Global Optimization of Integer and Mixed-Integer Bi-Level Programming Problems via Multi-Parametric Programming

Bi-level programming problems (BLPPs) arise very often in areas of engineering, transportation control. A key feature of such problems from a mathematical viewpoint is that even for the simplest linear case, a global optimization approach is typically necessary. In this work, we present two multi-parametric programming based algorithms for the solution of integer and mixed-integer bi-level programming problems. The first algorithm addresses the mixed-integer case of the BLPP and employs a reformulation linearization technique (Sherali and Adams, 1990, 1994; Adams and Sherali, 2005) and continuous multi-parametric programming for the solution of the inner problem. The second algorithm addresses the integer case of the BLPP and approaches the inner problem using global multi-parametric mixed-integer programming (Dua et al. 2004). In both algorithms the solution of the inner problem is embedded in the outer problem to form a set of single-level optimization problems that can be solved to global optimality using a global optimization software.