  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.
