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Smooth autonomous dynamical systems modeled by ordinary differentialequations (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 robustlysolve global optimization problems in such spaces. In turn, model-freeoptimization algorithms, which usually inherit their stability and convergenceproperties from their model-based counterparts, might also suffer from similartopological obstructions. For example, this is the case in zeroth-order methodsand perturbation-based techniques, where gradients and Hessians are usuallyestimated in real time via measurements or evaluations of the cost function. Toaddress this issue, we introduce a class of hybrid model-free optimizationdynamics that combine continuous-time and discrete-time feedback to overcomethe obstructions that emerge in traditional ODE-based optimization algorithmsevolving on smooth compact manifolds. In particular, we introduce a hybridcontroller that switches between different model-free feedback-laws obtained byapplying suitable exploratory geodesic dithers to a family of synergisticdiffeomorphisms adapted to the cost function that defines the optimizationproblem. The geodesic dithers enable exploration of the manifold whilepreserving its forward invariance, a property that is important for manypractical applications where physics-based constraints limit the feasibletrajectories of the system. The hybrid controller exploits the informationobtained from the geodesic dithers to achieve robust global practical stabilityof the set of minimizers of the cost. The proposed method is of model-freenature since it only requires measurements or evaluations of the cost function.Numerical results are presented to illustrate the main ideas and advantages ofthe method.

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