Safe Output-Regulation Flows for Constrained Optimization via Control Barrier Functions

Control-theoretic approaches to constrained optimization have been widely studied by interpreting optimization algorithms as dynamical systems. This paper formulates constrained optimization as an output-regulation problem with state constraints, where equality-constraint residuals are treated as regulated outputs and inequality constraints as state constraints. Proportional--integral feedback is employed to drive the equality-constraint residuals asymptotically to zero, while a control barrier function (CBF) safety filter is used to ensure inequality safety throughout the optimization process. We establish feasibility of the CBF filtering program throughout the inequality-feasible set and local exponential convergence of the resulting flow. Under stronger global conditions, we further prove convergence from an entire connected component of the inequality-feasible set. Since exact implementation of the flow requires solving a CBF quadratic program at each state, we further develop a flow that approximates the CBF solution through multiplier updates. Moreover, we provide a unified formulation encompassing the two proposed flows and projected primal--dual dynamics. Numerical examples demonstrate convergence, safety, and computational tradeoffs of the proposed flows.