Intrinsic formulation of KKT conditions and constraint qualifications on smooth manifolds

TL;DR

This paper extends the classical KKT conditions and constraint qualifications to optimization problems on smooth manifolds, establishing their relationships and implications in a geometric setting.

Contribution

It formulates KKT conditions and various constraint qualifications on smooth manifolds, paralleling Euclidean results and providing a numerical example on the sphere.

Findings

KKT conditions are valid under Guignard qualification on manifolds

Chain of implications among constraint qualifications is established

Numerical example demonstrates the theory on the Riemannian sphere

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.