Define a Non linear objective function with 2 decision variables

I have a matrix X (932x1404) and I want to find 2 matrices W(932x2) , H(2X1404) in order to minimize the error abs(X-WH) over W and H. It is similar to a non negative matrix factorization problem but I want to add specific constraints. The matrix X actually is a black and white image with values 0-255 and I want the W to have similar values (to represent the centroids of each class) and the matrix H to have values from 0 to 1 but with the sum of the values in each column to equal 1 (something like weights). So the problem I want to define in matlab is the following:

min_W,H (|X-W*H|) (o.f.)

under the constraints

Wik>=0 (c1)

Wik<=255 (c2)

Hkj>=0 (c3)

Hkj <= 1 (c4 , not neccessary I guess due ot the following constraint)

sum_k Hkj =1 for each column j of Matrix H (c5)

where i is the ith row and k the kth column of matrix W

and for matrix H the k denotes the kth row and j the jth column

the sum_k denotes the sum over all rows entries for a specific column j. for example for the first column we sum all the rows and so on.

Could you please let me know how I can formulate such a problem in matlab? I watched various tutorials for optimization toolbox but I have not figured it out yet.

Thank you in advance for your help

[Matt J]

Is it important that it be min(abs(error)) or would you be just as happy with least squares?

[Charis Nt]

for the moment any norm that could be used as 'termination criterio' or a norm in which we willl minimize would be fine.

Answer 1

[Matt J]

If you have R2109a or higher, the general set-up could be done like this. For simplicity, I demonstrate using a least squares objective.

W=optimvar('W',932,2,'LowerBound',0,'UpperBound',255);
H=optimvar('H',2,1404,'LowerBound',0,'UpperBound',1);
prob=optimproblem;
prob.Objective=sum(W(:))+sum(H(:));  %dummy objective - not used
prob.Constraints.sumH=sum(H,1)==1;


idxH=1:numel(H); idxW=numel(H)+1:numel(W)+numel(H);

p=rmfield(prob2struct(prob),{'f','f0','solver'});
p.fun=@myObjective(z,X,idxH,idxW);



p.x0([idxW;idxH])=[W0(:);H0(:)];

solution=mapSolution(prob, fmincon(p))

function err=myObjective(z,X,idxH,idxW)

  W=z(idxW);
  H=z(idxH);

  err=norm(X-W*H,'fro').^2;
  
end

[Charis Nt]

Thank you for your answer

I will try to run it and let you know. However I am using matlab 2018b, do you think that it will be an issue?

[Matt J]

Yes, you won't be able to use varindex in R2018b. I've rewritten the code to not use it.