Target tracking with dynamic convex optimization

Alec Koppel, Andrea Simonetto, Aryan Mokhtari, Geert Leus, Alejandro Ribeiro

Abstract: We develop a framework for trajectory tracking in dynamic settings, where an autonomous system is charged with the task of remaining close to an object of interest whose position varies continuously in time. We model this scenario as a convex optimization problem with a time-varying objective function and propose an adaptive discrete-time sampling prediction-correction scheme to find and track the solution trajectory while sampling the problem data at a constant rate of 1 /h. We propose approximate gradient trajectory (AGT) and approximate Newton trajectory tracking (ANT) as prediction-correction algorithms that (i) analyze the iso-residual dynamics of the optimality conditions in the prediction step, (ii) use gradient descent and Newton's method in the correction step, respectively, and (iii) approximate the partial derivative of the objective by a first-order backward derivative for the prediction step. We establish that the asymptotic error incurred by both proposed methods behaves as O(h <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ), and in some cases as O(h <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> ), which outperforms the state-of-the-art error bound of O(h) for correction-only methods in the gradient-correction step. The utility of the methods is demonstrated in an object tracking problem executed by an autonomous system.