NDLI: A framework of constraint preserving update schemes for optimization on Stiefel manifold

Content Provider	Springer Nature Link
Author	Dai, Yu Hong Jiang, Bo
Copyright Year	2014
Abstract	This paper considers optimization problems on the Stiefel manifold $$X^{\mathsf{T}}X=I_p$$ , where $$X\in \mathbb {R}^{n \times p}$$ is the variable and $$I_p$$ is the $$p$$ -by- $$p$$ identity matrix. A framework of constraint preserving update schemes is proposed by decomposing each feasible point into the range space of $$X$$ and the null space of $$X^{\mathsf{T}}$$ . While this general framework can unify many existing schemes, a new update scheme with low complexity cost is also discovered. Then we study a feasible Barzilai–Borwein-like method under the new update scheme. The global convergence of the method is established with an adaptive nonmonotone line search. The numerical tests on the nearest low-rank correlation matrix problem, the Kohn–Sham total energy minimization and a specific problem from statistics demonstrate the efficiency of the new method. In particular, the new method performs remarkably well for the nearest low-rank correlation matrix problem in terms of speed and solution quality and is considerably competitive with the widely used SCF iteration for the Kohn–Sham total energy minimization.
Ending Page	575
Page Count	41
Starting Page	535
File Format	PDF
ISSN	00255610
e-ISSN	14364646
Journal	Mathematical Programming
Issue Number	2
Volume Number	153
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2014-09-17
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Heterogeneous quadratic functions Sphere constraint Null space Theoretical, Mathematical and Computational Physics Range space Barzilai–Borwein-like method Kohn–Sham total energy minimization Adaptive nonmonotone line search Orthogonality constraint Feasible Mathematical Methods in Physics Stiefel manifold Calculus of Variations and Optimal Control; Optimization Mathematics of Computing Numerical Analysis Mathematical programming methods Nonlinear programming Combinatorics Low-rank correlation matrix
Content Type	Text
Resource Type	Article
Subject	Mathematics Software

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