  We show that the traveling salesman problem (TSP) and its many variants may
be modeled as functional optimization problems over a graph. In this
formulation, all vertices and arcs of the graph are functionals; i.e., a
mapping from a space of measurable functions to the field of real numbers. Many
variants of the TSP, such as those with neighborhoods, with forbidden
neighborhoods, with time-windows and with profits, can all be framed under this
construct. In sharp contrast to their discrete-optimization counterparts, the
modeling constructs presented in this paper represent a fundamentally new
domain of analysis and computation for TSPs and their variants. Beyond its
apparent mathematical unification of a class of problems in graph theory, the
main advantage of the new approach is that it facilitates the modeling of
certain application-specific problems in their home space of measurable
functions. Consequently, certain elements of economic system theory such as
dynamical models and continuous-time cost/profit functionals can be directly
incorporated in the new optimization problem formulation. Furthermore, subtour
elimination constraints, prevalent in discrete optimization formulations, are
naturally enforced through continuity requirements. The price for the new
modeling framework is nonsmooth functionals. Although a number of theoretical
issues remain open in the proposed mathematical framework, we demonstrate the
computational viability of the new modeling constructs over a sample set of
problems to illustrate the rapid production of end-to-end TSP solutions to
extensively-constrained practical problems.
