Smoothing l1-exact penalty method for intrinsically constrained Riemannian optimization problems

This paper deals with the Constrained Riemannian Optimization (CRO) problem, which involves minimizing a function subject to equality and inequality constraints on Riemannian manifolds. The study aims to advance optimization theory in the Riemannian setting by presenting and analyzing a penalty-type method for solving CRO problems. The proposed approach is based on techniques that involve smoothing the classical \(\ell_1\)-exact penalty function. This penalty-type method extends previous research by incorporating different smoothing functions, refining the penalty multipliers, and relaxing the constraints qualifications necessary for convergence. The method uses the extended Mangasarian-Fromovitz constraint qualification to ensure boundedness of Lagrange multipliers and global convergence to feasible and optimal solutions. In addition, under the assumption that the limit points are feasible, it is shown that these points satisfy the Approximate KKT (AKKT) conditions. Furthermore, when AKKT is combined with a weak constraint qualification, it is proved that the limit points satisfy the KKT conditions. Preliminary numerical experiments are conducted to demonstrate the effectiveness of the proposed method, which indicates that the method effectively addresses the complexity associated with CRO problems.