An exact penalty approach for optimization with nonnegative orthogonality constraints
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10107_2022_1794_ReferencePDF.pdf
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1.68 MB
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df23537f824ee2b4017fda60b9a95c08
Author(s)
Jiang, Bo
Meng, Xiang
Wen, Zaiwen
Chen, Xiaojun
Date Issued
March 25, 2022
Publisher
Springer Berlin Heidelberg
Citation
Jiang, Bo, Meng, Xiang, Wen, Zaiwen and Chen, Xiaojun. 2022. "An exact penalty approach for optimization with nonnegative orthogonality constraints."
Version
Author's final manuscript
Abstract
Abstract
Optimization with nonnegative orthogonality constraints has wide applications in machine learning and data sciences. It is NP-hard due to some combinatorial properties of the constraints. We first propose an equivalent optimization formulation with nonnegative and multiple spherical constraints and an additional single nonlinear constraint. Various constraint qualifications, the first- and second-order optimality conditions of the equivalent formulation are discussed. By establishing a local error bound of the feasible set, we design a class of (smooth) exact penalty models via keeping the nonnegative and multiple spherical constraints. The penalty models are exact if the penalty parameter is sufficiently large but finite. A practical penalty algorithm with postprocessing is then developed to approximately solve a series of subproblems with nonnegative and multiple spherical constraints. We study the asymptotic convergence and establish that any limit point is a weakly stationary point of the original problem and becomes a stationary point under some additional mild conditions. Extensive numerical results on the problem of computing the orthogonal projection onto nonnegative orthogonality constraints, the orthogonal nonnegative matrix factorization problems and the K-indicators model show the effectiveness of our proposed approach.
MIT Department
Massachusetts Institute of Technology. Operations Research Center
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1007/s10107-022-01794-8
