Solving an optimistic MILP-MILP bilevel problem with coin-or/MibS

Problem Definition

This example is taken from [10] and is a bilevel problem where the upper level is a mixed-integer linear program (MILP) and the lower level is a mixed-integer linear program (MILP).

The problem is formulated as follows:

\[\begin{split}\begin{align} \min_{x, y} \quad & -x + -10 y \\ \text{s.t.} \quad & x \in \mathbb Z_{\ge 0 },\\ & y\in \begin{array}[t]{rl} \displaystyle \underset{y}{\text{arg min}} \quad & y \\ \text{s.t.} \quad & -25 x + 20 y \leq 30, \\ & x + 2 y \leq 10, \\ & 2 x - y \leq 15, \\ & 2 x + 10 y \geq 15, \\ & y \geq 0, \\ & y \in \mathbb Z_{\ge 0}. \end{array} \end{align}\end{split}\]

Implementation

//
// Created by henri on 07.02.24.
//
#include <iostream>
#include "idol/modeling.h"
#include "idol/bilevel/optimizers/wrappers/MibS/MibS.h"
#include "idol/bilevel/modeling/Description.h"

using namespace idol;

int main(int t_argc, const char** t_argv) {

    /*
     This example is taken from "The Mixed Integer Linear Bilevel Programming Problem" (Moore and Bard, 1990).

        min -1 x + -10 y
        s.t.

        y in argmin { y :
            -25 x + 20 y <= 30,
            1 x + 2 y <= 10,
            2 x + -1 y <= 15,
            2 x + 10 y >= 15,
            y >= 0 and integer.
        }
        x >= 0 and integer.

     */

    Env env;

    // Define High Point Relaxation
    Model high_point_relaxation(env);

    auto x = high_point_relaxation.add_var(0, Inf, Integer, -1, "x");
    auto y = high_point_relaxation.add_var(0, Inf, Integer, -10, "y");

    auto follower_c1 = high_point_relaxation.add_ctr(-25 * x + 20 * y <= 30);
    auto follower_c2 = high_point_relaxation.add_ctr(x + 2 * y <= 10);
    auto follower_c3 = high_point_relaxation.add_ctr(2 * x - y <= 15);
    auto follower_c4 = high_point_relaxation.add_ctr(2 * x + 10 * y >= 15);

    // Prepare bilevel description
    Bilevel::Description description(env);
    description.set_lower_level_obj(y);
    description.make_lower_level(y);
    description.make_lower_level(follower_c1);
    description.make_lower_level(follower_c2);
    description.make_lower_level(follower_c3);
    description.make_lower_level(follower_c4);

    // Use coin-or/MibS as external solver
    high_point_relaxation.use(
                Bilevel::MibS(description)
                    .with_logs(true)
    );

    // Optimize and print solution
    high_point_relaxation.optimize();

    std::cout << save_primal(high_point_relaxation) << std::endl;

    return 0;
}