When I run your script I get the same results as you, so thank you for including all the relevant information.
Note that your constraint Aeq*x = beq does NOT preclude p(1) and p(4) from being nonzero. In fact, those variables can have any value; they don't affect the linear equality.
I then tried the following experiment:
% Obtain the result for i = 1
i = 1;
objective=@(p) 787.16-1.3217*(schedule(i,6)*p(1)+schedule(i,9)*p(4))-0.9312*(schedule(i,7)*p(2)+schedule(i,8)*p(3))+0.0092*((schedule(i,6)*p(1))^2+(schedule(i,9)*p(4))^2)+0.0062*((schedule(i,7)*p(2))^2+(schedule(i,8)*p(3))^2);
p0=[0,0,0,0];
A=[];
b=[];
Aeq=[schedule(i,6:9)];
beq=[load_data(i,2)];
lb=[0,0,0,0];
ub=[1200,600,600,1200];
nonlincon=@nlconProj;
[p,fval,ef,output]=fmincon(objective,p0,A,b,Aeq,beq,lb,ub,nonlincon);
schedule_final(i,:)=[p load_data(i,2)];
objzero = objective([0,p(2),p(3),0]) % Objective with 0 for p(1) and p(4)
fval - objzero
I get zero as the value of fval - objzero. This means that the objective function value is the same for these values as for the final value. In other words, fmincon sees no difference in the objective function, so there is no reason to prefer one of these final values over another.
If you want fmincon to return zero values, then make p(1) and p(4) part of the objective, such as
objective = @(p) (your expression) + p(1) + p(4);
I found that adding (p(1) + p(4))*1e-3 did not give me exactly 0 for p(1) and p(4). So I think that your objective might be somewhat sensitive to the values. And that is why fmincon returns nonzero values.
Alan Weiss
MATLAB mathematical toolbox documentation
