Finding a Hamiltonian cycle by finding the global minimizer of a linearly constrained problem

It has been shown that a global minimizer of smooth determinant matrix function corresponds to the largest cycle graph. When it exists, this is Hamiltonian cycle. Finding minimizers even challenge. The difficulty often exacerbated by existence many minimizers. One may think would help, but in case cycles ratio number local typically astronomically small. There are various equivalent forms problem and here we report on two. Although focus finding cycles, an interest itself, just proxy for class problems have discrete variables. solution relaxations these at degenerate vertex, neighborhood Hessian indefinite. form address virtue being ideal test algorithms designed nonlinear general. easy generate varying size character, they advantage able determine if found. A feature there solutions. For example, frequency assignment any permutation also solution. consequence common characteristic relaxed large numbers larger both minimizers, saddle points whose reduced only single negative eigenvalue. Efficient seek find type described. Results using BONMIN, solver with continuous variables, included.