Intrinsic formulation of KKT conditions and constraint qualifications on smooth manifolds
Provides a rigorous intrinsic framework for constrained optimization on manifolds, extending classical Euclidean theory to a broader geometric setting.
The paper formulates KKT conditions and constraint qualifications for optimization on smooth manifolds, proving that local minimizers admit Lagrange multipliers under the Guignard constraint qualification, and establishes the chain LICQ implies MFCQ implies ACQ implies GCQ, paralleling Euclidean results.
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.