In this paper, the aim is to compute Pareto efficient solutions of multi-objective optimization problems involving forbidden regions. More precisely, we assume that the vector-valued objective function is componentwise generalized-convex and acts between a real topological linear pre-image space and a finite-dimensional image space, while the feasible set is given by the whole pre-image space excepting some forbidden regions that are defined by convex sets. This leads us to a nonconvex multi-objective optimization problem. Using the recently proposed penalization approach by Günther and Tammer (2017), we show that the solution set of the original problem can be generated by solving a finite family of unconstrained multi-objective optimization problems. We apply our results to a special multi-objective location problem (known as point-objective location problem) where the aim is to locate a new facility in a continuous location space (a finite-dimensional Hilbert space) in the presence of a finite number of demand points. For the choice of the new location point, we are taking into consideration some forbidden regions that are given by open balls (defined with respect to the underlying norm). For such a nonconvex location problem, under the assumption that the forbidden regions are pairwise disjoint, we give complete geometrical descriptions for the sets of (strictly, weakly) Pareto efficient solutions by using the approach by Günther and Tammer (2017) and results derived by Jourani, Michelot and Ndiaye (2009).
C. Günther, Pareto efficient solutions in multi-objective optimization involving forbidden regions, Revista de Investigacion Operacional, Volume 39, Number 3, Pages 353-390, 2018 (see http://rev-inv-ope.univ-paris1.fr/volumes-since-2000/volume-39-2018/)