In this paper, a novel proportional integral (PI)-like estimator-based adaptive extremum seeking control (AdESC) algorithm is proposed for online optimization, where parameter convergence is achieved under a relaxed mathematical condition called initial excitation (IE). The proposed AdESC algorithm utilizes a new set of low-pass filter dynamics, which omits the requirement of switching mechanism in past literature for rank-checking while still ensuring parameter convergence. A detailed Lyapunov analysis is carried out using singular-perturbation like principle to establish closed-loop stability of the AdESC algorithm.