Using the given code, how can I optimize a matrix to minimize a cost function?

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Given S1, (1,K) vector, I want to optimize B (N,M) matrix to minimize the following cost function:
Subject to:
Where:
S2 is (1,K) vector and a function of matrix B.
S2 can be calculated after optimizing matrix B using the following system parameters and equations:
clc;
clear;
% Given system parameters:
N = 2;
K = 4;
M=2;
C_l = 4;
H = [0.1185 0.2811; 0.3550 0.8224; 0.3260 0.9644; 0.5333 0.6083]; % 4*2 matrix
A = [-2 1; -1 1]; % 2*2 matrix
C = [7 -3; 7 -3; -2 1; -2 1]; % 4*2 matrix
P = [25000000 0; 0 25000000]; % 4*4 matrix
S1 = [3.1683 3.1686 1.8716 1.8898]; % 1*4 vector
S2 = zeros(1,K); % intial value
B = zeros(N,M); % intial value
% How can we optimize the value of B matrix to achieve our goal?
%calculate S2 from B and the other given inputs
for j=1:1:N
d(j) = (B(j,:)*P*B(j,:)')/((2^(2*C_l))-(norm(A(:,j))^2));
end
D_d = diag(d);
for i=1:1:K
V_d(i)=C(i,:)*P*B'*H(i,:)'*inv(1+H(i,:)*(A'*D_d*A+B*P*B')*H(i,:)');
sigma_d(i)=norm((V_d(i)*H(i,:)*B-C(i,:))*(P^(1/2)))^2+(V_d(i)^2)*(1+H(i,:)*A'*D_d*A*H(i,:)');
S2(i)=0.5*log2((P(1,1))/sigma_d(:,i));
end
  6 Comments

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Accepted Answer

Matt J
Matt J on 18 Nov 2020
Well, the way you would approach it is using fmincon as below. Using your initial guess of zeros(N,M), fmincon is unable to find a solution so, assuming a feasible solution does exist, you will need to choose a less arbitrary initial guess. I urge you to get familiar with the fmincon documentation.
B_Optimal=doOptimization
Converged to an infeasible point. fmincon stopped because the size of the current step is less than the value of the step size tolerance but constraints are not satisfied to within the value of the constraint tolerance.
B_Optimal = 2×2
46.2078 -19.7949 8.1964 -3.4964
function B_Optimal=doOptimization
% Given system parameters:
N = 2;
K = 4;
M=2;
C_l = 4;
H = [0.1185 0.2811; 0.3550 0.8224; 0.3260 0.9644; 0.5333 0.6083]; % 4*2 matrix
A = [-2 1; -1 1]; % 2*2 matrix
C = [7 -3; 7 -3; -2 1; -2 1]; % 4*2 matrix
P = [25000000 0; 0 25000000]; % 4*4 matrix
S1 = [3.1683 3.1686 1.8716 1.8898]; % 1*4 vector
Binitial = zeros(N,M); % intial value
% How can we optimize the value of B matrix to achieve our goal?
B_Optimal=fmincon( @(B)norm( S2(B)-S1).^2, Binitial,[],[],[],[],[],[],@nlcon);
function [c,ceq]=nlcon(B)
ceq=[];
c=S1-S2(B);
end
function out=S2(B)
%calculate S2 from B and the other given inputs
for j=N:-1:1
d(j) = (B(j,:)*P*B(j,:)')/((2^(2*C_l))-(norm(A(:,j))^2));
end
D_d = diag(d);
for i=K:-1:1
V_d(i)=C(i,:)*P*B'*H(i,:)'*inv(1+H(i,:)*(A'*D_d*A+B*P*B')*H(i,:)');
sigma_d(i)=norm((V_d(i)*H(i,:)*B-C(i,:))*(P^(1/2)))^2+(V_d(i)^2)*(1+H(i,:)*A'*D_d*A*H(i,:)');
out(i)=0.5*log2((P(1,1))/sigma_d(:,i));
end
end
end
  2 Comments
Rik
Rik on 18 Nov 2020
How certain is it that such a B actually exists? It is not my field, so algebraically determining is such a matrix is possible in the first place is beyond my skills. However, I don't see anything to indicate that it would be impossible to find this out.

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