Safety-Critical Control as a Design Paradigm for Anytime Solvers of Variational Inequalities

This paper shows that safety-critical control methods for the synthesis of safeguarding controllers also have an important role to play in optimization theory. We are motivated by applications where the solution of a variational inequality is used to regulate a dynamically evolving plant. We design algorithms to solve these problems by blending techniques from safety-critical control with tools from monotone operator theory. These algorithms are anytime, meaning that, if initialized in the feasible set, they are guaranteed to return a feasible solution to the optimization problem regardless of when they are terminated, which makes them particularly suitable for real-time applications. In some cases, our approach leads to reinterpretations of well-known algorithms from the lens of control theory, and in other cases we derive entirely novel algorithms. Our results demonstrate the promising potential of safety-critical control for both the analysis and design of optimization algorithms.