Minimizing a differentiable function restricted to the set of matrices with orthonormal columns finds applications in several fields of science and engineering. In this paper, we propose to solve this problem through a nonmonotone variation of the inexact restoration method consisting of two phases: restoration phase, aimed to improve feasibility, and minimization phase, aimed to decrease the function value in the tangent set of the constraints. For this, we give a suitable characterization of the tangent set of the orthogonality constraints, allowing us to (i) deal with the minimization phase efficiently and (ii) employ the Cayley transform to bring a point in the tangent set back to feasibility, leading to a SVD-free restoration phase. Based on previous global convergence results for the nonmonotone inexact restoration algorithm, it follows that any limit point of the sequence generated by the new algorithm is stationary. Moreover, numerical experiments on three different classes of the problem indicate that the proposed method is reliable and competitive with other recently developed methods.

将具有正交列的矩阵的可微函数最小化可以在科学和工程学的多个领域中找到应用。在本文中，我们提出了通过非精确还原方法的非单调变化来解决此问题的方法，该方法包括两个阶段：旨在改善可行性的还原阶段和旨在减小约束切线集合中的函数值的最小化阶段。为此，我们对正交约束的切线集进行了适当的刻画，使我们能够（i）有效地处理最小化阶段，并且（ii）使用Cayley变换将切线集中的点恢复为可行性，从而到无SVD的恢复阶段。根据先前针对非单调不精确还原算法的全局收敛结果，因此，新算法生成的序列的任何极限点都是固定的。此外，对三种不同类别问题的数值实验表明，所提出的方法是可靠的，并且与其他最近开发的方法竞争。