Issues in the Development of Global Optimization Algorithms for Bilevel Programs with a Nonconvex Inner Program

The co-operative formulation of a nonlinear bilevel program involving nonconvex functions is considered and two equivalent reformulations to simpler programs are presented. It is shown that previous literature proposals for the global solution of such programs are not generally valid for nonconvex inner programs and several consequences of nonconvexity in the inner program are identified. In particular, issues with the computation of lower and upper bounds as well as with the choice of branching variables in a branch-andbound framework are analyzed. This analysis lays the foundation for the development of rigorous algorithms and some algorithmic expectations are established. Key-words Bilevel program; nonconvex; global optimization; GSIP; branch-and-bound AMS Classification http://www.ams.org/msc/ 65K05=mathematical programming algorithms 90C26=nonconvex programming 90c33=complementarity programming 90C31=sensitivity,stability,parametric optimization