Fixed-Time Stable Gradient Flows: Applications to Continuous-Time Optimization

Continuous-time optimization is currently an active field of research in optimization theory; prior work in this area has yielded useful insights and elegant methods for proving stability and convergence properties of the continuous-time optimization algorithms. This article proposes novel gradient-flow schemes that yield convergence to the optimal point of a convex optimization problem within a fixed time from any given initial condition for unconstrained optimization, constrained optimization, and min–max problems. It is shown that the solution of the modified gradient-flow dynamics exists and is unique under certain regularity conditions on the objective function, while fixed-time convergence to the optimal point is shown via Lyapunov-based analysis. The application of the modified gradient flow to unconstrained optimization problems is studied under the assumption of gradient dominance, a relaxation of strong convexity. Then, a modified Newton's method is presented that exhibits fixed-time convergence under some mild conditions on the objective function. Building upon this method, a novel technique for solving convex optimization problems with linear equality constraints that yields convergence to the optimal point in fixed time is developed. Finally, the general min–max problem is considered, and a modified saddle-point dynamics to obtain the optimal solution in fixed time is developed.