A Derivative-Free Geometric Algorithm for Optimization on a Sphere

Optimization on a unit sphere finds crucial applications in science and engineering. However, derivatives of the objective function may be difficult to compute or corrupted by noises, or even not available in many applications. Hence, we propose a Derivative-Free Geometric Algorithm (DFGA) which, to the best of our knowledge, is the first derivative-free algorithm that takes trust region framework and explores the spherical geometry to solve the optimization problem with a spherical constraint. Nice geometry of the spherical surface allows us to pursue the optimization at each iteration in a local tangent space of the sphere. Particularly, by applying Householder and Cayley transformations, DFGA builds a quadratic trust region model on the local tangent space such that the local optimization can essentially be treated as an unconstrained optimization. Under mild assumptions, we show that there exists a subsequence of the iterates generated by DFGA converging to a stationary point of this spherical optimization. Furthermore, under the Lojasiewicz property, we show that all the iterates generated by DFGA will converge with at least a linear or sublinear convergence rate. Our numerical experiments on solving the spherical location problems, subspace clustering and image segmentation problems resulted from hypergraph partitioning, indicate DFGA is very robust and efficient for solving optimization on a sphere without using derivatives.