Manifold optimization is an emerging field of contemporary optimization thatnbsp;constructs efficient and robust algorithms by exploiting the specific geometricalnbsp;structure of the search space. In our case the search space takes the form of anbsp;manifold.nbsp;Manifold optimization methods mainly focus on adapting existing optimizationnbsp;methods from the usual ?easy-to-deal-with? Euclidean search spaces to manifoldsnbsp;whose local geometry can be defined e.g. by a Riemannian structure. In this waynbsp;the form of the adapted algorithms can stay unchanged. However, to accommodatenbsp;the adaptation process, assumptions on the search space manifold often have tonbsp;be made. In addition, the computations and estimations are confined by the localnbsp;geometry.This book presents a framework for population-based optimization on Riemanniannbsp;manifolds that overcomes both the constraints of locality and additional assumptions.nbsp;Multi-modal, black-box manifold optimization problems on Riemannian manifoldsnbsp;can be tackled using zero-order stochastic optimization methods from a geometricalnbsp;perspective, utilizing both the statistical geometry of the decision spacenbsp;and Riemannian geometry of the search space.This monograph presents in a self-contained manner both theoretical and empiricalnbsp;aspects ofnbsp;stochastic population-based optimization on abstract Riemanniannbsp;manifolds.
