Breaking Limitation of Convergence in Continuous Optimization: Predefined-Time Gradient Flows

This paper investigates the continuous-time unconstrained optimization via dynamical gradient flow systems. For strongly convex objective functions, a gradient flow with predefined time convergence is proposed, which is suitable for relaxing the objective function to only satisfy the PolyakLojasiewicz ($\mathbf{P}-\mathrm{L}$) condition. A predefined-time convergent Newton’s flow is designed for the minimization of strictly convex functions. The proposed result is extended for strongly convex objective functions with time-varying parameters. In our schemes, the upper bound of convergence time can be used as a prior choice without involving system initialization and complex parameter calculations. The convergence of algorithms is rigorously proved by using the Lyapunov method. Numerical simulation illustrates the effectiveness of the proposed methods.