智能反射面—流形优化

使用Manopt工具箱适合优化最小化问题，如果你的优化问题是最大化问题，那么需要将其转换为最小化问题然后使用Manopt工具箱求解。

具体安装过程

Matlab添加Manopt - 知乎 (zhihu.com)

优化问题

clc,clear;
close all;
s=rng(1);%rand seed
N=10;
GR_num=1e3;% number, should be large enough
A = rand(N)+1i*rand(N);
A = (A*A'); % semidefinite matrix

% SDR
cvx_begin 
variable X(N,N) hermitian semidefinite
minimize (real(trace(A*X)))
subject to
    diag(X)==ones(N,1);
 cvx_end

 
% gassian random
obj = zeros(GR_num,1);
v=zeros(N,GR_num);
[V1,D1]=eig(X);
for ii=1:GR_num
    v(:,ii) = V1*D1^(1/2)*sqrt(1/2)*(randn(N,1) + 1i*randn(N,1));
    v(:,ii) = exp(1i*angle(v(:,ii)));% guarantee constant modulus
    obj(ii)=real(trace(A*(v(:,ii)*v(:,ii)')));
end
[~,idx] = min(obj);
v_opt = v(:,idx);
% check constant modulus
abs(v_opt)
% check optimal value
real(cvx_optval)

real(trace(v_opt'*A*v_opt ))



%采用方法二进行求解
% Create the problem structure.
v_init = ones(N,1);
manifold = complexcirclefactory(N);
problem.M = manifold;
options.verbosity=0;
% Define the problem cost function and its Euclidean gradient.
problem.cost  = @(x) trace(x'*A*x);
problem.egrad = @(x) 2*A*x;                   % notice the 'e' in 'egrad' for Euclidean

% Solve.
[x, xcost, info, options] = conjugategradient(problem,v_init,options);
 
% Display some statistics.
figure;
semilogy([info.iter], [info.gradnorm], '.-','linewidth',1);
xlabel('Iteration number');
ylabel('Norm of the gradient of f');
grid on

figure;
semilogy([info.iter], real([info.cost]), 'b.-','linewidth',1.2);
xlabel('Iteration number');
ylabel('cost');
grid on
set(gca,'fontsize',11);

real(x'*A*x)