Projection-based Reformulation and Decomposition Algorithm for A Class of Mixed-Integer Bilevel Linear Programs

We propose an efficient algorithm for solving mixed integer bilevel linear programs (MIBLPs). The MIBLPs addressed in this work involve continuous and integer variables in both upper- and lower-level programs. In addition, the upper-level constraints are allowed to be dependent on the lower-level solutions. We first reformulate the original MIBLP into an equivalent single-level optimization problem. The issue of relatively complete response is tackled using the disjunctive programming approach. Based on this single-level reformulation, a decomposition algorithm is developed that converges to the global optimal solution in finite iterations. The master problem provides a valid lower bound, while two subproblems are used to provide a valid upper bound or to test the feasibility. A KKT-condition-based cut is generated according to the solutions to the subproblems and added to the master problem at the end of each iteration, so that non-decreasing lower bounds can be obtained successively. An implementation of the algorithm is described and illustrative examples are presented.