Interior-point and sequential quadratic programming give me the same answer. Is there an error with my code?

I am attempting to minimise a least squares regression model and compare answers given by SQP and IP, and the system is constrained by x(1) + x(2) + x(3) = 1, and each variable cannot be smaller than 0 or bigger than 1. However, I consistently get the same numerical answers for both methods, which should not be the case. I am wondering if I missed a step or something along those lines? My code is as follows;

fun=@(x) (x(1)-147/505).^2+(x(1)-167/530).^2+(x(2)-504/2525).^2+(x(2)-471/2120).^2+(x(3)-1011/2120).^2+(x(3)-1256/2525).^2;
x0=[0.3, 0.5, 0.2];
Aeq=[1, 1, 1];
beq=1;
lb=[0, 0, 0];
ub=[1, 1, 1];
IP=fmincon(fun,x0,[],[],Aeq,beq,lb,ub);
fprintf('%.13f\n',IP)
options = optimoptions('fmincon','Algorithm','sqp');
SQP=fmincon(fun,x0,[],[],Aeq,beq,lb,ub);
fprintf('%.13f\n',SQP)

These are my results;

IP:

0.3027134331491

0.2105086010671

0.4867779657839

SQP:

0.3027134331491

0.2105086010671

0.4867779657839

Any help or pointers would be appreciated. Thank you in advance.

[Torsten]

However, I consistently get the same numerical answers for both methods, which should not be the case.

Why ?

[Aisel]

The question requires me to compare both answers, implying one is more accurate. Are they supposed to be the same?

[Torsten]

Maybe one needs more iterations with one of the methods (use 'Display','iter' in the options structure). But if you prescribe the accuracy of the solution to be equal for both methods (as you implicitly do in your code because you didn't change them in the optimoptions structure), in theory both solutions should be equally precise, shouldn't they ?

[Aisel]

I see, that makes sense. Thank you.

Answer 1

[Torsten]

I just saw in your code that you don't pass the options structure to fmincon. Thus in both calls, the same solver is used.

Use

fun=@(x) (x(1)-147/505).^2+(x(1)-167/530).^2+(x(2)-504/2525).^2+(x(2)-471/2120).^2+(x(3)-1011/2120).^2+(x(3)-1256/2525).^2;
x0=[0.3, 0.5, 0.2];
Aeq=[1, 1, 1];
beq=1;
lb=[0, 0, 0];
ub=[1, 1, 1];
options = optimoptions('fmincon','Algorithm','interior-point','Display','iter');
IP=fmincon(fun,x0,[],[],Aeq,beq,lb,ub,[],options);

                                            First-order      Norm of
 Iter F-count            f(x)  Feasibility   optimality         step
    0       4    3.328630e-01    0.000e+00    1.455e+00
    1       8    9.097657e-02    0.000e+00    1.369e+00    6.098e-01
    2      12    2.579621e-03    0.000e+00    9.931e-02    1.857e-01
    3      16    1.181553e-03    0.000e+00    7.284e-02    3.823e-02
    4      20    1.098100e-03    0.000e+00    4.139e-02    2.558e-02
    5      24    7.552129e-04    0.000e+00    2.454e-03    1.287e-02
    6      28    7.552951e-04    0.000e+00    1.000e-03    8.613e-04
    7      32    7.545440e-04    0.000e+00    2.015e-04    5.086e-04
    8      36    7.545172e-04    0.000e+00    2.077e-06    1.144e-04
    9      40    7.545172e-04    0.000e+00    2.001e-08    1.263e-06

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.

fprintf('%.13f\n',IP)

0.3027134331492
0.2105086010669
0.4867779657839

options = optimoptions('fmincon','Algorithm','sqp','Display','iter');
SQP=fmincon(fun,x0,[],[],Aeq,beq,lb,ub,[],options);

 Iter  Func-count            Fval   Feasibility   Step Length       Norm of   First-order  
                                                                       step    optimality
    0           4    3.328630e-01     0.000e+00     1.000e+00     0.000e+00     1.156e+00  
    1           9    2.478717e-01     1.110e-16     7.000e-01     6.930e-01     1.335e+00  
    2          13    2.415543e-01     0.000e+00     1.000e+00     6.044e-01     1.139e+00  
    3          17    5.519559e-02     1.110e-16     1.000e+00     4.532e-01     5.134e-01  
    4          21    4.792452e-03     0.000e+00     1.000e+00     1.976e-01     1.459e-01  
    5          25    7.959527e-04     1.110e-16     1.000e+00     4.792e-02     1.485e-02  
    6          29    7.546824e-04     2.220e-16     1.000e+00     4.766e-03     8.850e-04  
    7          33    7.545173e-04     2.220e-16     1.000e+00     2.919e-04     1.947e-05  
    8          37    7.545172e-04     2.220e-16     1.000e+00     6.953e-06     3.359e-07  

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in 
feasible directions, to within the value of the optimality tolerance,
and constraints are satisfied to within the value of the constraint tolerance.

fprintf('%.13f\n',SQP)

0.3027134886159
0.2105086203988
0.4867778909853

[Aisel]

Thank you @Torsten, you've been a fantastic help.