  We discuss a dynamical systems perspective on discrete optimization.
Departing from the fact that many combinatorial optimization problems can be
reformulated as finding low energy spin configurations in corresponding Ising
models, we derive a penalized rank-two relaxation of the Ising formulation. It
turns out that the associated gradient flow dynamics exactly correspond to a
type of hardware solvers termed oscillator-based Ising machines. We also
analyze the advantage of adding angle penalties by leveraging random rounding
techniques. Therefore, our work contributes to a rigorous understanding of
oscillator-based Ising machines by drawing connections to the penalty method in
constrained optimization and providing a rationale for the introduction of
sub-harmonic injection locking. Furthermore, we characterize a class of
coupling functions between oscillators, which ensures convergence to discrete
solutions. This class of coupling functions avoids explicit penalty terms or
rounding schemes, which are prevalent in other formulations.
