Optimization of matrix with constraints

I have written a prototype function to minimize a matrix equation with code below. The input N is an even number, A and H0 are 2Nx2N matrices with real entries but complex eigenvalues.

I found the documentation of Optimization Toolbox not easy to follow, so I listed all the constraints and the equation to be minimized in the code to give my best to anyone whom would help me.

function H=extractor(N,A,H0)
A=real(A);
T0=A(1:2,1:2);
LHP = optimvar('LHP',N-1,4,'LowerBound',-2,'UpperBound',2);
RHP = optimvar('RHP',N-1,4,'LowerBound',-2,'UpperBound',2);
for q=1:N/2
    if q~=N/2
        UPPER(1:2,2*q-1:2*q)=[LHP(q,2),LHP(q,4);LHP(q,1),LHP(q,3)]+[RHP(N-q,2),RHP(N-q,4);RHP(N-q,1),RHP(N-q,3)];
        LOWER(2*q-1:2*q,1:2)=[RHP(q,2),RHP(q,1);RHP(q,4),RHP(q,3)]+[LHP(N-q,2),LHP(N-q,1);LHP(N-q,4),LHP(N-q,3)];
    elseif q==N/2
        UPPER(1:2,2*q-1:2*q)=[LHP(q,2),LHP(q,4);LHP(q,1),LHP(q,3)];
        LOWER(2*q-1:2*q,1:2)=[RHP(q,2),RHP(q,1);RHP(q,4),RHP(q,3)];
    end
end
for q=1:N/2-1
    DECAYL(q,1:4)=LHP(q,1:4)-LHP(N-q,1:4);
    DECAYR(q,1:4)=RHP(q,1:4)-RHP(N-q,1:4);
end
for i=1:N
    for j=1:N
        d=abs(j-i);
        if i<j %upper triangle 
            if d>N/2 % higher order
                H(2*i-1:2*i,2*j-1:2*j)=[LHP(d,2),LHP(d,1);LHP(d,4),LHP(d,3)];
            elseif d==N/2 % central order
                H(2*i-1:2*i,2*j-1:2*j)=[LHP(d,2),LHP(d,4);LHP(d,1),LHP(d,3)];
            elseif d<N/2 % lower order
                H(2*i-1:2*i,2*j-1:2*j)=[LHP(d,2),LHP(d,4);LHP(d,1),LHP(d,3)];
            end
        elseif i>j %lower triangle
            if d>N/2 % higher order
                H(2*i-1:2*i,2*j-1:2*j)=[RHP(d,2),RHP(d,4);RHP(d,1),RHP(d,3)];
            elseif d==N/2 % central order
                H(2*i-1:2*i,2*j-1:2*j)=[RHP(d,2),RHP(d,1);RHP(d,4),RHP(d,3)];
            elseif d<N/2 % lower order
                H(2*i-1:2*i,2*j-1:2*j)=[RHP(d,2),RHP(d,1);RHP(d,4),RHP(d,3)];
            end
        elseif i==j %diagonal
                H(2*i-1:2*i,2*j-1:2*j)=T0;
        end   
    end
end
%constraint1: A(1:2,3:N+2)==UPPER && A(3:N+2,1:2)==LOWER;
%constraint2: abs(DECAYL)>0 && abs(DECAYR)>0
%minimize this equation: sum(abs(sort(real(eig(H0)))-sort(real(eig(H)))))+sum(abs(sort(imag(eig(H0)))-sort(imag(eig(H)))))

Answer 1

[Akshat Dalal]

Hi Zhenghao,

I understand that you want to solve an optimization problem. You can use the ‘Optimize’ function provided in the Optimization Toolbox. You can use the following approach to solve your problem:

• In your script, insert an ‘Optimize Live Editor task’ by clicking on the Insert tab and then, in the Code section, selecting Task > Optimize.
• Select the objective and Constraints type as per your data and then select the recommended solver.
• Define the objective function for your problem. The inputs would be ‘H’ and ‘H0’ and the output would the value of the equation you want to minimize. You can refer the attached examples to see how to pass multiple inputs.
• Define the constraints function for your problem. The inputs would be ‘A’, ‘N’, ‘UPPER’, ‘LOWER’, ‘DECAYL’, and ‘DECAYR’. Declare the inequality constraints in the ‘c’ structure and the equality constraints in the ‘ceq’ structure. For example,

    c(1) = -abs(DECAYL);
    ceq = [A(1:2,3:N+2) - UPPER, A(3:N+2,1:2) - LOWER];

You can refer the attached examples to learn more about constraint functions.

• Select the objective function and constraint function you wrote in previous steps as inputs to the problem data, and then run the solver to get the desired results.

You could refer the following documentation to learn more about the ‘Optimize’ function - https://in.mathworks.com/help/releases/R2021b/optim/ug/optimize.html

You could also refer to the following example on using the ‘Optimize’ function –

1. https://in.mathworks.com/help/releases/R2021b/optim/ug/get-started-optimize-live-editor-task.html
2. https://in.mathworks.com/help/releases/R2021b/optim/ug/example-nonlinear-constrained-minimization.html
3. https://in.mathworks.com/help/releases/R2021b/optim/ug/nonlinear-equality-and-inequality-constraints.html