IF it is ok to violate the sum to 1, then do as David has shown. The solution he wrote will allow that equation to be violated.
HOWEVER, if that must be treated as inviolable, then you need to use an equality constraint.
% r and R represent equationions that can be violted.
r = [.95,.89,.33,.31,.08;.95,.86,.036,.32,.08;.95,.73,.35,.21,.08;.87,.46,.32,.13,.08];
R=[.49;.47;.42;.19];
% the equality constraint, that the unknowns sum to 1
Aeq = [1 1 1 1 1];
beq = 1;
lb = zeros(1,5);
ub = ones(1,5);
ABCDE = lsqlin(r,R,[],[],Aeq,beq,lb,ub);
You can unpack the values of A,B,C,D,E into scalar variables. However, my suggestion is to keep them in vector form.
Finally since the problem you have written is an underdetermined one, it MAY have an exact solution, however the bound constraints may prevent that from happening. If the array r has more than 4 rows, then this will usually be a least squares problem. In that case, then you would usually want to keep the equality constraint separate.
Usually the best way to solve problems of this sort is to keep the equality constraints separate from the equations that can allow some amount of violation.
