This problem seems to be optimizing different objectives(column-wise sum). It would make sense to optimize them separately as for this use-case the column-wise sum would depend only on the respective columns of 'B'(the unknown in this case).
However, if you are interested in maximizing something like
sum(sum(A.*B))
% which is optimizing the sum of column-wise sum of
% element-wise-product. It should be maximum for the
% maximum values of column-wise-sum
Then I have tried to make small working example code for such a situation using 'fmincon' function. You can always use any other similar function for doing so.
A = magic (3)
con = @constraints % we define a function file "constraints.m" that contains the constraints
min = @(B) -(sum(sum(A.*B))) % since all optimisation in Matlab can find minimum so we are trying to minimise the negetive of the objective.
a = [];
b = [];
Aeq = [];
beq = [];
lb = [];
ub = [];
x0 = zeros(3); % starting point for optimization
[sol, fval] = fmincon(min,x0,a,b,Aeq,beq,lb,ub,con) % This would give you 'B' and value of 'sum' (negetively minimised)
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% the contents of the file "constraints.m" (kept in same folder) are as following:-
function [c,ceq] = constraints(x)
c = x*[1;2;3] - [51;62;37];
ceq = [];
end
% [1;2;3] corresponds to 'c' in the question
% [51;62;37] corresponds to 'd' in the question
% imply Bc <= d (in question its Bc < d (without equalto))
But if you still want to optimize different objectives at once, you can take a look at the following functions like:
fgoalatain
ga
gamultiobj
patternsearch
Kindly note that the results may differ due to implimentation of different algorithms for different functions.
