Semivectorial bilevel optimization problem: Penalty approach

We consider a bilevel optimization problem where the upper level is a scalar optimization problem and the lower level is a vector optimization problem. For the lower level, we deal with weakly efficient solutions. We approach our problem using a suitable penalty function which vanishes over the weakly efficient solutions of the lower-level vector optimization problem and which is nonnegative over its feasible set. Then, we use an exterior penalty method.