How to automatically constrain some of the unknown values in an optimization problem?

I have an optimization problem with several unknown values (x) which is a part of an array:

A=[0 0 x(1) x(2) 0 0 0 x(3) x(4) x(5) 0 0 0 x(6) 0 0 x(7) x(8)];

The x-values are to be optimized but each x-'group', which is separated by the zeros, are to be equal to each other. ex.: x(1)=x(2) and x(3)=x(4)=x(5) and so on. x(6) is not to be equal to another x because it is not in a 'group' with other x's, therfore x(6) will not be constrained.

I am supposed to write this constraint as ceq(x)=[..] in a constraint function.

Is there any automatic coding that can constrain these x-values in the way i explained?

Answer 1

[Torsten]

If you know in advance which elements of the x-vector are grouped together, you can simply use Aeq and beq to define the equality constraints. No need to use the nonlinear constraints option in ceq.

[Rikke]

Thanks for your respond! I know which one is grouped together, but it will be time consuming because I actually have an array of 1740 elements with 55 of them as the unknown x-values. And I want the optimization model to work with any kind of array not only the one I have for the moment.

[Matt J]

@Rikke,

How are the groupings of the 55 unknowns specified as input? Do you have some sort of binary map?

Answer 2

[Matt J]

I will assume you have some binary vector v which indicates the groupings of the variables, so for example this,

[0 0 x(1) x(2) 0 0 0 x(3) x(4) x(5) 0 0 0 x(6) 0 0 x(7) x(8)];

would be input as the vector

v=[0 0 1 1 0 0 0 1 1 1 0 0 0 1 0 0 1 1];

Then you can construct the linear equality constraint matrices Aeq, beq as follows

Aeq=diff(speye(numel(v))  ,1,1);

   discard=(v(1:end-1)==0)|(diff(v)~=0);
   
Aeq(discard,:)=[];
Aeq(:,~v)=[];   

beq=zeros(size(Aeq,1),1);

[Matt J]

For the 8-variable example that you posted, this should result in

>> full(Aeq), full(beq).'

ans =

    -1     1     0     0     0     0     0     0
     0     0    -1     1     0     0     0     0
     0     0     0    -1     1     0     0     0
     0     0     0     0     0     0    -1     1


ans =

     0     0     0     0

indicating that the first two unknowns will be equal, the next 3 unknowns will be equal and the final two unknowns will be equal.

[Rikke]

Thank you so much Matt J, it worked!