```julia
model = BilevelModel()

@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
    c1,  x +  y <= 8
    c2, 4x +  y >= 8
    c3, 2x +  y <= 13
    c4, 2x - 7y <= 0
end)
```

--------------------------------

### Define a bilevel model with DualOf

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/tutorials/lower_duals

Constructs a bilevel model where the upper-level variable lambda is linked to the lower-level demand_equilibrium constraint.

```julia
model = BilevelModel()
@variable(Upper(model), 0 <= qS <= 100)
@variable(Lower(model), 0 <= gS <= 100)
@variable(Lower(model), 0 <= gR1 <= 40)
@variable(Lower(model), 0 <= gR2 <= 40)
@variable(Lower(model), 0 <= gD <= 100)
@objective(Lower(model), Min, 50gR1 + 100gR2 + 1000gD)
@constraint(Lower(model), gS <= qS)
@constraint(Lower(model), demand_equilibrium, gS + gR1 + gR2 + gD == 100)
@variable(Upper(model), lambda, DualOf(demand_equilibrium))
@objective(Upper(model), Max, lambda*gS)
```

--------------------------------

### Configure BigMMode with HiGHS

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/tutorials/modes

Sets the solver to HiGHS and configures the BigMMode parameters for a bilevel model.

```julia
set_optimizer(model, HiGHS.Optimizer)

BilevelJuMP.set_mode(model,
    BilevelJuMP.BigMMode(primal_big_M = 100, dual_big_M = 100))

optimize!(model)

objective_value(model)
@assert abs(objective_value(model) - (3 * (7/2 * 8/15) + 8/15)) < 1e-1 # src
```

--------------------------------

### Solve with MibS

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/examples/MibS_example1

Executes the solver using the MibS backend.

```julia
solution = BilevelJuMP.solve_with_MibS(model, MibS_jll.mibs)
```

--------------------------------

### Define a Bilevel Model with Conic Constraints

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/tutorials/conic_lower

Uses BilevelJuMP to define a bilevel program where the lower level includes a SecondOrderCone constraint.

```julia
using BilevelJuMP
model = BilevelModel()
@variable(Upper(model), x)
@variable(Lower(model), y[i=1:3])
@objective(Upper(model), Min, x + 3y[1])
@constraint(Upper(model), x >= 2)
@constraint(Upper(model), x <= 6)
@objective(Lower(model), Min, - y[1])
@constraint(Lower(model), con1, x +  y[1] <=  8)
@constraint(Lower(model), con2, x + 4y[1] >=  8)
@constraint(Lower(model), con3, x + 2y[1] <= 12)
@constraint(Lower(model), con4, y in SecondOrderCone())
```

--------------------------------

### Model Status Queries

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/examples/PHTP_example1

Commands to retrieve the primal and termination status of the optimization model.

```julia
primal_status(model)
```

```text
FEASIBLE_POINT::ResultStatusCode = 1
```

```julia
termination_status(model)
```

```text
LOCALLY_SOLVED::TerminationStatusCode = 4
```

--------------------------------

### Set Dual Hints

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/examples/FOBP_example2

Provides dual bound hints for the lower level constraints to assist the solver.

```julia
for c in [l1, l2, l3, l4]
    BilevelJuMP.set_dual_upper_bound_hint(c, 15)
    BilevelJuMP.set_dual_lower_bound_hint(c, -15)
end
```

--------------------------------

### Optimize the Bilevel Model

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/examples/FOBP_example4

Executes the optimization process for the defined model.

```julia
optimize!(model)
```

--------------------------------

### Solver Statistics

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/examples/PHTP_example2

Final summary of the optimization run, including objective value, constraint violations, and evaluation counts.

```text
Number of Iterations....: 33

                                   (scaled)                 (unscaled)
Objective...............:  -3.2290220275399406e-08   -3.2290220275399406e-08
Dual infeasibility......:   1.0289546992225951e-12    1.0289546992225951e-12
Constraint violation....:   1.4210854715202004e-14    1.4210854715202004e-14
Variable bound violation:   4.8166666662694828e-09    4.8166666662694828e-09
Complementarity.........:   2.5157965842376463e-09    2.5157965842376463e-09
Overall NLP error.......:   2.5157965842376463e-09    2.5157965842376463e-09


Number of objective function evaluations             = 40
Number of objective gradient evaluations             = 34
Number of equality constraint evaluations            = 40
Number of inequality constraint evaluations          = 40
Number of equality constraint Jacobian evaluations   = 34
Number of inequality constraint Jacobian evaluations = 34
Number of Lagrangian Hessian evaluations             = 33
Total seconds in IPOPT                               = 0.017

EXIT: Optimal Solution Found.
```

--------------------------------

### Set Primal Hints

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/examples/FOBP_example2

Provides primal bound hints for variables to assist the solver.

```julia
BilevelJuMP.set_primal_lower_bound_hint(x, -10)
BilevelJuMP.set_primal_upper_bound_hint(x, 6)
BilevelJuMP.set_primal_lower_bound_hint(y, -1)
BilevelJuMP.set_primal_upper_bound_hint(y, 9)
```

--------------------------------

### SOS1 Reformulation Mode

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/reference

Uses SOS1 constraints to convert complementarity constraints into mixed-integer constraints.

```julia
SOS1Mode()
```

--------------------------------

### Set Dual Bounds

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/examples/SOCBLP_example1

Provides dual bound hints for the lower level constraints to assist the solver.

```julia
BilevelJuMP.set_dual_upper_bound_hint(soc_lw, +[5.0, 5.0])
BilevelJuMP.set_dual_lower_bound_hint(soc_lw, -[5.0, 5.0])
```

```julia
for con in [con1, con3]
    BilevelJuMP.set_dual_lower_bound_hint(con, -15)
end
```

```julia
for con in [lb_y_1, lb_y_2, con2]
    BilevelJuMP.set_dual_upper_bound_hint(con, +15)
end
```

--------------------------------

### Solve MISOCP with Mixed Mode

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/tutorials/conic_lower

Configures a MISOCP solver with binary expansions and sets specific reformulation modes for conic and non-conic constraints.

```julia
using Xpress
using QuadraticToBinary
set_optimizer(model,
    ()->QuadraticToBinary.Optimizer{Float64}(Xpress.Optimizer(),lb=-10,ub=10))
BilevelJuMP.set_mode(model,
    BilevelJuMP.MixedMode(default = BilevelJuMP.IndicatorMode()))
BilevelJuMP.set_mode(con4, BilevelJuMP.ProductMode(1e-5))
optimize!(model)
```

--------------------------------

### Obtain Primal and Dual Solutions

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/tutorials/getting_started

Extracts the primal variable values and dual values for specific constraints.

```julia
julia> value(x)
1.8666666666666667

julia> value(y)
0.5333333333333332

julia> dual(l1)
1-element Vector{Float64}:
 0.0

julia> dual(l2)
1-element Vector{Float64}:
 49.75
```

--------------------------------

### IPOPT Final Statistics

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/examples/PHTP_example1

Summary of the final objective value, constraint violations, and solver performance metrics.

```text
Number of Iterations....: 33

                                   (scaled)                 (unscaled)
Objective...............:   1.6999999980907578e+01    1.6999999980907578e+01
Dual infeasibility......:   8.9958831495096092e-09    8.9958831495096092e-09
Constraint violation....:   2.2204460492503131e-16    2.2204460492503131e-16
Variable bound violation:   4.0833333485561435e-09    4.0833333485561435e-09
Complementarity.........:   4.8883842466698459e-09    4.8883842466698459e-09
Overall NLP error.......:   8.9958831495096092e-09    8.9958831495096092e-09


Number of objective function evaluations             = 38
Number of objective gradient evaluations             = 34
Number of equality constraint evaluations            = 38
Number of inequality constraint evaluations          = 38
Number of equality constraint Jacobian evaluations   = 34
Number of inequality constraint Jacobian evaluations = 34
Number of Lagrangian Hessian evaluations             = 33
Total seconds in IPOPT                               = 0.015

EXIT: Optimal Solution Found.
```

--------------------------------

### Create anonymous dual variables

Source: https://joaquimg.github.io/BilevelJuMP.jl/stable/tutorials/lower_duals

Iteratively create anonymous dual variables for constraints using a loop.

```julia
my_duals = []
for i in 1:3
    var = @variable(Upper(model), variable_type = DualOf(reserves[i]))
    push!(my_duals, var)
end
my_duals # a vector of anonimous variables
```