2. Symmetric Optimization Problems

In this chapter we study several classes of symmetric optimization problems which are identified with the corresponding spaces of objective functions, equipped with appropriate complete metrics. Using the Baire category approach, for any of these classes, we show the existence of subset of the space of functions, which is a countable intersection of open and everywhere dense sets, such that for every objective function from this intersection the corresponding symmetric optimization problem possesses a solution. These results are obtained as realizations of a general variational principle which is established in this chapter. We extend these results for certain classes of symmetric optimization problems using a porosity notion. We identify a class of symmetric minimization problems with a certain complete metric space of functions, study the set of all functions for which the corresponding minimization problem has a solution, and show that the complement of this set is not only of the first category but also a σ-porous set.