This paper considers a bilevel program, which has many applications in
practice. To develop effective numerical algorithms, it is generally necessary
to transform the bilevel program into a single-level optimization problem. The
most popular approach is to replace the lower-level program by its KKT
conditions and then the bilevel program can be reformulated as a mathematical
program with equilibrium constraints (MPEC for short). However, since the MPEC
does not satisfy the Mangasarian-Fromovitz constraint qualification at any
feasible point, the well-developed nonlinear programming theory cannot be
applied to MPECs directly. In this paper, we apply the Wolfe duality to show
that, under very mild conditions, the bilevel program is equivalent to a new
single-level reformulation (WDP for short) in the globally and locally optimal
sense. We give an example to show that, unlike the MPEC reformulation, WDP may
satisfy the Mangasarian-Fromovitz constraint qualification at its feasible
points. We give some properties of the WDP reformulation and the relations
between the WDP and MPEC reformulations. We further propose a relaxation method
for solving WDP and investigate its limiting behavior. Comprehensive numerical
experiments indicate that, although solving WDP directly does not perform very
well in our tests, the relaxation method based on the WDP reformulation is
quite efficient