Mathematical programming formulation for neural combinatorial optimization algorithms

This paper considers the analog neural solution of the combinatorial optimization problem. The solution method is analyzed based on the Lagrange multiplier method for the continuous relaxation problem of 0-1 integer programming. It is shown that the solution process can be interpreted as the solution by the gradient method for the saddle point of the Lagrange function. An improved Hopfield net is derived from the formulation as the pure integer programming. The elastic net and the generalized deformable model are derived from the mixed integer programming problem. Based on those results, an interpretation of the deterministic annealing is derived from the viewpoint of mathematical programming. It is shown that the Lagrange function can work as the Lyapunov function for the solution process and the convergence property of those neural methods of solution is analyzed.