  This paper proposes a novel landscape smoothing method for the symmetric
Traveling Salesman Problem (TSP). We first define the Homotopic Convex (HC)
transformation of a TSP as a convex combination of a well-constructed simple
TSP and the original TSP. The simple TSP, called the convex-hull TSP, is
constructed by transforming a known local or global optimum. We observe that
controlled by the coefficient of the convex combination, with local or global
optimum, (i) the landscape of the HC transformed TSP is smoothed in terms that
its number of local optima is reduced compared to the original TSP; (ii) the
fitness distance correlation of the HC transformed TSP is increased. Further,
we observe that the smoothing effect of the HC transformation depends highly on
the quality of the used optimum. A high-quality optimum leads to a better
smoothing effect than a low-quality optimum. We then propose an iterative
algorithmic framework in which the proposed HC transformation is combined
within a heuristic TSP solver. It works as an escaping scheme from local optima
aiming to improve the global search ability of the combined heuristic. Case
studies using the 3-Opt and the Lin-Kernighan local search as the heuristic
solver show that the resultant algorithms significantly outperform their
counterparts and two other smoothing-based TSP heuristic solvers on most of the
test instances with up to 20,000 cities.
