Combinatorial optimization using the Lagrange primal-dual dynamics of

Coupled-bistable-oscillator machines have recently generated significant interest due to their observed ability to rapidly produce high-quality solutions to nondeterministic-polynomial-time-complete optimization problems. While the dynamics of such systems are often derived in the literature, it has hitherto been unclear why exactly the system dynamics perform optimization so well. This paper answers this question by presenting a complete equivalence between coupled-oscillator machines and the primal-dual method of Lagrange multipliers. This equivalence explains how coupled-oscillator solvers implement the correct optimization constraints and find high-quality solutions. The equivalence also provides precise mathematical meaning to each system component and enables the principled design of systems that implement more-sophisticated optimization algorithms. We simulate the system dynamics and demonstrate (1) that its performance is competitive with performance of the best-known digital algorithms, (2) that the circuit is robust with regard to large component variations, hinting that the traditional shortcomings of analog computation may be less important for these applications, and (3) that the circuit consumes extremely low amounts of power (on the order of milliwatts) and energy (approximately 100 nJ) per optimization even for problems of 2000 variables.