# A non-monotone Inexact Restoration approach for minimization with orthogonality constraints

In this work we consider the problem of minimizing a differentiable functional restricted to the set of $n\times p$ matrices with orthonormal columns. This problem appears in several fields such as statistics, signal processing, global positioning system, machine learning, physics, chemistry and others. We present an algorithm based on a recent non-monotone variation of the inexact restoration method for nonlinear programming along with its implementation details. We give a simple characterization of the set of tangent directions (with respect to the orthogonality constraints) and we use it for dealing with the minimization (tangent) phase. For the restoration phase we employ the well-known Cayley transform for bringing the computed point back to the feasible set (i.e., the restoration phase is exact). Under standard assumptions we prove that any limit point of the sequence generated by the algorithm is a stationary point. A numerical comparison with a well established algorithm is also presented on three different classes of the problem.

## Citation

Federal University of Santa Catarina, Department of Mathematics. Florianópolis. SC. Brazil. 88040-900