Nonmonotone inexact restoration approach for minimization with orthogonality constraints


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.

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D.S.G thanks CAPES/Print Processes 88881.310538/2018-01 and 88887.465828/2019-00 which allow him to present part of this work at ICCOPT 2019 and supported his period as visiting professor at École Polytechnique where part of this work was carried out. We also thank anonymous referees whose comments helped to improve the quality of this work.


J.B.F., D.S.G, and F.S.V.B are grateful to CNPq by the financial support (Grants n. 421386/2016-9 and 308523/2017-2). L.L.T.P would like to thank CAPES for the scholarship during her Ph.D. at Universidade Federal de Santa Catarina. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.

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Correspondence to Juliano B. Francisco.

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A Tables of numerical results

This appendix gathers the tables with numerical results for the experiments discussed in Sections 6.36.4, and 6.5. The symbol “–” means that the solver could not achieve \(\|\tilde {P}_{S_{k}} (\nabla f(Y_{k})) \| < \varepsilon = 10^{-6}\), using less than IT\(_{\max \limits }=2,000\) iterations and FE\(_{\max \limits }=4,000\) function evaluations.

Only for test problems of Table 7, the maximum number of iterations was increased to IT\(_{\max \limits }=5,000\) and function evaluations to IT\(_{\max \limits }=10,000\). Moreover, for this last case, we set M = 0 and \(\delta _{CG} = 10^{-4} \| P_{S_{0}} (Y_{0}) \|\) in IRNOM.

Table 1 CPU time (seconds) in random instances of LEP
Table 2 CPU time (seconds) in random instances of NEP with α = 1
Table 3 CPU time (seconds) in random instances of NEP with α = 10
Table 4 CPU time (seconds) in random instances of NEP with α = 100
Table 5 CPU time in random instances of OPP (uniformly distributed singular values)
Table 6 CPU time in random instances of OPP (equally spaced singular values)
Table 7 CPU time in random instances of OPP (clustered eigenvalues)

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Francisco, J.B., Gonçalves, D.S., Bazán, F.S.V. et al. Nonmonotone inexact restoration approach for minimization with orthogonality constraints. Numer Algor (2020).

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  • Inexact restoration
  • Orthogonality constraints
  • Cayley transform
  • Conjugate gradient

Mathematics subject classification (2010)

  • 49Q99
  • 65K05
  • 90C22
  • 90C26
  • 90C30