Global Optimization on the Sphere: A Stochastic Hybrid Systems Approach

Abstract

We cast a stochastic, hybrid algorithm for global optimization on the unit sphere in the framework of stochastic hybrid inclusions. The algorithm contains two hybrid features. First, it includes hysteresis switching between two coordinate charts in order to be able to fully explore the sphere by flowing without encountering singularities in the flow vector field. Secondly, it combines gradient flow with jumps that aim to escape singular points of the function to minimize, other than those singular points corresponding to global minima. The algorithm is stochastic because the jumps involve random probing on the sphere. Solutions are not unique because the jumps are governed by a set-valued mapping, i.e., an inclusion. Regarding the coordinate charts employed, we discuss both the use of spherical coordinates as well as stereographic projection. By using the framework of stochastic hybrid inclusions, we provide a detailed stability characterization of the optimization algorithm. In particular, we establish uniform global asymptotic stability in probability of the set of global minimizers for arbitrary continuously differentiable functions defined on the sphere.