Equivalence of coupled parametric oscillator dynamics to Lagrange multiplier primal-dual optimization

There has been a recent surge of interest in physics-based solvers for combinatorial optimization problems. We present a dynamical solver for the Ising problem that is comprised of a network of coupled parametric oscillators and show that it implements Lagrange multiplier constrained optimization. We show that the pump depletion effect, which is intrinsic to parametric oscillators, enforces binary constraints and enables the system's continuous analog variables to converge to the optimal binary solutions to the optimization problem. Moreover, there is an exact correspondence between the equations of motion for the coupled oscillators and the update rules in the primal-dual method of Lagrange multipliers. Though our analysis is performed using electrical LC oscillators, it can be generalized to any system of coupled parametric oscillators. We simulate the dynamics of the coupled oscillator system and demonstrate that the performance of the solver on a set of benchmark problems is comparable to the best-known results obtained by digital algorithms in the literature.