Solving for each row of X is an independent optimization problem that can be solved easily with LSQNONNEG (available from the Optimization Toolbox). Use a loop to solve for each row independently.
Example 1 (test when know the exact answer):
% Set up some data
A = rand(5);
Xtrue = rand(5);
B = Xtrue*A;
% Solve for each row of X using LSQNONNEG
X = [];
for k = 1:size(B,1)
X(k,:) = lsqnonneg(A',B(k,:)');
end
% Verify the result
X - Xtrue
Example 2:
A = rand(6,3);
B = rand(6,3);
X = [];
for k = 1:size(B,1)
X(k,:) = lsqnonneg(A',B(k,:)');
end
% Verify that all X are positive
X
Note that if your data is very big, this algorithm could easily be sped up by running it in parallel.
