Finite- and Fixed-Time Gradient Flows for Constrained Optimization via Control Barrier Functions

This letter presents a novel finite-and fixed-time convergent framework for solving constrained convex optimization problems using safe gradient flow dynamics. In the existing literature, finite-and fixed-time gradient flow methods have primarily addressed equality constraints by formulating Lagrangian-based gradient dynamics or a scaled augmented Lagrangian formulation. In contrast, the proposed method augments the gradient flow with constraint-driven control inputs derived using control barrier function (CBF) techniques, enabling simultaneous handling of equality and inequality constraints without relying on penalty function formulations. This formulation treats decision variables as system states and Lagrange multipliers as control inputs, enabling the feasible set to act as an invariant safe set throughout the system’s evolution. The approach ensures strict feasibility and guarantees convergence to the optimal solution in both finite-and fixed-time. Rigorous Lyapunov-based analysis establishes theoretical stability guarantees, and numerical simulations demonstrate rapid, exact convergence for problems involving both equality and inequality constraints.