- A Framework of Constraint Preserving Update Schemes for Optimization on Stiefel Manifold Bo Jiang (jiangbolsec.cc.ac.cn) Yu-Hong Dai (dyhlsec.cc.ac.cn) Abstract: This paper considers optimization problems on the Stiefel manifold $X^TX=I_p$, 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 $X^T$. 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 (BB)-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 significantly better than some state-of-the-art algorithms for the nearest low-rank correlation matrix problem and is considerably competitive with the widely used SCF iteration for the Kohn-Sham total energy minimization. Keywords: Stiefel manifold, orthogonality constraint, sphere constraint, range space, null space, Barzilai-Borwein-like method, feasible, global convergence, adaptive nonmonotone line search, low-rank correlation matrix, Kohn-Sham total energy minimization, heterogeneous quadratic functions Category 1: Nonlinear Optimization Category 2: Applications -- Science and Engineering Citation: @TECHREPORT{StiManOpt-Jiang-Dai2012, author = {Jiang, Bo and Dai, Yu-Hong}, title = {A Framework of Constraint Preserving Update Schemes for Optimization on Stiefel Manifold}, institution = {Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences}, year = {2012} } Download: [PDF]Entry Submitted: 01/01/2013Entry Accepted: 01/01/2013Entry Last Modified: 12/25/2013Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.