Orbital independence in symmetric mathematical programs

It is well known that symmetric mathematical programs are harder to solve to global optimality using Branch-and-Bound type algorithms, since the solution symmetry is reflected in the size of the Branch-and-Bound tree. It is also well known that some of the solution symmetries are usually evident in the formulation, making it possible to attempt to deal with symmetries as a preprocessing step. One of the easiest approaches is to “break” symmetries by adjoining some symmetry-breaking constraints to the formulation, thereby removing some symmetric global optima, then solve the reformulation with a generic solver. Sets of such constraints can be generated from each orbit of the action of the symmetries on the variable index set. It is unclear, however, whether and how it is possible to choose two or more separate orbits to generate symmetry-breaking constraints which are compatible with each other (in the sense that they do not make all global optima infeasible). In this paper we discuss a new concept of orbit independence which clarifies this issue.