A Brief Introduction to Manifold Optimization
Jiang Hu, Xin Liu, Zaiwen Wen, Ya-xiang Yuan
Abstract
Abstract 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.
Topics & Concepts
Manifold (fluid mechanics)ConvexityMathematical optimizationOptimization problemManifold alignmentComputer sciencePerspective (graphical)MathematicsClass (philosophy)Pseudo-Riemannian manifoldNonlinear dimensionality reductionArtificial intelligenceGeometryRicci curvatureEngineeringMechanical engineeringCurvatureEconomicsFinancial economicsDimensionality reductionAdvanced Optimization Algorithms ResearchMetaheuristic Optimization Algorithms ResearchAdvanced Multi-Objective Optimization Algorithms