Gradient-free optimization using delayed discrete-time measurements

This paper investigates a kind of gradient-free optimization algorithm called unbiased extremum seeking (uES) for general strongly convex maps subject to discrete-time measurements with time-varying delay. Two algorithms are proposed: a classical uES scheme and a bounded uES scheme. Square-wave perturbation signals are employed to accommodate the discrete-time measurement nature of the system, and tuning exponential functions are introduced to eliminate steady-state oscillation errors, ensuring convergence to the exact optimizer. The sampling process is modeled as a time-varying delay, and a time-delay approach to averaging is developed for stability analysis. This approach transforms the original extremum seeking system into a perturbed gradient descent system whose perturbation vanishes exponentially. A Lyapunov-based analysis establishes exponential stability of both algorithms for general strongly convex cost functions and rigorously demonstrates their robustness with respect to time-varying measurement delay. Numerical examples are provided to validate the effectiveness of the proposed schemes.