How to find a constrained distribution within an array that results in an optimum.

Hi,

This is my first time posting here so please bear with me. I have been struggling with an optimization problem for a while now and was hoping somebody here might have a solution or can steer me in the right direction.

I have a function: y = a * b * c * x. a, b, c are arrays of 20 elements which are constants and I am trying to optimize y. The problem now lies with x.

x is also an array with 20 elements, however each value can take a value ranging from 0 to a certain maximum value. The problem now is, is that the array x summed up has to be a certain value. I am thus trying to optimize y for a certain distribution of values over the array x, while the summed value stays the same.

x is also an array with 20 elements, however each value can take a value ranging from 0 to a certain maximum value. The problem now is, is that the array x summed up has to be a certain value. I am thus trying to optimize y for a certain distribution of values over the array x, while the summed value stays the same.

I hope my explanation is clear enough and if you have any questions please let me know.

Any help would be greatly appreciated.

Answer 1

[Alan Weiss]

I suppose that when you say that you want to "optimize" y, you want to minimize or maximize the sum of the elements in that multiplication. And I suppose that the multiplication is element-wise, meaning the objective function has to do with

 sum_I(a(I)*b(I)*c(I)*x(I))

where the a, b, and c are given vectors of numbers. Your constraints are that each x(I) must lie in a certain range, and the sum of the x(I) must equal a given number.

If I am correct about all these suppositions, then you can use the linprog solver in Optimization Toolbox to solve your problem. If I am wrong about something, then please clarify where I misunderstood.

Alan Weiss

MATLAB mathematical toolbox documentation

[Jordy van der Pol]

Yes those assumptions are correct. Thanks for the suggestion! I will take a look at the linprog solver.

[Jordy van der Pol]

Hi,

I have now tried to solve it by use of the linprog solver according to the following code

q = rho * g .* dpot;
s = ones(1,Nlayers);

f = [q];
A = [];
b = [];
Aeq = [s];

beq = [T_p];
lb = zeros(1,Nlayers);
ub = pot_uptake;
options = optimoptions('linprog','Display','iter');

[x,fval,exitflag,output,lambda] = linprog(f, A, b, Aeq, beq, lb, ub, options)

where rho, g and dpot are constants which I have chosen to combine into one constant q, which is a column of 20 elements.

T_p is the value that sum(x(i)) should be at all times, lb and ub subsequently represent the range of values x(i) is allowed to take.

However, when it runs and I sum up x(i) that comes out of the solver, it is not equal to T_p.

So I was now wondering where the error lies in my code or understanding of the linprog solver?

Thanks in advance!

For what it's worth, the output of the solver is:

output =

struct with fields:

         iterations: 0
    constrviolation: 6.9750e-08
            message: 'Optimal solution found.'
          algorithm: 'dual-simplex'
      firstorderopt: 0

[Alan Weiss]

I am not sure that you set a maximum value on all components of x, but that should not affect whether your solution satisfies the constraints. The exit message indicates a small constraint violation. Is that what you mean when you say that the sum of the output variables is not T_p? Is it off by about 1e-7?

Also, did you realize the linprog always minimizes? To maximize the objective function, take the negative of q.

Alan Weiss

MATLAB mathematical toolbox documentation

[Jordy van der Pol]

Hi,

yes that is what I mean, although the violation seems small, the constrain violation has the same value as T_p (T_p = 6.9750e-08)

The algorithm gives for each x a value of 0 at the moment. Maybe that the problem lies with the fact that the numbers the algorithm has to deal with are so small?

Yes I did realize that it minimizes, which is wat I needed.

[Alan Weiss]

Your problem might be poorly scaled. Try rescaling it.

You could also try lowering the ConstraintTolerance option to, say, 1e-9.

Alan Weiss

MATLAB mathematical toolbox documentation