is it possible to optimize a function that have a matrix output with fmincon ?

I want to minimize a function which the output is a matrix with an equality constraint. Matlab displays the following error message:

 Error using fmincon (line 289) Aeq must have 440 column(s).
 Error in objective2 (line 52) x = fmincon(fun,x0,A,b,Aeq,beq)

Below I write down my the example to solve:

A1=[-3 0;
    0 -4];
B=[2;
   1];

C=[1 0;
   0.9 1;];

Ed=[0.7 0.7;
    0.3 0.3];

Ef=[1;
    0];

x0=zeros(22,20);

Hf=[zeros(2,11);
    C*Ef zeros(2,10);
    C*A1*Ef C*Ef zeros(2,9);
    C*A1^2*Ef C*A1*Ef C*Ef zeros(2,8);
    C*A1^3*Ef C*A1^2*Ef C*A1*Ef  C*Ef zeros(2,7);
    C*A1^4*Ef C*A1^3*Ef C*A1^2*Ef C*A1*Ef C*Ef zeros(2,6);
    C*A1^5*Ef C*A1^4*Ef C*A1^3*Ef C*A1^2*Ef C*A1*Ef C*Ef zeros(2,5);
    C*A1^6*Ef C*A1^5*Ef C*A1^4*Ef C*A1^3*Ef C*A1^2*Ef C*A1*Ef C*Ef zeros(2,4);
    C*A1^7*Ef C*A1^6*Ef C*A1^5*Ef C*A1^4*Ef C*A1^3*Ef C*A1^2*Ef C*A1*Ef C*Ef zeros(2,3);
    C*A1^8*Ef C*A1^7*Ef C*A1^6*Ef C*A1^5*Ef C*A1^4*Ef C*A1^3*Ef C*A1^2*Ef C*A1*Ef C*Ef zeros(2,2);
    C*A1^9*Ef C*A1^8*Ef C*A1^7*Ef C*A1^6*Ef C*A1^5*Ef C*A1^4*Ef C*A1^3*Ef C*A1^2*Ef C*A1*Ef C*Ef zeros(2,1); 
    ];

Hd=[zeros(2,22);
    C*Ed zeros(2,20);
    C*A1*Ed C*Ed zeros(2,18);
    C*A1^2*Ed C*A1*Ed C*Ed zeros(2,16);
    C*A1^3*Ed C*A1^2*Ed C*A1*Ed  C*Ed zeros(2,14);
    C*A1^4*Ed C*A1^3*Ed C*A1^2*Ed C*A1*Ed C*Ed zeros(2,12);
    C*A1^5*Ed C*A1^4*Ed C*A1^3*Ed C*A1^2*Ed C*A1*Ed C*Ed zeros(2,10);
    C*A1^6*Ed C*A1^5*Ed C*A1^4*Ed C*A1^3*Ed C*A1^2*Ed C*A1*Ed C*Ed zeros(2,8);
    C*A1^7*Ed  C*A1^6*Ed C*A1^5*Ed C*A1^4*Ed C*A1^3*Ed C*A1^2*Ed C*A1*Ed C*Ed zeros(2,6);
    C*A1^8*Ed C*A1^7*Ed  C*A1^6*Ed C*A1^5*Ed C*A1^4*Ed C*A1^3*Ed C*A1^2*Ed C*A1*Ed C*Ed zeros(2,4);
    C*A1^9*Ed  C*A1^8*Ed C*A1^7*Ed  C*A1^6*Ed C*A1^5*Ed C*A1^4*Ed C*A1^3*Ed C*A1^2*Ed C*A1*Ed C*Ed zeros(2,2); 
    ];

Ho=[C;C*A1;C*A1^2;C*A1^3;C*A1^4;C*A1^5;C*A1^6;C*A1^7;C*A1^8;C*A1^9;C*A1^10];

%--------------------------------------------------------------------------
fun = @(x)(x*Hd*Hd'*x')/(x*Hf*Hf'*x');

A = [];
b = [];
Aeq = Ho';
beq = 0;

x = fmincon(fun,x0,A,b,Aeq,beq)
x

[Torsten]

We need to see your MATLAB code to say what's going wrong.

From the error message it seems that you are trying to solve a problem with 440 unknowns, but that the matrix to define the equality constraint has more or less than 440 columns (which is not possible).

Best wishes

Torsten.

[Marya]

Yes, the variable to optimise is a matrix of size (20x22), so we have 440 unknowns. But here we have the following constaint: x.Ho=0 (x and Ho are orthogonal) or in fmincon the equality constraint must be in the form Aeq.x=beq, I apply the transpose for (x.Ho=0)' ,we obtain Ho'.x'=0 and we return to the form Aeq.y=beq with Aeq=Ho' y=x' .

we find enclosed the code :

A1=[-3 0;
    0 -4];

B=[2;
   1];

C=[1 0;
   0.9 1;];

Ed=[0.7 0.7;
    0.3 0.3];

Ef=[1;
    0];

x0=zeros(22,20);

Hf=[zeros(2,11);
    C*Ef zeros(2,10);
    C*A1*Ef C*Ef zeros(2,9);
    C*A1^2*Ef C*A1*Ef C*Ef zeros(2,8);
    C*A1^3*Ef C*A1^2*Ef C*A1*Ef  C*Ef zeros(2,7);
    C*A1^4*Ef C*A1^3*Ef C*A1^2*Ef C*A1*Ef C*Ef zeros(2,6);
    C*A1^5*Ef C*A1^4*Ef C*A1^3*Ef C*A1^2*Ef C*A1*Ef C*Ef zeros(2,5);
    C*A1^6*Ef C*A1^5*Ef C*A1^4*Ef C*A1^3*Ef C*A1^2*Ef C*A1*Ef C*Ef zeros(2,4);
    C*A1^7*Ef C*A1^6*Ef C*A1^5*Ef C*A1^4*Ef C*A1^3*Ef C*A1^2*Ef C*A1*Ef C*Ef zeros(2,3);
    C*A1^8*Ef C*A1^7*Ef C*A1^6*Ef C*A1^5*Ef C*A1^4*Ef C*A1^3*Ef C*A1^2*Ef C*A1*Ef C*Ef zeros(2,2);
    C*A1^9*Ef C*A1^8*Ef C*A1^7*Ef C*A1^6*Ef C*A1^5*Ef C*A1^4*Ef C*A1^3*Ef C*A1^2*Ef C*A1*Ef C*Ef zeros(2,1); 
    ];

Hd=[zeros(2,22);
    C*Ed zeros(2,20);
    C*A1*Ed C*Ed zeros(2,18);
    C*A1^2*Ed C*A1*Ed C*Ed zeros(2,16);
    C*A1^3*Ed C*A1^2*Ed C*A1*Ed  C*Ed zeros(2,14);
    C*A1^4*Ed C*A1^3*Ed C*A1^2*Ed C*A1*Ed C*Ed zeros(2,12);
    C*A1^5*Ed C*A1^4*Ed C*A1^3*Ed C*A1^2*Ed C*A1*Ed C*Ed zeros(2,10);
    C*A1^6*Ed C*A1^5*Ed C*A1^4*Ed C*A1^3*Ed C*A1^2*Ed C*A1*Ed C*Ed zeros(2,8);
    C*A1^7*Ed  C*A1^6*Ed C*A1^5*Ed C*A1^4*Ed C*A1^3*Ed C*A1^2*Ed C*A1*Ed C*Ed zeros(2,6);
    C*A1^8*Ed C*A1^7*Ed  C*A1^6*Ed C*A1^5*Ed C*A1^4*Ed C*A1^3*Ed C*A1^2*Ed C*A1*Ed C*Ed zeros(2,4);
    C*A1^9*Ed  C*A1^8*Ed C*A1^7*Ed  C*A1^6*Ed C*A1^5*Ed C*A1^4*Ed C*A1^3*Ed C*A1^2*Ed C*A1*Ed C*Ed zeros(2,2); 
    ];

Ho=[C;C*A1;C*A1^2;C*A1^3;C*A1^4;C*A1^5;C*A1^6;C*A1^7;C*A1^8;C*A1^9;C*A1^10];

%----------------------------------

fun = @(x)(x*Hd*Hd'*x')/(x*Hf*Hf'*x');

A = [];
b = [];
Aeq = Ho';
beq = zeros(2,20);

x = fmincon(fun,x0,A,b,Aeq,beq)
x

[Torsten]

x must be defined as a vector, not as a matrix.

How else can you expect that "fun" makes sense for x being a matrix ?

Best wishes

Torsten.