Computing the optimal partition of variables in multi-homogeneous homotopy methods

The multi-homogenous homotopy continuation method is one of the most efficient approaches in finding all isolated solutions of polynomial systems. A different partition of variables leads to a different homotopy system. The homotopy using the optimal partition of variables reduces the computational cost in curve following to the minimum. However, finding the optimal variable partition is likely an NP hard problem. An approximate algorithm is introduced in this paper to avoid exhaustive search in finding the (approximate) optimal variable partition. The global convergence of this algorithm is proved with Markov chain theory. Numerical comparisons with algorithms existed show the efficiency of the new method.