A Unified Landing Framework for Equality-Constrained Optimization

In this paper, we provide an extension of the so-called "landing algorithms", recently introduced for solving optimization problems over Stiefel manifolds. With a Riemannian optimization perspective, we introduce a class of infeasible algorithms, extending landing algorithms to equality-constrained problems and improving their theoretical foundations by connecting it to SQP (sequential quadratic programming). We develop a backtracking-based landing method for deterministic optimization problems, addressing the limitation of fixed stepsize schemes of previous work in the Stiefel case. We also generalize the stochastic version of the landing algorithm to handle stochastic optimization problems with general equality constraints by employing a decaying stepsize. We finally provide numerical illustrations of the landing algorithms for solving problems over the generalized Stiefel manifold and the symplectic manifold.