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- Title
Error bound and exact penalty method for optimization problems with nonnegative orthogonal constraint.
- Authors
Qian, Yitian; Pan, Shaohua; Xiao, Lianghai
- Abstract
This paper is concerned with a class of optimization problems with the non-negative orthogonal constraint, in which the objective function is |$L$| -smooth on an open set containing the Stiefel manifold |$\textrm {St}(n,r)$|. We derive a locally Lipschitzian error bound for the feasible points without zero rows when |$n>r>1$| , and when |$n>r=1$| or |$n=r$| achieve a global Lipschitzian error bound. Then, we show that the penalty problem induced by the elementwise |$\ell _1$| -norm distance to the non-negative cone is a global exact penalty, and so is the one induced by its Moreau envelope under a lower second-order calmness of the objective function. A practical penalty algorithm is developed by solving approximately a series of smooth penalty problems with a retraction-based nonmonotone line-search proximal gradient method, and any cluster point of the generated sequence is shown to be a stationary point of the original problem. Numerical comparisons with the ALM [Wen, Z. W. & Yin, W. T. (2013, A feasible method for optimization with orthogonality constraints. Math. Programming , 142 , 397–434),] and the exact penalty method [Jiang, B. Meng, X. Wen, Z. W. & Chen, X. J. (2022, An exact penalty approach for optimization with nonnegative orthogonality constraints. Math. Programming. https://doi.org/10.1007/s10107-022-01794-8)] indicate that our penalty method has an advantage in terms of the quality of solutions despite taking a little more time.
- Publication
IMA Journal of Numerical Analysis, 2024, Vol 44, Issue 1, p120
- ISSN
0272-4979
- Publication type
Article
- DOI
10.1093/imanum/drac084