Manifold constrained steepest descent for smooth and closed-set optimization

We study minimization of smooth functions over feasible sets that have smooth embedded-manifold structure throughout or only on selected regions, using linear minimization oracles (LMOs) to determine search directions under user-chosen norms. Restricting an LMO to a tangent space, however, can require an iterative inner solve. We propose Manifold Constrained Steepest Descent (MCSD) and a tangent-projected variant, MCSD-TP, which avoid solving tangent-space LMO subproblems iteratively. The spectral-norm specialization of MCSD on the Stiefel manifold yields SPEL, which admits an efficient implementation using matrix-sign computations. Under standard regularity assumptions, both methods attain an O(\log T/\sqrt T) best-iterate stationarity bound. For closed feasible sets, the hybrid MCSD--PGD combines either smooth method with projected gradient descent; under the corresponding local smoothness condition, every accumulation point is Bouligand stationary. Experiments on Stiefel-constrained PCA, weighted low-rank approximation, and sparse phase retrieval illustrate the proposed methods.