Journal of Applied and Numerical Optimization

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DOI: 10.23952/jano.7.2025.1.06
Volume 7, Issue 1, 1 April 2025, Pages 85-95

 

Abstract. In this paper, we introduce a two-stage algorithm for a large-scale least squares problem with a simplex constraint. The basic idea is to use the accelerated proximal gradient (APG) method to generate a relatively good precision solution for the primal problem, and take this point as the initial point of the second stage. In the second stage, we use the semismooth Newton (SsN) based on the augmented Lagrangian method (ALM) to solve the dual problem, that is, the ALM is used to solve the dual problem, and the SsN method is used to solve the subproblem. It is worth mentioning that, due to the piecewise linearity of the subproblem, the accuracy of the SsN method decreases relatively slowly in the early stage, and the APG method can effectively reduce the time required in this stage. In addition, we also prove the convergence and convergence rate of the proposed algorithm under certain conditions. Numerical experiments demonstrate that our algorithm outperforms the current state-of-the-art algorithms for the least squares problem with a simplex constraint.

 

How to Cite this Article:
C. Xiang, C. Wang, P. Shi, P. Tang, A. Xu, Solving a large-scale least squares problem with a simplex constraint by a two-stage algorithm, J. Appl. Numer. Optim. 7 (2025), 85-95.