Effects of local minima and bifurcation delay on combinatorial optimization with continuous variables

Combinatorial optimization problems can be mapped onto Ising models, and their ground state is generally difficult to find. A lot of heuristics for these problems have been proposed, and one promising approach is to use continuous variables. In recent years, one such algorithm has been implemented by using parametric oscillators known as coherent Ising machines. Although these algorithms have been confirmed to have high performance through many experiments, unlike other familiar algorithms such as simulated annealing, their computational ability has not been fully investigated. In this paper, we propose a simple heuristic based on continuous variables whose static and dynamical properties are easy to investigate. Through the analyses of the proposed algorithm, we find that many local minima in the early stage of the optimization and bifurcation delay reduce its performance in a certain class of Ising models.