  This paper synthesizes anytime algorithms, in the form of continuous-time
dynamical systems, to solve monotone variational inequalities. We introduce
three algorithms that solve this problem: the projected monotone flow, the safe
monotone flow, and the recursive safe monotone flow. The first two systems
admit dual interpretations: either as projected dynamical systems or as
dynamical systems controlled with a feedback controller synthesized using
techniques from safety-critical control. The third flow bypasses the need to
solve quadratic programs along the trajectories by incorporating a dynamics
whose equilibria precisely correspond to such solutions, and interconnecting
the dynamical systems on different time scales. We perform a thorough analysis
of the dynamical properties of all three systems. For the safe monotone flow,
we show that equilibria correspond exactly with critical points of the original
problem, and the constraint set is forward invariant and asymptotically stable.
The additional assumption of convexity and monotonicity allows us to derive
global stability guarantees, as well as establish the system is contracting
when the constraint set is polyhedral. For the recursive safe monotone flow, we
use tools from singular perturbation theory for contracting systems to show KKT
points are locally exponentially stable and globally attracting, and obtain
practical safety guarantees. We illustrate the performance of the flows on a
two-player game example and also demonstrate the versatility for
interconnection and regulation of dynamical processes of the safe monotone flow
in an example of a receding horizon linear quadratic dynamic game.
