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References

The solvers implemented here follow the constraint dissolving and penalty-free line of work on Stiefel manifold optimization.

  • Xiao, N., Liu, X., and Yuan, Y. (2022). A class of smooth exact penalty function methods for optimization problems with orthogonality constraints. Optimization Methods and Software, 37(4), 1205-1241.

  • Xiao, N., Liu, X., and Yuan, Y. (2022). Exact penalty function for \(\ell_{2,1}\) norm minimization over the Stiefel manifold. SIAM Journal on Optimization, 31(4), 3097-3126.

  • Xiao, N., Liu, X., and Toh, K.-C. (2023). Dissolving constraints for Riemannian optimization. Mathematics of Operations Research.

  • Barzilai, J. and Borwein, J. M. (1988). Two-point step size gradient methods. IMA Journal of Numerical Analysis, 8(1), 141-148.

  • Edelman, A., Arias, T. A., and Smith, S. T. (1998). The geometry of algorithms with orthogonality constraints. SIAM Journal on Matrix Analysis and Applications, 20(2), 303-353.

  • Absil, P.-A., Mahony, R., and Sepulchre, R. (2008). Optimization Algorithms on Matrix Manifolds. Princeton University Press.

  • Golub, G. H. and Van Loan, C. F. (2013). Matrix Computations, 4th edition. Johns Hopkins University Press. Chapter 8 covers the cyclic Jacobi eigenvalue algorithm used for the polar factor.

Provenance

The algorithm ported here originates in the STOP toolbox by Nachuan Xiao, Lei Wang, Bin Gao, Xin Liu and Ya-xiang Yuan, distributed at https://stmopt.gitee.io/. smopt re-implements its numerics in Fortran 77 behind the same solver interface.