Anytime Solvers for Variational Inequalities: the (Recursive) Safe Monotone Flows

Abstract

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.

Figures (3)

• Figure 1: Illustration of the notion of (a) $\alpha$-restricted tangent set and (b) tangent cone. The gray-shaded region represents the set $\mathcal{C}$. The colored regions depict either type of set, which consists of vectors centered various points $x_i$. The dashed border indicates directions in which the magnitude of vectors in the set are unbounded. In (a), note that the $\alpha$-restricted tangent set is nonempty at $x_2 \not\in \mathcal{C}$, however because the region does not overlap with the point $x_2$, the set $T^{(\alpha)}_\mathcal{C}(x_2)$ does not contain the zero vector, and all vectors point strictly toward $\mathcal{C}$. In (b), note that the tangent cone is empty at points outside $\mathcal{C}$.
• Figure 2: Implementation of (a) projected monotone flow, (b) safe monotone flow ($\alpha=1.0$), and (c) recursive safe monotone flow ($\tau=0.25$) in a two-player game. The shaded region shows the constraint set $\mathcal{C}$ and the colored paths represent trajectories of the corresponding flow starting from various initial condition. (d) shows a zoomed-in portion of the boundary of $\mathcal{C}$ to illustrate the practical safety of the recursive safe monotone flow.
• Figure 3: Receding horizon implementation of the safe monotone flow solving a linear quadratic dynamic game for different choices of termination time $t_f$. The closed-loop implementation of the exact solution corresponds to $t_f = \infty$ (dashed lines). (a) We plot the evolution of $\left\lVert z(s)\right\rVert$ in green. (b) We plot the evolution of the first component of $w_1(s)$ in blue-green (scale in left $y$-axis) and the first component of $w_2(x)$ in red-orange (scale in right $y$-axis).

Theorems & Definitions (46)

• Lemma 2.1: Projection-based Safeguarding Feedback
• Lemma 2.2: VCBF-based Safeguarding Feedback
• Lemma 4.1
• Proposition 4.2
• proof
• Theorem 4.3
• Lemma 5.1: Vector Control Barrier Function for \ref{['eq:variational-system']}
• Proposition 5.2
• proof
• Proposition 5.3