Model-Free Optimization on Smooth Compact Manifolds: Overcoming Topological Obstructions using Zeroth-Order Hybrid Dynamics

Abstract

Smooth autonomous dynamical systems modeled by ordinary differential equations (ODEs) cannot robustly and globally stabilize a point in compact, boundaryless manifolds. This obstruction, which is topological in nature, implies that traditional smooth optimization dynamics are not able to robustly solve global optimization problems in such spaces. In turn, model-free optimization algorithms, which usually inherit their stability and convergence properties from their model-based counterparts, might also suffer from similar topological obstructions. For example, this is the case in zeroth-order methods and perturbation-based techniques, where gradients and Hessians are usually estimated in real time via measurements or evaluations of the cost function. To address this issue, we introduce a class of hybrid model-free optimization dynamics that combine continuous-time and discrete-time feedback to overcome the obstructions that emerge in traditional ODE-based optimization algorithms evolving on smooth compact manifolds. In particular, we introduce a hybrid controller that switches between different model-free feedback-laws obtained by applying suitable exploratory geodesic dithers to a family of synergistic diffeomorphisms adapted to the cost function that defines the optimization problem. The geodesic dithers enable exploration of the manifold while preserving its forward invariance, a property that is important for many practical applications where physics-based constraints limit the feasible trajectories of the system. The hybrid controller exploits the information obtained from the geodesic dithers to achieve robust global practical stability of the set of minimizers of the cost. The proposed method is of model-free nature since it only requires measurements or evaluations of the cost function. Numerical results are presented to illustrate the main ideas and advantages of the method.