Stochastic Adaptive Optimization with Dithers

Optimization problems are essential for a broad spectrum of applications, and have been extensively studied in many aspects. Most existing literature on optimization algorithms does not consider systems that involve parameter uncertainties. This paper studies a class of stochastic adaptive optimization problems in which identification of unknown parameters and search of the optimal solutions must be performed simultaneously. Due to a fundamental conflict between parameter identifiability and optimality in such problems, we introduce a method of adding stochastic dither signals into the system, which provides sufficient excitation for estimating the unknown parameters, leading to adaptive optimization algorithms. Joint identification and optimization algorithms are developed and their simultaneous convergence properties of parameter estimation and optimization variable updates are proved. Under both noise-free and noisy observations, the corresponding convergence rates are established. The main results of this paper reveal certain fundamental relationships and trade-off among updating step sizes, dither magnitudes, parameter estimation errors, optimization accuracy, and convergence rates. Simulation case studies are used to illustrate the adaptive optimization algorithms and their main properties.