Uniform Framework for Unconstrained and Constrained Optimization: Optimization on Riemannian Manifolds

Research on unconstrained and constrained optimization has been separately conducted for a long time. Normally, constrained optimization, especially the nonlinear equality constrained optimization problem is much more difficult to be researched. People became to recognize that both unconstrained optimization and constrained optimization are actually the optimization problem on Riemannian manifolds since 1982. Therefore, unconstrained optimization methods can be extended directly to constrained optimization cases under the condition of Riemannian manifolds. Recently, more and more scholars realize the importance of the optimization methods on Riemannian manifold and it has become a new direction of research on nonlinear optimization. This article reviews the development and new applications of the optimization problem on Riemannian manifolds and adduces correlative arithmetic and some examples in addition.