  In this paper we develop new applications of variational analysis and
generalized differentiation to the following optimization problem and its
specifications: given n closed subsets of a Banach space, find such a point for
which the sum of its distances to these sets is minimal. This problem can be
viewed as an extension of the celebrated Fermat-Torricelli problem: given three
points on the plane, find another point such that the sum of its distances to
the designated points is minimal. The generalized Fermat-Torricelli problem
formulated and studied in this paper is of undoubted mathematical interest and
is promising for various applications including those frequently arising in
location science, optimal networks, etc. Based on advanced tools and recent
results of variational analysis and generalized differentiation, we derive
necessary as well as necessary and sufficient optimality conditions for the
extended version of the Fermat-Torricelli problem under consideration, which
allow us to completely solve it in some important settings. Furthermore, we
develop and justify a numerical algorithm of the subgradient type to find
optimal solutions in convex settings and provide its numerical implementations.
