  Karush-Kuhn-Tucker (KKT) conditions for equality and inequality constrained
optimization problems on smooth manifolds are formulated. Under the Guignard
constraint qualification, local minimizers are shown to admit Lagrange
multipliers. The linear independence, Mangasarian-Fromovitz, and Abadie
constraint qualifications are also formulated, and the chain "LICQ implies MFCQ
implies ACQ implies GCQ" is proved. Moreover, classical connections between
these constraint qualifications and the set of Lagrange multipliers are
established, which parallel the results in Euclidean space. The constrained
Riemannian center of mass on the sphere serves as an illustrating numerical
example.
