Hybrid Gradient Descent for Robust Global Optimization on the Circle

We further develop the concept of synergistic potential functions and corresponding hybrid feedback laws, from [1], in the context of optimization on the circle. Our hybrid gradient-descent algorithm solves the problem of globally optimizing an objective function with unknown minimum by using an adjustable diffeomorphism to warp the function at points where it exceeds a known threshold. We use two diffeomorphisms to split the objective function into a pair of potential functions for control, each having the same minimum as the objective function, but with their peaks shifted around the circle. Hysteretic switching between these potential functions produces robust global asymptotic convergence to the set of minima of the objective function. We also present a model-free algorithm which solves the problem of global optimization on the circle without access to the gradient of the objective function. This algorithm estimates the gradient by dithering the state in a neighborhood of the circle, resulting in robust global stability of a neighborhood of the optimizer.