Optimization on Manifolds: Context and Applications

This lecture introduces optimization on manifolds, focusing on the context and applications. It covers classical unconstrained optimization in linear spaces, then transitions to optimization on smooth manifolds with Riemannian structures. The instructor discusses the gradient projection method along geodesics, its practicality since the 1990s, and its mainstream adoption. The lecture also mentions the popularization of optimization algorithms on matrix manifolds in the 2010s. Key references include works by Luenberger, Edelman, Arias, and Smith, highlighting the evolution of optimization techniques over the years.