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This paper is concerned with the class of nonconvex optimization problems with orthogonality constraints. We develop computationally efficient relaxations that transform non-convex orthogonality constrained problems into polynomial-time solvable surrogates. A novel penalization technique is used to enforce feasibility and derive certain conditions under which the constraints of the original non-convex problem are guaranteed to be satisfied. Moreover, we extend our approach to a feasibility-preserving sequential scheme that solves penalized relaxation to obtain near-globally optimal points. Experimental results on synthetic and real datasets demonstrate the effectiveness of the proposed approach on two practical applications in machine learning.

Non-smooth Optimization over Stiefel Manifolds with Applications to Dimensionality Reduction and Graph Clustering