How to use matlab to find an optimized matrix with regarding to minimization of a norm?

I’m trying to solve a minimization problem whose purpose is to optimize a matrix, where the multiplication with a vector is close to another vector. But I have some problems to execute it by using Matlab. The problem is illustrated as follows:

The summation goes to 15, because there are 15 unique vector pairs of u and b. Which function in Matlab can I use to solve this issue? I thought “fmincon” is an option, but I could not get the input parameters correct

X0=ones(1,9)/3;
Aeq=
beq=ones(3,1);
ub=2*ones(1,9);
lb=-2*ones(1,6);
x = fmincon(????,x0,[],[],Aeq,beq,lb,ub);

[Torsten]

Which vector norm do you mean by

|Q*u_i - b_i|

?

Best wishes

Torsten.

[Flora]

Sorry, forgot a 2

I mean the 2-nom.

Answer 1

[Torsten]

Try

function main
Aeq=[1 1 1 0 0 0 0 0 0 
     0 0 0 1 1 1 0 0 0
     0 0 0 0 0 0 1 1 1];
beq=ones(3,1);
lb=-2*ones(1,9);
ub=2*ones(1,9);
U=horzcat(u1,u2,u3,u4,u5,u6,u7,u8,u9,u10,u11,u12,u13,u14,u15);
B=horzcat(b1,b2,b3,b4,b5,b6,b7,b8,b9,b10,b11,b12,b13,b14,b15);
x0=ones(1,9)/3;
fun=@(x)(norm([x(1) x(2) x(3);x(4) x(5) x(6);x(7) x(8) x(9)]*U-B,'fro'));
x=fmincon(fun,x0,[],[],Aeq,beq,lb,ub);

Best wishes

Torsten.

[Flora]

Thanks, it works.

The result is not that optimal as I wished, but probably I have to change the criteria. Or need to find another way to do this. For different vector pairs the difference between vector Q*ui and bi is more than 15 for a vector element. This is also observed for the matrix Q, when it is optimized for each vector pair. For each element the value could vary with 1.