Fast convergence to an approximate solution by message-passing for complex optimizations

Message-passing (MP) is a powerful tool for finding an approximate solution in optimization. We generalize it in product-sum form, and numerically show the fast convergence. From the linearity of MP in a logarithmic space, it is derived that an equilibrium solution exists in a neighborhood of random initial values. These results will give one of the reason why the convergence is very fast in collective computation based on a common mathematical background.