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We show that the traveling salesman problem (TSP) and its many variants maybe modeled as functional optimization problems over a graph. In thisformulation, all vertices and arcs of the graph are functionals; i.e., amapping from a space of measurable functions to the field of real numbers. Manyvariants of the TSP, such as those with neighborhoods, with forbiddenneighborhoods, with time-windows and with profits, can all be framed under thisconstruct. In sharp contrast to their discrete-optimization counterparts, themodeling constructs presented in this paper represent a fundamentally newdomain of analysis and computation for TSPs and their variants. Beyond itsapparent mathematical unification of a class of problems in graph theory, themain advantage of the new approach is that it facilitates the modeling ofcertain application-specific problems in their home space of measurablefunctions. Consequently, certain elements of economic system theory such asdynamical models and continuous-time cost/profit functionals can be directlyincorporated in the new optimization problem formulation. Furthermore, subtourelimination constraints, prevalent in discrete optimization formulations, arenaturally enforced through continuity requirements. The price for the newmodeling framework is nonsmooth functionals. Although a number of theoreticalissues remain open in the proposed mathematical framework, we demonstrate thecomputational viability of the new modeling constructs over a sample set ofproblems to illustrate the rapid production of end-to-end TSP solutions toextensively-constrained practical problems.

An Optimal Control Theory for the Traveling Salesman Problem and Its Variants