README.m

% A feasible method for optimization with orthogonality constraints
%
% -------------------------------------------------------------------------
% 1. Problems and solvers
%
% The package contains codes for the following two problems:
% (1) 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).
% 
% (2) min F(X), S.t., X'*X = I_k, where X \in R^{n,k}
%
%     Solver: OptStiefelGBB.m
%
%     Solver demo: test_eig_rand_demo.m, computing leading eigenvalues
%
%
% 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  Tr(CX), s.t., X_{ii}=1, X \succeq 0
%
% - SDP: \min Tr(CX), s.t., 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, \; rank(X) \le p.
%
% - The Bose–Einstein condensates (BEC) problem
%
% - Linear eigenvalue problems: 
%   \min Tr(X^{\top}AX), s.t., X^{\top}X =I 
%
% - The electronic structure calculation: 
%   the Kohn-Sham total energy minimization 
%   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
% -------------------------------------------------------------------------
% 2. Reference
%
%  (1) Zaiwen Wen and Wotao Yin. A feasible method for optimization with 
%   orthogonality constraints. Mathematical Programming (2013): 397-434.
%
%  (2) Jiang Hu, Andre Milzarek, Zaiwen Wen, Yaxiang Yuan. Adaptive 
%   Regularized Newton Method for Riemannian Optimization. 
%   SIAM Journal on Scientific Computing
% 
%  (3) 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.
%
%  (4) 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.
%
%  (5) 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
%
%  (6) 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
%
%
% -------------------------------------------------------------------------
% 3. 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
%
% Enjoy!
% Zaiwen and Wotao
%
% -------------------------------------------------------------------------
% 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/>