OptM beta 1.0

A feasible method for optimization with orthogonality constraints

Problems and solvers

The package contains codes for the following three problems:

• $\min F(X), s.t., ||X_i||_2 = 1$

  Solver: OptManiMulitBallGBB.m Solver demo: Test_maxcut_demo.m, solving the max-cut problem

  The constraints can even be a single sphere: $||X||_F = 1$

  Solver demo: GPE_SP1d_Func.m, solving the BEC problem (the data is not provided).

• $\min F(X), S.t., X^{\top} X = I_k$, where $X \in R^{n,k}$

  Solver: OptStiefelGBB.m
  Solver demo: test_eig_rand_demo.m, computing leading eigenvalues

  QAP: TestQAPLIB_orth_fixmu_improve_selected.m. To run the code, first download qaplib from (https://www.opt.math.tugraz.at/qaplib/), then run GenQAPLIB.m to generate the data matrices in the matlab format.

Applications have been solved by these solvers:

• Homogeneous polynomial optimization problems with multiple spherical constraints： $$\max ; \sum_{1\le i\le n_1, 1\le j \le n_2, 1 \le k \le n_3, 1\le l \le n_4} a_{ijkl} x_i y_j z_k w_l ; s.t., |x|_2 = |y|_2 = |z|_2 = |w|2= 1,$$ where $A = (a{ijkl})$ is a fourth-order tensor of size $n\times n \times n\times n$.

• Maxcut SDP: $\min \mathrm{Tr}(CX), s.t., X_{ii}=1, X \succeq 0$

• SDP: $\min \mathrm{Tr}(CX), s.t., \mathrm{Tr}(X)=1, X \succeq 0 $

• Low-Rank Nearest Correlation Estimation: $ \min_{ X \succeq 0} ; \frac{1}{2} | H \odot (X - C) |F^2, ; X{ii} = 1, ; i = 1, \ldots, n, ; \mathrm{rank}(X) \le p.$

• The Bose–Einstein condensates (BEC) problem

• Linear eigenvalue problems: $\min \mathrm{Tr}(X^{\top}AX), s.t., X^{\top}X =I $

• The electronic structure calculation: the Kohn-Sham total energy minimization and the Hartree-Fock total energy minimization

• Quadratic assignment problem

• Harmonic energy minimization

  For more general problems and solvers, see: Adaptive Regularized Newton Method for Riemannian Optimization https://github.com/wenstone/ARNT

References

• Zaiwen Wen and Wotao Yin. A feasible method for optimization with orthogonality constraints. Mathematical Programming (2013): 397-434.

• Jiang Hu, Andre Milzarek, Zaiwen Wen, Yaxiang Yuan. Adaptive Regularized Newton Method for Riemannian Optimization. SIAM Journal on Scientific Computing

• Zaiwen Wen, Andre Milzarek, Michael Ulbrich and Hongchao Zhang, Adaptive regularized self-consistent field iteration with exact Hessian for electronic structure calculation. SIAM Journal on Scientific Computing (2013), A1299-A1324.

• Xinming Wu, Zaiwen Wen, and Weizhu Bao. A regularized Newton method for computing ground states of Bose–Einstein condensates. Journal of Scientific Computing (2017): 303-329.

• X. Zhang, J. Zhu, Z. Wen, A. Zhou, Gradient-type Optimization Methods for Electronic Structure Calculation, SIAM Journal on Scientific Computing, Vol. 36, No. 3 (2014), pp. C265-C289

• R. Lai, Z. Wen, W. Yin, X. Gu, L. Lui, Folding-Free Global Conformal Mapping for Genus-0 Surfaces by Harmonic Energy Minimization, Journal of Scientfic Computing, 58(2014), 705-725

The Authors

We hope that the package is useful for your application. If you have any bug reports or comments, please feel free to email one of the toolbox authors:

• Zaiwen Wen, wendouble@gmail.com
• Wotao Yin, wotao.yin@gmail.com

Copyright

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Copyright (C) 2018, Zaiwen Wen and Wotao Yin Copyright (C) 2010, Zaiwen Wen and Wotao Yin

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program. If not, see http://www.gnu.org/licenses/