Minimization of a quasi-concave function over an efficient set

We introduce a projective approach for studying symmetric travelling salesman polytopes (STSPs). Thesymmetric travelling salesman polytope STSP(V) (resp.,Hamiltonian path polytope HP(V)) is the convex hull of incidence vectors of all Hamiltonian cycles (resp., paths) on the complete undirected graph with node setV. For any nodeh V, HP(V) is aprojection of STSP(V {h}). We show that HP(V) and STSP(V {h}) are isomorphic, and HP(V) is of full dimension minus one. By this projective approach, we obtain generalclique-lifting results, all based on simple conditions, for deriving large new classes of STSP facets. These results apply to all known non-trivial STSP facets, and generalize clique-lifting results of Maurras (1975), Grötschel and Padberg (1979) and Naddef and Rinaldi (1988).This research is supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada to the first author.