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

fminconmatrixnonlinearoptimizationvector

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

Best Answer

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
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