Equivalence of Coupled Parametric Oscillator Dynamics to Lagrange Multiplier Optimization

Physical machines can solve optimization problems, employing the maximization or minimization principles that are built into physics. We present a dynamical solver for the Ising problem that is comprised of a network of coupled bistable parametric oscillators and show that it implements Lagrange multiplier constrained optimization. We show that the $2\omega$ pump depletion effect, which is intrinsic to parametric oscillators, automatically enforces the binary Ising constraints. This enables the system's continuous analog variables to converge to high-quality binary solutions to the optimization problem. Moreover, we establish 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 on benchmark problems and demonstrate that its performance is comparable to the best-known results.