Sequential optimality conditions for nonlinear optimization on Riemannian manifolds and a globally convergent augmented Lagrangian method

• Published in: Computational optimization and applications Vol. 81; no. 2; pp. 397 - 421
• Main Authors: Yamakawa, Yuya, Sato, Hiroyuki
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.03.2022; Springer Nature B.V
• Subjects: Approximation; Convergence; Convex and Discrete Geometry; Management Science; Mathematics; Mathematics and Statistics; Operations Research; Operations Research/Decision Theory; Optimization; Riemann manifold; Statistics; Theorems; Augmented Lagrangian method; Riemannian manifolds; Global convergence; Approximate KKT conditions; Riemannian optimization; Sequential optimality conditions
• ISSN: 0926-6003; 1573-2894
• DOI: 10.1007/s10589-021-00336-w

Recently, the approximate Karush–Kuhn–Tucker (AKKT) conditions, also called the sequential optimality conditions, have been proposed for nonlinear optimization in Euclidean spaces, and several methods to find points satisfying such conditions have been developed by researchers. These conditions are known as genuine necessary optimality conditions because all local optima satisfy them with no constraint qualification (CQ). In this paper, we extend the AKKT conditions to nonlinear optimization on Riemannian manifolds and propose an augmented Lagrangian (AL) method that globally converges to points satisfying such conditions. In addition, we prove that the AKKT and KKT conditions are indeed equivalent under a certain CQ. Finally, we examine the effectiveness of the proposed AL method via several numerical experiments.