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Karush-Kuhn-Tucker (KKT) conditions for equality and inequality constrainedoptimization problems on smooth manifolds are formulated. Under the Guignardconstraint qualification, local minimizers are shown to admit Lagrangemultipliers. The linear independence, Mangasarian-Fromovitz, and Abadieconstraint qualifications are also formulated, and the chain "LICQ implies MFCQimplies ACQ implies GCQ" is proved. Moreover, classical connections betweenthese constraint qualifications and the set of Lagrange multipliers areestablished, which parallel the results in Euclidean space. The constrainedRiemannian center of mass on the sphere serves as an illustrating numericalexample.

Intrinsic formulation of KKT conditions and constraint qualifications on smooth manifolds