Optimal Trajectory Tracking Using Newton and Levenberg‐Marquardt‐Like Algorithms for Constrained Time‐Varying Optimization

ABSTRACT In this paper, we propose continuous‐time algorithms for tracking the optimal trajectory of time‐varying optimization problems with time‐varying constraints in a predefined time, where the time of convergence is selected a priori. A robust Newton‐like approach is developed for the cases, where the exact knowledge of the rate of change of the gradient of objective function is unknown. Levenberg‐Marquardt‐like algorithm is proposed for the cases when the Hessian of the objective function is singular or near singular. Lyapunov‐based convergence analysis is discussed for the proposed algorithms. Simulation results for time‐varying optimization problems show the efficacy of the proposed approach. We demonstrate the applicability of the proposed method through its use in a collision‐free robot navigation problem.