Anytime solvers for variational inequalities: The (recursive) safe monotone flows

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 specified by a quadratic program derived 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.