Pseudo-Riemannian manifold

Perspective (graphical)

Mathematics

Manifold (fluid mechanics)

Financial economics

Mathematical optimization

Ricci curvature

Nonlinear dimensionality reduction

Engineering

Optimization problem

Manifold alignment

Artificial intelligence

Economics

Curvature

Class (philosophy)

Geometry

Computer science

Dimensionality reduction

Convexity

Mechanical engineering

Journal of the Operations Research Society of China

Manifold optimization is ubiquitous in computational and applied mathematics, statistics, engineering, machine learning, physics, chemistry, etc. One of the main challenges usually is the non-convexity of the manifold constraints. By utilizing the geometry of manifold, a large class of constrained optimization problems can be viewed as unconstrained optimization problems on manifold. From this perspective, intrinsic structures, optimality conditions and numerical algorithms for manifold optimization are investigated. Some recent progress on the theoretical results of manifold optimization is also presented.

A Brief Introduction to Manifold Optimization