Intrinsic formulation of KKT conditions and constraint qualifications on smooth manifolds

Abstract

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.

References in corpus (3)

• Restoration of Manifold-Valued Images by Half-Quadratic Minimization
• A Parallel Douglas Rachford Algorithm for Minimizing ROF-like Functionals on Images with Values in Symmetric Hadamard Manifolds
• Simple algorithms for optimization on Riemannian manifolds with constraints

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• Hamiltonian of polymatrix zero-sum games
• An inexact proximal point method for variational inequality on Hadamard manifolds
• Stochastic Augmented Lagrangian Method in Riemannian Shape Manifolds