Modes overview

There are several ways to solve bilevel problems with BilevelJuMP. The main difference between them is the way the complementarity constraints are reformulated. The reformulation method is set with the mode option of the BilevelModel constructor.

The modes available are:

BilevelJuMP.SOS1Mode(): uses SOS1 constraints to model complementarity constraints and solve the problem with MIP solvers (Cbc, Xpress, Gurobi, CPLEX, SCIP).
BilevelJuMP.IndicatorMode(): uses Indicator constraints to model complementarity constraints and solve the problem with MIP solvers (Cbc, Xpress, Gurobi, CPLEX, SCIP).
BilevelJuMP.BigMMode(): uses the Fortuny-Amat and McCarl reformulation 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 good bounds are provided, 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, respectively, 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.
BilevelJuMP.ProductMode(): 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.
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(constraint, mode), where constraint is a constraint of the bilevel problem and mode is one of the modes described above. If no mode is set for a constraint, the default mode is used instead. To set a mode to reformulate variables bounds use: BilevelJuMP.set_mode(variable, mode).
BilevelJuMP.StrongDualityMode: this mode is not a complementarity reformulation method, instead, all complementarity constraints are replaced by constraints enforcing the strong duality. This mode is especially amenable for NLP solvers (Ipopt, KNITRO). MIP solvers can also be used but they will have to deal with the products of variables crated or the will have to be used in conjunction with QuadraticToBinary.jl.

The following example shows how to solve a classic bilevel problem.

We start loading all libraries needed for this example.

using BilevelJuMP, HiGHS, Ipopt, SCIP

Now we create a bilevel model with the BilevelModel constructor with no solver nor mode specified.

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)

(c1 : x + y ≤ 8, c2 : 4 x + y ≥ 8, c3 : 2 x + y ≤ 13, c4 : 2 x - 7 y ≤ 0)

`BigMMode` and `HiGHS.Optimizer`

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

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)

`SOS1Mode` and `SCIP.Optimizer`

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

presolving:
(round 1, fast)       0 del vars, 3 del conss, 0 add conss, 3 chg bounds, 0 chg sides, 0 chg coeffs, 0 upgd conss, 0 impls, 0 clqs
(round 2, fast)       0 del vars, 3 del conss, 0 add conss, 8 chg bounds, 0 chg sides, 0 chg coeffs, 0 upgd conss, 0 impls, 0 clqs
(round 3, exhaustive) 0 del vars, 6 del conss, 0 add conss, 8 chg bounds, 3 chg sides, 0 chg coeffs, 0 upgd conss, 0 impls, 0 clqs
   (0.0s) symmetry computation skipped: there exist constraints that cannot be handled by symmetry methods.
presolving (4 rounds: 4 fast, 2 medium, 2 exhaustive):
 0 deleted vars, 6 deleted constraints, 0 added constraints, 8 tightened bounds, 0 added holes, 3 changed sides, 0 changed coefficients
 0 implications, 0 cliques
presolved problem has 10 variables (0 bin, 0 int, 0 impl, 10 cont) and 10 constraints
      4 constraints of type <SOS1>
      6 constraints of type <linear>
Presolving Time: 0.00

 time | node  | left  |LP iter|LP it/n|mem/heur|mdpt |vars |cons |rows |cuts |sepa|confs|strbr|  dualbound   | primalbound  |  gap   | compl.
  0.0s|     1 |     0 |     5 |     - |   639k |   0 |  10 |  10 |   6 |   0 |  0 |   0 |   0 | 6.133333e+00 |      --      |    Inf | unknown
  0.0s|     1 |     0 |     6 |     - |   639k |   0 |  10 |  10 |   7 |   1 |  2 |   0 |   0 | 6.133333e+00 |      --      |    Inf | unknown
  0.0s|     1 |     0 |     7 |     - |   639k |   0 |  10 |  10 |   8 |   2 |  4 |   0 |   0 | 6.133333e+00 |      --      |    Inf | unknown
* 0.0s|     1 |     0 |     7 |     - |    LP  |   0 |  10 |  10 |   8 |   2 |  6 |   0 |   0 | 6.133333e+00 | 6.133333e+00 |   0.00%| unknown

SCIP Status        : problem is solved [optimal solution found]
Solving Time (sec) : 0.00
Solving Nodes      : 1
Primal Bound       : +6.13333333333333e+00 (1 solutions)
Dual Bound         : +6.13333333333333e+00
Gap                : 0.00 %

Warning

SCIP requires a non-standard installation procedure in windows. See SCIP.jl for more details.

`IndicatorMode` and `SCIP.Optimizer`

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

presolving:
(round 1, fast)       8 del vars, 11 del conss, 0 add conss, 11 chg bounds, 0 chg sides, 0 chg coeffs, 0 upgd conss, 0 impls, 0 clqs
(round 2, fast)       8 del vars, 11 del conss, 0 add conss, 14 chg bounds, 0 chg sides, 0 chg coeffs, 0 upgd conss, 0 impls, 0 clqs
(round 3, fast)       8 del vars, 11 del conss, 0 add conss, 16 chg bounds, 0 chg sides, 0 chg coeffs, 0 upgd conss, 0 impls, 0 clqs
   (0.0s) probing cycle finished: starting next cycle
(round 4, exhaustive) 8 del vars, 11 del conss, 0 add conss, 17 chg bounds, 0 chg sides, 0 chg coeffs, 0 upgd conss, 46 impls, 3 clqs
   (0.0s) probing cycle finished: starting next cycle
   (0.0s) symmetry computation skipped: there exist constraints that cannot be handled by symmetry methods.
presolving (5 rounds: 5 fast, 2 medium, 2 exhaustive):
 8 deleted vars, 11 deleted constraints, 0 added constraints, 17 tightened bounds, 0 added holes, 0 changed sides, 0 changed coefficients
 46 implications, 3 cliques
presolved problem has 14 variables (4 bin, 0 int, 0 impl, 10 cont) and 17 constraints
      8 constraints of type <SOS1>
      9 constraints of type <linear>
Presolving Time: 0.00

 time | node  | left  |LP iter|LP it/n|mem/heur|mdpt |vars |cons |rows |cuts |sepa|confs|strbr|  dualbound   | primalbound  |  gap   | compl.
p 0.0s|     1 |     0 |     5 |     - |  clique|   0 |  14 |  17 |   9 |   0 |  0 |   0 |   0 | 0.000000e+00 | 1.800000e+01 |    Inf | unknown
p 0.0s|     1 |     0 |    11 |     - |   locks|   0 |  14 |  17 |   9 |   0 |  0 |   0 |   0 | 0.000000e+00 | 6.133333e+00 |    Inf | unknown
  0.0s|     1 |     0 |    17 |     - |   754k |   0 |  14 |  15 |   9 |   0 |  0 |   0 |   0 | 6.133333e+00 | 6.133333e+00 |   0.00%| unknown

SCIP Status        : problem is solved [optimal solution found]
Solving Time (sec) : 0.00
Solving Nodes      : 1
Primal Bound       : +6.13333333333333e+00 (2 solutions)
Dual Bound         : +6.13333333333333e+00
Gap                : 0.00 %

Warning

SCIP requires a non-standard installation procedure in windows. See SCIP.jl for more details.

`ProductMode` and `Ipopt.Optimizer`

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

This is Ipopt version 3.14.4, running with linear solver MUMPS 5.4.1.

Number of nonzeros in equality constraint Jacobian...:        4
Number of nonzeros in inequality constraint Jacobian.:       30
Number of nonzeros in Lagrangian Hessian.............:        8

Total number of variables............................:        6
                     variables with only lower bounds:        1
                variables with lower and upper bounds:        0
                     variables with only upper bounds:        3
Total number of equality constraints.................:        1
Total number of inequality constraints...............:       11
        inequality constraints with only lower bounds:        2
   inequality constraints with lower and upper bounds:        0
        inequality constraints with only upper bounds:        9

iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
   0  0.0000000e+00 8.00e+00 1.25e+00  -1.0 0.00e+00    -  0.00e+00 0.00e+00   0
   1  1.1715810e+00 6.47e+00 7.43e+00  -1.0 4.20e+00    -  2.30e-02 1.94e-01h  1
   2  3.3572931e+00 3.62e+00 7.92e+00  -1.0 3.51e+00    -  1.78e-01 4.34e-01h  1
   3  4.0212202e+00 2.75e+00 7.49e+00  -1.0 1.98e+00    -  3.33e-01 2.33e-01h  1
   4  5.0194310e+00 1.45e+00 2.08e+01  -1.0 1.53e+00    -  2.24e-01 4.53e-01h  1
   5  5.4214626e+00 9.28e-01 1.48e+01  -1.0 8.61e-01    -  2.39e-01 3.24e-01h  1
   6  5.5118789e+00 8.11e-01 1.01e+02  -1.0 4.61e+00    -  8.91e-02 7.14e-02H  1
   7  5.8180392e+00 4.11e-01 1.05e+03  -1.0 1.38e+01    -  5.31e-02 1.21e-01f  4
   8  5.8109187e+00 4.20e-01 1.01e+03  -1.0 1.08e+00   0.0 1.00e+00 4.07e-02h  1
   9  6.1545183e+00 5.33e-02 1.99e+03  -1.0 1.31e+00  -0.5 1.00e+00 9.94e-01h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  10  6.1580596e+00 6.24e-02 5.04e+03  -1.0 3.56e-01  -1.0 1.00e+00 1.25e-01h  1
  11  6.1546734e+00 4.75e-02 4.15e+03  -1.0 7.14e-01    -  1.00e+00 6.44e-01h  1
  12  6.1446716e+00 2.01e-02 9.53e+03  -1.0 1.21e+00    -  1.00e+00 5.79e-01h  1
  13  6.1404036e+00 9.88e-03 2.21e+04  -1.0 9.75e-01    -  1.00e+00 5.87e-01h  1
  14  6.1376399e+00 4.50e-03 5.26e+04  -1.0 9.21e-01    -  1.00e+00 5.88e-01h  1
  15  6.1361150e+00 2.11e-03 1.25e+05  -1.0 7.41e-01    -  1.00e+00 5.91e-01h  1
  16  6.1351608e+00 9.63e-04 2.89e+05  -1.0 5.95e-01    -  1.00e+00 5.98e-01h  1
  17  6.1345596e+00 4.31e-04 6.32e+05  -1.0 4.38e-01    -  1.00e+00 6.15e-01h  1
  18  6.1344562e+00 3.57e-04 2.89e+06  -1.0 3.08e-01    -  1.00e+00 1.64e-01f  3
  19  6.1339299e+00 6.40e-05 5.29e+05  -1.0 3.58e-01    -  1.00e+00 9.04e-01h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  20  6.1339131e+00 5.84e-05 5.21e+06  -1.0 2.91e-02    -  1.00e+00 3.48e-01f  2
  21  6.1336199e+00 1.02e-05 3.75e+05  -1.0 1.00e-01    -  1.00e+00 1.00e+00h  1
  22  6.1335417e+00 4.72e-06 1.16e+06  -1.0 2.95e-02    -  1.00e+00 6.66e-01h  1
  23  6.1334913e+00 2.46e-06 3.69e+06  -1.0 2.73e-02    -  1.00e+00 3.97e-01f  2
  24  6.1334055e+00 4.88e-07 5.89e+05  -1.0 1.34e-02    -  1.00e+00 1.00e+00h  1
  25  6.1333506e+00 1.59e-08 8.75e+04  -1.0 8.26e-03    -  1.00e+00 1.00e+00h  1
  26  6.1333604e+00 5.74e-08 3.18e+05  -1.0 4.19e-03    -  1.00e+00 3.61e-01f  2
  27  6.1333559e+00 3.78e-08 2.30e+05  -1.0 6.97e-04    -  1.00e+00 8.11e-01h  1
  28  6.1333447e+00 3.25e-09 1.96e+06  -1.0 2.95e-03    -  1.00e+00 4.96e-01f  2
  29  6.1333430e+00 1.11e-16 2.44e+03  -1.0 2.20e-04    -  1.00e+00 1.00e+00h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  30  6.1333427e+00 0.00e+00 3.07e+03  -2.5 3.24e-05    -  1.00e+00 1.00e+00f  1
  31  6.1333393e+00 1.11e-16 2.13e+01  -2.5 3.73e-05   4.0 1.00e+00 1.00e+00f  1
  32  6.1333392e+00 0.00e+00 3.41e+00  -3.8 1.25e-06   3.5 1.00e+00 1.00e+00h  1
  33  6.1333388e+00 1.82e-13 3.70e-03  -3.8 3.43e-06   3.0 1.00e+00 1.00e+00f  1
  34  6.1333354e+00 0.00e+00 8.38e+03  -5.7 7.43e-04    -  6.86e-01 1.00e+00h  1
  35  6.1333342e+00 1.11e-16 5.38e+03  -5.7 3.43e-03    -  6.74e-01 1.00e+00h  1
  36  6.1333345e+00 0.00e+00 1.46e+03  -5.7 9.77e-07   2.6 7.78e-01 1.00e+00h  1
  37  6.1333339e+00 0.00e+00 1.41e+03  -5.7 4.86e-05   2.1 1.00e+00 7.74e-02f  2
  38  6.1333339e+00 0.00e+00 1.49e+03  -5.7 2.74e-03    -  1.00e+00 3.81e-06h 19
  39  6.1333341e+00 0.00e+00 2.25e-02  -5.7 7.27e-06   1.6 1.00e+00 1.00e+00h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  40  6.1333340e+00 2.37e-08 8.97e+01  -5.7 3.45e-03    -  7.32e-01 1.00e+00f  1
  41  6.1333339e+00 1.77e-09 1.48e+02  -5.7 9.82e-03    -  1.00e+00 2.34e-01h  2
  42  6.1333338e+00 0.00e+00 6.66e-03  -5.7 4.46e-06   1.1 1.00e+00 1.00e+00h  1
  43  6.1333335e+00 0.00e+00 3.61e+00  -5.7 8.11e-03    -  1.00e+00 1.00e+00f  1
  44  6.1333336e+00 2.22e-16 8.36e+00  -5.7 1.83e+01    -  1.32e-03 6.70e-05h  5
  45  6.1333339e+00 0.00e+00 1.21e-03  -5.7 5.06e-06   0.6 1.00e+00 1.00e+00h  1
  46  6.1333345e+00 3.18e-08 1.18e+02  -5.7 1.52e-02    -  6.91e-01 1.00e+00H  1
  47  6.1333348e+00 2.60e-08 2.55e+02  -5.7 1.86e-04   0.2 1.00e+00 1.25e-01h  4
  48  6.1333347e+00 1.57e-08 1.38e+02  -5.7 1.69e-03    -  1.00e+00 2.50e-01h  3
  49  6.1333345e+00 8.02e-09 1.61e+02  -5.7 4.53e-03    -  1.00e+00 2.50e-01h  3
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  50  6.1333344e+00 2.28e-09 1.19e+02  -5.7 4.18e-03    -  1.00e+00 2.50e-01h  3
  51  6.1333341e+00 2.22e-16 6.69e+01  -5.7 5.38e-03    -  1.00e+00 5.00e-01h  2
  52  6.1333334e+00 4.44e-16 6.91e-01  -5.7 1.10e-02    -  1.00e+00 1.00e+00h  1
  53  6.1333337e+00 0.00e+00 1.71e-02  -5.7 5.65e-05  -0.3 1.00e+00 1.00e+00h  1
  54  6.1333334e+00 0.00e+00 4.49e+00  -5.7 1.71e-02    -  9.61e-01 1.00e+00H  1
  55  6.1333335e+00 0.00e+00 1.96e-02  -5.7 1.10e-04  -0.8 1.00e+00 1.00e+00h  1
  56  6.1333333e+00 0.00e+00 7.61e+01  -5.7 7.61e-02    -  6.30e-01 1.00e+00H  1
  57  6.1333334e+00 1.11e-16 9.52e-02  -5.7 2.78e-04  -1.3 1.00e+00 1.00e+00h  1
  58  6.1333333e+00 0.00e+00 1.93e+01  -5.7 2.38e+00    -  4.20e-02 2.97e-02f  2
  59  6.1333333e+00 1.11e-16 9.73e-03  -5.7 5.06e-04  -1.7 1.00e+00 1.00e+00h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  60  6.1333334e+00 2.22e-16 7.05e+01  -5.7 4.25e-01    -  8.23e-01 1.00e+00F  1
  61  6.1333334e+00 6.42e-09 7.18e+01  -5.7 1.03e+00    -  4.63e-01 1.36e-01h  2
  62  6.1333333e+00 0.00e+00 7.16e+01  -5.7 2.26e-01    -  1.00e+00 3.74e-01h  2
  63  6.1333333e+00 0.00e+00 2.05e+02  -5.7 6.89e-01    -  7.09e-01 1.00e+00h  1
  64  6.1333333e+00 8.88e-16 1.68e+03  -5.7 2.10e+00    -  4.86e-01 1.00e+00h  1
  65  6.1333333e+00 0.00e+00 4.68e-02  -5.7 1.12e-04  -2.2 1.00e+00 1.00e+00h  1
  66  6.1333333e+00 0.00e+00 1.38e+03  -5.7 3.45e+00    -  9.13e-01 1.00e+00f  1
  67  6.1333333e+00 5.63e-08 1.69e+04  -5.7 3.21e+01    -  3.28e-01 1.00e+00h  1
  68  6.1333333e+00 0.00e+00 5.53e-02  -5.7 1.25e-04  -2.7 1.00e+00 1.00e+00h  1
  69  6.1333333e+00 0.00e+00 2.78e+03  -5.7 4.04e+01    -  9.63e-01 1.00e+00f  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  70  6.1333333e+00 0.00e+00 2.23e+04  -5.7 8.16e+02    -  1.25e-01 1.00e+00h  1
  71  6.1333333e+00 0.00e+00 4.77e-06  -5.7 1.75e-06  -3.2 1.00e+00 1.00e+00h  1
  72  6.1333333e+00 0.00e+00 3.36e+00  -8.6 1.19e+00    -  9.93e-01 1.00e+00h  1
  73  6.1333333e+00 0.00e+00 2.97e-01  -8.6 6.06e+01    -  1.00e+00 1.00e+00h  1
  74  6.1333333e+00 0.00e+00 2.52e+00  -8.6 6.30e+02    -  8.99e-01 1.00e+00h  1
  75  6.1333333e+00 0.00e+00 1.15e+01  -8.6 5.88e+03    -  4.30e-01 1.00e+00h  1
  76  6.1333333e+00 0.00e+00 2.74e-09  -8.6 1.21e-05  -3.6 1.00e+00 1.00e+00h  1

Number of Iterations....: 76

                                   (scaled)                 (unscaled)
Objective...............:   6.1333332741882947e+00    6.1333332741882947e+00
Dual infeasibility......:   2.7371810682683953e-09    2.7371810682683953e-09
Constraint violation....:   0.0000000000000000e+00    0.0000000000000000e+00
Variable bound violation:   4.4274809187413012e-09    4.4274809187413012e-09
Complementarity.........:   3.3792751059460505e-09    3.3792751059460505e-09
Overall NLP error.......:   3.3792751059460505e-09    3.3792751059460505e-09


Number of objective function evaluations             = 141
Number of objective gradient evaluations             = 77
Number of equality constraint evaluations            = 141
Number of inequality constraint evaluations          = 141
Number of equality constraint Jacobian evaluations   = 77
Number of inequality constraint Jacobian evaluations = 77
Number of Lagrangian Hessian evaluations             = 76
Total seconds in IPOPT                               = 0.044

EXIT: Optimal Solution Found.

`StrongDualityMode` and `Ipopt.Optimizer`

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

This is Ipopt version 3.14.4, running with linear solver MUMPS 5.4.1.

Number of nonzeros in equality constraint Jacobian...:        4
Number of nonzeros in inequality constraint Jacobian.:       35
Number of nonzeros in Lagrangian Hessian.............:        8

Total number of variables............................:        6
                     variables with only lower bounds:        1
                variables with lower and upper bounds:        0
                     variables with only upper bounds:        3
Total number of equality constraints.................:        1
Total number of inequality constraints...............:        9
        inequality constraints with only lower bounds:        3
   inequality constraints with lower and upper bounds:        0
        inequality constraints with only upper bounds:        6

iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
   0  0.0000000e+00 8.00e+00 1.37e+00  -1.0 0.00e+00    -  0.00e+00 0.00e+00   0
   1  6.9373007e-01 7.10e+00 4.04e+00  -1.0 4.21e+00    -  2.29e-02 1.15e-01h  1
   2  4.1458199e+00 2.60e+00 3.57e+00  -1.0 3.84e+00    -  1.18e-01 6.25e-01h  1
   3  4.8949366e+00 1.62e+00 3.58e+00  -1.0 1.45e+00    -  9.38e-01 3.62e-01h  1
   4  5.7107572e+00 6.46e-01 1.78e+00  -1.0 1.27e+00    -  3.22e-01 6.17e-01h  1
   5  5.9360287e+00 3.84e-01 4.61e+00  -1.0 8.08e-01    -  7.52e-01 4.57e-01h  1
   6  6.0003610e+00 1.73e-01 7.61e-01  -1.0 5.81e+00    -  2.74e-01 3.48e-01h  1
   7  6.1244252e+00 1.29e-01 6.99e-01  -1.7 1.39e+00    -  8.70e-01 9.43e-01h  1
   8  6.1300137e+00 8.68e-02 1.19e+02  -1.7 4.08e-01  -2.0 1.00e+00 3.26e-01h  1
   9  6.1336984e+00 2.63e-02 1.44e+02  -1.7 2.77e+00    -  1.00e+00 6.20e-01h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  10  6.1317629e+00 9.14e-03 2.51e+02  -1.7 1.45e+01    -  6.09e-01 3.74e-01h  1
  11  6.1326207e+00 8.08e-03 3.75e+02  -1.7 7.78e+01    -  5.24e-01 3.34e-01f  1
  12  6.1330471e+00 4.49e-03 5.42e+02  -1.7 5.06e+01    -  5.84e-01 4.01e-01H  1
  13  6.1330042e+00 4.19e-03 2.38e+02  -1.7 2.41e+02    -  3.12e-01 4.80e-01F  1
  14  6.1332031e+00 4.03e-03 2.43e+02  -1.7 3.17e+02    -  3.74e-01 3.74e-01H  1
  15  6.1332221e+00 6.18e-04 7.32e+02  -1.7 5.37e+02    -  5.47e-01 4.62e-01H  1
  16  6.1332962e+00 6.48e-04 5.28e+02  -1.7 9.70e+02    -  4.48e-01 4.99e-01H  1
  17  6.1333059e+00 6.51e-03 1.37e+04  -1.7 1.92e+03    -  7.43e-01 3.38e-01f  2
  18  6.1333095e+00 6.49e-03 3.75e+04  -1.7 3.05e+03    -  6.23e-01 1.25e-01h  4
  19  6.1333111e+00 6.32e-03 7.22e+04  -1.7 4.20e+03    -  7.09e-01 6.25e-02h  5
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  20  6.1333127e+00 6.18e-03 1.06e+05  -1.7 4.53e+03    -  6.98e-01 6.25e-02h  5
  21  6.1333141e+00 6.05e-03 1.40e+05  -1.7 4.98e+03    -  7.11e-01 6.25e-02h  5
  22  6.1333155e+00 5.93e-03 1.73e+05  -1.7 5.43e+03    -  7.12e-01 6.25e-02h  5
  23  6.1333167e+00 5.81e-03 2.06e+05  -1.7 5.95e+03    -  7.18e-01 6.25e-02h  5
  24  6.1333179e+00 5.70e-03 2.39e+05  -1.7 6.52e+03    -  7.21e-01 6.25e-02h  5
  25  6.1333190e+00 5.60e-03 2.71e+05  -1.7 7.14e+03    -  7.25e-01 6.25e-02h  5
  26  6.1333200e+00 5.50e-03 3.03e+05  -1.7 7.81e+03    -  7.28e-01 6.25e-02h  5
  27  6.1333348e+00 6.40e-02 2.04e+04  -1.7 8.54e+03    -  7.31e-01 1.00e+00w  1
  28  6.1333342e+00 1.11e-02 1.79e+05  -1.7 4.06e+03    -  1.00e+00 8.78e-01w  1
  29  6.1333333e+00 8.76e-03 1.17e+05  -1.7 6.43e+03    -  1.00e+00 5.33e-01w  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  30  6.1333204e+00 5.39e-03 3.47e+05  -1.7 3.49e+03    -  7.31e-01 3.12e-02h  5
  31  6.1333209e+00 5.28e-03 3.91e+05  -1.7 9.06e+03    -  7.54e-01 3.12e-02h  6
  32  6.1333211e+00 5.22e-03 4.39e+05  -1.7 9.36e+03    -  7.46e-01 1.56e-02h  7
  33  6.1333213e+00 5.15e-03 4.87e+05  -1.7 9.68e+03    -  7.64e-01 1.56e-02h  7
  34  6.1333214e+00 5.13e-03 5.38e+05  -1.7 9.81e+03    -  7.57e-01 3.91e-03h  9
  35  6.1333222e+00 5.08e-03 5.53e+05  -1.7 9.95e+03    -  7.71e-01 6.25e-02h  5
  36  6.1333230e+00 5.01e-03 5.65e+05  -1.7 1.05e+04    -  7.30e-01 6.25e-02h  5
  37  6.1333237e+00 4.95e-03 5.79e+05  -1.7 1.17e+04    -  7.55e-01 6.25e-02h  5
  38  6.1333244e+00 4.88e-03 5.92e+05  -1.7 1.27e+04    -  7.45e-01 6.25e-02h  5
  39  6.1333250e+00 4.81e-03 6.05e+05  -1.7 1.39e+04    -  7.55e-01 6.25e-02h  5
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  40  6.1333340e+00 6.12e-02 1.79e+04  -1.7 1.51e+04    -  7.55e-01 1.00e+00w  1
  41  6.1333339e+00 1.21e-02 1.51e+05  -1.7 1.01e+04    -  1.00e+00 8.38e-01w  1
  42  6.1333333e+00 1.18e-02 4.68e+05  -1.7 1.61e+04    -  1.00e+00 7.36e-01w  1
  43  6.1333335e+00 2.87e-03 1.85e+05  -1.7 6.35e+03    -  1.00e+00 8.95e-01h  1
  44  6.1333334e+00 3.03e-07 3.16e-02  -1.7 1.83e-03  -2.5 1.00e+00 1.00e+00f  1
  45  6.1333333e+00 4.87e-07 3.54e+02  -8.6 1.85e+00    -  1.00e+00 1.00e+00h  1
  46  6.1333333e+00 0.00e+00 1.62e+03  -8.6 2.25e+01    -  9.99e-01 1.00e+00h  1
  47  6.1333333e+00 0.00e+00 3.32e-09  -8.6 9.69e-08  -3.0 1.00e+00 1.00e+00h  1
  48  6.1333333e+00 3.32e-05 8.33e-01  -9.0 3.63e+03    -  7.14e-01 1.00e+00h  1
  49  6.1333333e+00 3.64e-12 7.72e-09  -9.0 2.08e-05  -3.4 1.00e+00 1.00e+00h  1

Number of Iterations....: 49

                                   (scaled)                 (unscaled)
Objective...............:   6.1333332728272785e+00    6.1333332728272785e+00
Dual infeasibility......:   7.7153502585296349e-09    7.7153502585296349e-09
Constraint violation....:   3.6379788070917130e-12    3.6379788070917130e-12
Variable bound violation:   0.0000000000000000e+00    0.0000000000000000e+00
Complementarity.........:   1.4317498150955074e-09    1.4317498150955074e-09
Overall NLP error.......:   7.7153502585296349e-09    7.7153502585296349e-09


Number of objective function evaluations             = 179
Number of objective gradient evaluations             = 50
Number of equality constraint evaluations            = 179
Number of inequality constraint evaluations          = 179
Number of equality constraint Jacobian evaluations   = 50
Number of inequality constraint Jacobian evaluations = 50
Number of Lagrangian Hessian evaluations             = 49
Total seconds in IPOPT                               = 0.030

EXIT: Optimal Solution Found.

`MixedMode` and `SCIP.Optimizer`

set_optimizer(model, SCIP.Optimizer)

BilevelJuMP.set_mode(model, BilevelJuMP.MixedMode(default = BilevelJuMP.SOS1Mode()))

BilevelJuMP.set_mode(c1, BilevelJuMP.IndicatorMode())

BilevelJuMP.set_mode(c3, BilevelJuMP.SOS1Mode())

optimize!(model)

objective_value(model)

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

presolving:
(round 1, fast)       2 del vars, 5 del conss, 0 add conss, 5 chg bounds, 0 chg sides, 0 chg coeffs, 0 upgd conss, 0 impls, 0 clqs
(round 2, fast)       2 del vars, 5 del conss, 0 add conss, 10 chg bounds, 0 chg sides, 0 chg coeffs, 0 upgd conss, 0 impls, 0 clqs
(round 3, exhaustive) 2 del vars, 7 del conss, 0 add conss, 10 chg bounds, 2 chg sides, 0 chg coeffs, 0 upgd conss, 0 impls, 0 clqs
   (0.0s) probing cycle finished: starting next cycle
   (0.0s) symmetry computation skipped: there exist constraints that cannot be handled by symmetry methods.
presolving (4 rounds: 4 fast, 2 medium, 2 exhaustive):
 2 deleted vars, 7 deleted constraints, 0 added constraints, 10 tightened bounds, 0 added holes, 2 changed sides, 0 changed coefficients
 7 implications, 0 cliques
presolved problem has 11 variables (1 bin, 0 int, 0 impl, 10 cont) and 12 constraints
      5 constraints of type <SOS1>
      7 constraints of type <linear>
Presolving Time: 0.00

 time | node  | left  |LP iter|LP it/n|mem/heur|mdpt |vars |cons |rows |cuts |sepa|confs|strbr|  dualbound   | primalbound  |  gap   | compl.
  0.0s|     1 |     0 |    10 |     - |   641k |   0 |  11 |  12 |   7 |   0 |  0 |   0 |   0 | 6.133333e+00 |      --      |    Inf | unknown
  0.0s|     1 |     0 |    11 |     - |   645k |   0 |  11 |  12 |   8 |   1 |  1 |   0 |   0 | 6.133333e+00 |      --      |    Inf | unknown
  0.0s|     1 |     0 |    12 |     - |   655k |   0 |  11 |  12 |   9 |   2 |  2 |   0 |   0 | 6.133333e+00 |      --      |    Inf | unknown
  0.0s|     1 |     0 |    13 |     - |   655k |   0 |  11 |  12 |  10 |   3 |  4 |   0 |   0 | 6.133333e+00 |      --      |    Inf | unknown
* 0.0s|     1 |     0 |    13 |     - |    LP  |   0 |  11 |  12 |  10 |   3 |  6 |   0 |   0 | 6.133333e+00 | 6.133333e+00 |   0.00%| unknown

SCIP Status        : problem is solved [optimal solution found]
Solving Time (sec) : 0.00
Solving Nodes      : 1
Primal Bound       : +6.13333333333333e+00 (1 solutions)
Dual Bound         : +6.13333333333333e+00
Gap                : 0.00 %

Warning

SCIP requires a non-standard installation procedure in windows. See SCIP.jl for more details.

Info

View this file on Github.

This page was generated using Literate.jl.

Modes overview

BigMMode and HiGHS.Optimizer

SOS1Mode and SCIP.Optimizer

IndicatorMode and SCIP.Optimizer

ProductMode and Ipopt.Optimizer

StrongDualityMode and Ipopt.Optimizer

MixedMode and SCIP.Optimizer

`BigMMode` and `HiGHS.Optimizer`

`SOS1Mode` and `SCIP.Optimizer`

`IndicatorMode` and `SCIP.Optimizer`

`ProductMode` and `Ipopt.Optimizer`

`StrongDualityMode` and `Ipopt.Optimizer`

`MixedMode` and `SCIP.Optimizer`