  Bilevel optimization problems embed the optimality of a subproblem as a
constraint of another optimization problem. We introduce the concept of
near-optimality robustness for bilevel optimization, protecting the upper-level
solution feasibility from limited deviations from the optimal solution at the
lower level. General properties and necessary conditions for the existence of
solutions are derived for near-optimal robust versions of general bilevel
optimization problems. A duality-based solution method is defined when the
lower level is convex, leveraging the methodology from the robust and bilevel
literature. Numerical results assess the efficiency of exact and heuristic
methods and the impact of valid inequalities on the solution time.
