  In this paper, we exploit the so-called value function reformulation of the
bilevel optimization problem to develop duality results for the problem. Our
approach builds on Fenchel-Lagrange-type duality to establish suitable results
for the bilevel optimization problem. First, we overview some standard duality
results to show that they are not applicable to our problem. Secondly, via the
concept of partial calmness, we establish weak and strong duality results. In
particular, Lagrange, Fenchel-Lagrange, and Toland-Fenchel- Lagrange duality
concepts are investigated for this type of problems under some suitable
conditions. Thirdly, based on the use of some regularization of our bilevel
program, we establish sufficient conditions ensuring strong duality results
under a generalized Slater-type condition without convexity assumptions and
without the partial calmness condition. Finally, without the Slater condition,
a strong duality result is constructed for the bilevel optimization problem
with geometric constraint.
