FMINCON showing x0 not between lb and ub, though my x0 is strictly between limits.

I am using FMINCON with Interior Point algorithm.

My initial guess x0=[-5.5, -0.5, -0.5].

Lower bound: lb=[-6, -1, -1]

Upper bound: ub=[-4, 1, 1]

While running the code, I am getting the message "Your initial point x0 is not between bounds lb and ub; FMINCON shifted x0 to strictly satisfy the bounds".

Can someone please explain why?

[Mike Croucher]

Could you post more of your code please? Enough for us to reproduce the error message.

[Debapriya Sengupta]

x0=[-5.5,-0.5,-0.5];

opt=optimoptions('fmincon','Display','iter');
gs=GlobalSearch;
problem=createOptimProblem('fmincon','x0',x0,'objective',@Reg_3D_OF_Func,'lb',[-6,-1,-1],'ub',[-4,1,1],'options',opt);
[x,fval,exitflag,output]=run(gs,problem);

Reg_3D_OF_Func is an user defined function.

Answer 1

[Walter Roberson]

It has something to do with the fact you are using GlobalSearch: some of the runs might be generating initial values out of bounds while others might generate in-bounds x0.

x0=[-5.5,-0.5,-0.5];

opt=optimoptions('fmincon','Display','iter');
gs=GlobalSearch;
problem=createOptimProblem('fmincon','x0',x0,'objective',@Reg_3D_OF_Func,'lb',[-6,-1,-1],'ub',[-4,1,1],'options',opt);
[x,fval,exitflag,output]=run(gs,problem)

                                            First-order      Norm of
 Iter F-count            f(x)  Feasibility   optimality         step
    0       4    2.344400e+00    0.000e+00    2.388e+00
    1       8    7.338240e-02    0.000e+00    5.972e-01    1.786e+00
    2      12    1.891233e-02    0.000e+00    9.811e-02    3.629e-01
    3      16    1.723423e-03    0.000e+00    1.019e-01    1.307e-01
    4      20    4.826181e-05    0.000e+00    1.861e-02    4.727e-02
    5      24    5.945924e-06    0.000e+00    1.411e-03    7.126e-03
    6      28    7.053477e-06    0.000e+00    1.056e-03    1.646e-03
    7      32    5.933509e-06    0.000e+00    9.970e-04    7.821e-04
    8      36    2.540379e-07    0.000e+00    2.068e-04    1.932e-03
    9      40    3.664163e-11    0.000e+00    2.461e-06    4.980e-04
   10      44    4.245787e-15    0.000e+00    2.007e-08    5.998e-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.

                                            First-order      Norm of
 Iter F-count            f(x)  Feasibility   optimality         step
    0       4    9.659986e-02    0.000e+00    4.182e-01
    1       8    1.373969e-01    0.000e+00    4.445e-01    5.725e-01
    2      12    1.675187e-02    0.000e+00    1.052e-01    2.537e-01
    3      16    1.883819e-03    0.000e+00    5.712e-02    1.062e-01
    4      20    1.664789e-04    0.000e+00    2.225e-02    5.203e-02
    5      24    5.425752e-06    0.000e+00    9.961e-04    1.257e-02
    6      28    5.672983e-07    0.000e+00    9.679e-04    2.218e-03
    7      32    1.803324e-07    0.000e+00    4.072e-04    8.821e-04
    8      36    2.577797e-07    0.000e+00    2.000e-04    2.684e-04
    9      40    5.690331e-09    0.000e+00    5.334e-05    4.345e-04
   10      44    9.687839e-09    0.000e+00    4.000e-05    3.019e-05
   11      48    8.163145e-13    0.000e+00    1.504e-06    9.793e-05
   12      52    9.816544e-13    0.000e+00    4.000e-07    9.274e-07

Feasible point with lower objective function value found.


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.

                                            First-order      Norm of
 Iter F-count            f(x)  Feasibility   optimality         step
    0       4    1.923754e-01    0.000e+00    5.760e-01
    1       8    3.124102e-02    0.000e+00    3.611e-01    5.540e-01
    2      12    1.376073e-01    0.000e+00    4.439e-01    4.675e-01
    3      16    2.552247e-02    0.000e+00    1.070e-01    2.382e-01
    4      20    3.334620e-03    0.000e+00    1.013e-01    1.334e-01
    5      24    1.621608e-03    0.000e+00    2.093e-02    4.580e-02
    6      28    2.315814e-05    0.000e+00    3.357e-03    3.594e-02
    7      32    3.626316e-07    0.000e+00    2.483e-04    4.225e-03
    8      36    4.298216e-11    0.000e+00    2.681e-06    5.956e-04
    9      40    3.301282e-15    0.000e+00    3.891e-08    6.500e-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.

                                            First-order      Norm of
 Iter F-count            f(x)  Feasibility   optimality         step
    0       4    2.248363e-01    0.000e+00    8.371e-01
    1       8    9.776205e-02    0.000e+00    4.498e-01    6.073e-01
    2      12    1.634787e-02    0.000e+00    2.020e-01    2.769e-01
    3      16    2.197283e-02    0.000e+00    1.001e-01    1.125e-01
    4      20    2.536506e-03    0.000e+00    9.986e-02    1.215e-01
    5      24    1.608425e-03    0.000e+00    2.065e-02    3.801e-02
    6      28    2.049303e-05    0.000e+00    4.480e-03    3.626e-02
    7      32    3.651080e-07    0.000e+00    2.420e-04    4.042e-03
    8      36    2.369817e-07    0.000e+00    2.000e-04    1.831e-04
    9      40    9.597326e-09    0.000e+00    4.028e-05    3.889e-04
   10      44    1.067244e-12    0.000e+00    4.174e-07    9.693e-05

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.

                                            First-order      Norm of
 Iter F-count            f(x)  Feasibility   optimality         step
    0       4    2.195452e-01    0.000e+00    8.921e-01
    1       8    3.881739e-02    0.000e+00    3.965e-01    5.323e-01
    2      12    2.166919e-02    0.000e+00    1.997e-01    2.560e-01
    3      16    2.968060e-02    0.000e+00    1.018e-01    1.131e-01
    4      20    2.854896e-03    0.000e+00    6.308e-02    1.207e-01
    5      24    1.741922e-03    0.000e+00    2.028e-02    1.909e-02
    6      28    2.393553e-05    0.000e+00    4.120e-03    3.692e-02
    7      32    3.704782e-07    0.000e+00    2.511e-04    4.288e-03
    8      36    4.331027e-11    0.000e+00    2.692e-06    6.022e-04
    9      40    4.271830e-15    0.000e+00    2.290e-08    6.528e-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.

                                            First-order      Norm of
 Iter F-count            f(x)  Feasibility   optimality         step
    0       4    3.903371e-01    0.000e+00    1.078e+00
    1       8    5.483460e-02    0.000e+00    3.487e-01    4.704e-01
    2      12    7.626129e-02    0.000e+00    2.851e-01    4.456e-01
    3      16    2.313944e-02    0.000e+00    1.091e-01    1.864e-01
    4      20    6.339247e-04    0.000e+00    1.160e-01    1.280e-01
    5      24    1.571817e-03    0.000e+00    2.002e-02    1.838e-02
    6      28    2.770342e-05    0.000e+00    2.391e-03    3.481e-02
    7      32    3.482794e-07    0.000e+00    2.568e-04    4.749e-03
    8      36    2.392309e-07    0.000e+00    2.000e-04    1.556e-04
    9      40    8.592744e-09    0.000e+00    4.029e-05    3.980e-04
   10      44    2.570428e-12    0.000e+00    2.226e-06    9.195e-05
   11      48    9.670676e-13    0.000e+00    4.000e-07    1.411e-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.

                                            First-order      Norm of
 Iter F-count            f(x)  Feasibility   optimality         step
    0       4    3.450835e-01    0.000e+00    1.079e+00
    1       8    4.913316e-02    0.000e+00    4.267e-01    3.732e-01
    2      12    4.413770e-02    0.000e+00    2.294e-01    3.715e-01
    3      16    3.405702e-02    0.000e+00    1.189e-01    1.674e-01
    4      20    2.015277e-02    0.000e+00    1.001e-01    6.395e-02
    5      24    2.548019e-03    0.000e+00    5.578e-02    9.849e-02
    6      28    1.755198e-03    0.000e+00    2.053e-02    2.039e-02
    7      32    3.730993e-05    0.000e+00    2.688e-03    3.600e-02
    8      36    4.033678e-07    0.000e+00    2.622e-04    5.479e-03
    9      40    1.351920e-10    0.000e+00    1.808e-05    6.297e-04
   10      44    2.522677e-11    0.000e+00    2.285e-06    1.133e-05
   11      48    2.429539e-11    0.000e+00    2.000e-06    1.246e-06
   12      52    9.799547e-13    0.000e+00    4.000e-07    3.940e-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.

                                            First-order      Norm of
 Iter F-count            f(x)  Feasibility   optimality         step
    0       4    3.414518e-01    0.000e+00    1.094e+00
    1       8    1.571625e-01    0.000e+00    4.879e-01    5.064e-01
    2      13    1.919028e-02    0.000e+00    9.678e-02    3.292e-01
    3      17    1.740166e-03    0.000e+00    2.157e-02    9.812e-02
    4      21    7.027629e-05    0.000e+00    7.857e-03    3.443e-02
    5      25    1.486300e-05    0.000e+00    6.852e-03    8.832e-03
    6      29    4.590223e-06    0.000e+00    1.000e-03    3.919e-03
    7      33    9.110524e-08    0.000e+00    4.304e-04    1.845e-03
    8      37    2.417750e-07    0.000e+00    2.000e-04    2.068e-04
    9      41    3.721944e-11    0.000e+00    5.716e-06    4.867e-04
   10      45    2.429729e-11    0.000e+00    2.000e-06    3.516e-06
   11      49    4.266043e-15    0.000e+00    2.004e-08    4.875e-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.

                                            First-order      Norm of
 Iter F-count            f(x)  Feasibility   optimality         step
    0       4    3.371769e-01    0.000e+00    1.131e+00
    1       8    1.417682e-01    0.000e+00    5.145e-01    3.859e-01
    2      12    7.935906e-02    0.000e+00    4.096e-01    6.270e-01
    3      16    2.145820e-02    0.000e+00    1.195e-01    2.836e-01
    4      20    2.811002e-03    0.000e+00    3.154e-02    9.566e-02
    5      24    1.798861e-03    0.000e+00    2.030e-02    1.242e-02
    6      28    6.810898e-05    0.000e+00    2.034e-02    4.149e-02
    7      32    1.693050e-05    0.000e+00    7.976e-03    1.235e-02
    8      36    2.407902e-07    0.000e+00    2.000e-04    4.190e-03
    9      40    9.810262e-09    0.000e+00    4.028e-05    3.917e-04
   10      44    1.068018e-12    0.000e+00    4.176e-07    9.801e-05

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.

                                            First-order      Norm of
 Iter F-count            f(x)  Feasibility   optimality         step
    0       4    3.318605e-01    0.000e+00    1.131e+00
    1       8    1.929770e-01    0.000e+00    5.145e-01    4.638e-01
    2      13    2.105123e-02    0.000e+00    1.053e-01    3.858e-01
    3      17    1.788712e-03    0.000e+00    2.104e-02    1.047e-01
    4      21    6.576564e-05    0.000e+00    9.479e-03    3.555e-02
    5      25    1.780964e-05    0.000e+00    7.656e-03    9.251e-03
    6      29    4.870876e-06    0.000e+00    1.000e-03    4.078e-03
    7      33    9.647318e-08    0.000e+00    3.964e-04    1.899e-03
    8      37    2.414362e-07    0.000e+00    2.000e-04    1.906e-04
    9      41    3.464390e-11    0.000e+00    2.432e-06    4.857e-04
   10      45    3.853556e-15    0.000e+00    2.216e-08    5.833e-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.

                                            First-order      Norm of
 Iter F-count            f(x)  Feasibility   optimality         step
    0       4    3.305586e-01    0.000e+00    1.131e+00
    1       8    1.114903e-01    0.000e+00    5.145e-01    3.607e-01
    2      12    6.405262e-02    0.000e+00    3.427e-01    5.568e-01
    3      16    2.461008e-02    0.000e+00    1.162e-01    2.339e-01
    4      20    1.902676e-03    0.000e+00    6.747e-02    1.150e-01
    5      24    1.740198e-03    0.000e+00    2.014e-02    1.012e-02
    6      28    3.214181e-05    0.000e+00    6.520e-03    3.696e-02
    7      32    9.852330e-07    0.000e+00    1.572e-03    5.678e-03
    8      36    1.925062e-07    0.000e+00    2.001e-04    9.362e-04
    9      40    7.551514e-09    0.000e+00    4.022e-05    3.544e-04
   10      44    1.044779e-12    0.000e+00    4.152e-07    8.588e-05

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.

                                            First-order      Norm of
 Iter F-count            f(x)  Feasibility   optimality         step
    0       4    3.281169e-01    0.000e+00    1.131e+00
    1       8    1.335371e-01    0.000e+00    5.145e-01    3.879e-01
    2      12    1.142470e-01    0.000e+00    4.635e-01    6.760e-01
    3      16    2.382899e-02    0.000e+00    1.203e-01    3.172e-01
    4      20    2.139229e-03    0.000e+00    7.781e-02    1.108e-01
    5      24    1.681621e-03    0.000e+00    2.016e-02    2.013e-02
    6      28    3.397309e-05    0.000e+00    2.680e-03    3.532e-02
    7      32    3.980299e-07    0.000e+00    2.605e-04    5.210e-03
    8      36    4.547410e-11    0.000e+00    2.754e-06    6.242e-04
    9      40    4.328082e-15    0.000e+00    2.009e-08    6.689e-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.

                                            First-order      Norm of
 Iter F-count            f(x)  Feasibility   optimality         step
    0       4    3.278485e-01    0.000e+00    1.131e+00
    1       8    1.242883e-01    0.000e+00    5.145e-01    3.629e-01
    2      12    8.976789e-02    0.000e+00    4.435e-01    6.287e-01
    3      16    2.481430e-02    0.000e+00    1.180e-01    2.843e-01
    4      20    1.999250e-03    0.000e+00    7.820e-02    1.150e-01
    5      24    1.695534e-03    0.000e+00    2.016e-02    1.746e-02
    6      28    3.419336e-05    0.000e+00    2.679e-03    3.550e-02
    7      32    3.971558e-07    0.000e+00    2.605e-04    5.232e-03
    8      36    4.704326e-11    0.000e+00    3.784e-06    6.237e-04
    9      40    2.428228e-11    0.000e+00    2.000e-06    2.667e-06
   10      44    9.310731e-16    0.000e+00    5.788e-08    4.899e-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.

Your initial point x0 is not between bounds lb and ub; FMINCON
shifted x0 to strictly satisfy the bounds.

                                            First-order      Norm of
 Iter F-count            f(x)  Feasibility   optimality         step
    0       4    6.742875e-01    0.000e+00    8.324e-01
    1       8    3.544975e-02    0.000e+00    6.734e-01    9.545e-01
    2      12    1.125477e-02    0.000e+00    9.724e-02    2.542e-01
    3      16    6.950022e-04    0.000e+00    9.089e-02    1.097e-01
    4      20    4.282315e-05    0.000e+00    1.602e-02    3.286e-02
    5      24    9.212134e-06    0.000e+00    1.389e-03    7.960e-03
    6      28    5.130918e-06    0.000e+00    1.000e-03    1.367e-03
    7      32    2.294501e-07    0.000e+00    2.059e-04    1.805e-03
    8      36    3.515817e-11    0.000e+00    2.439e-06    4.731e-04
    9      40    4.364979e-15    0.000e+00    2.007e-08    5.876e-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.


GlobalSearch stopped because it analyzed all the trial points.

All 14 local solver runs converged with a positive local solver exit flag.

x = 1×3

-4.7000 0.8000 -0.6200

fval = 4.2458e-15

exitflag = 1

output = struct with fields:
                funcCount: 2631
         localSolverTotal: 14
       localSolverSuccess: 14
    localSolverIncomplete: 0
    localSolverNoSolution: 0
                  message: 'GlobalSearch stopped because it analyzed all the trial points.↵↵All 14 local solver runs converged with a positive local solver exit flag.'

function y = Reg_3D_OF_Func(x)
   y = sum((x - [-4.7 .8 -.62]).^2);
end

[Matt J]

some of the runs might be generating initial values out of bounds while others might generate in-bounds x0.

Probably not out of bounds, but some almost certainly are exactly on the boundaries. I.e., the bounds are not satisfied strictly, as interior-point requies, but they are satisfied.

[Debapriya Sengupta]

Could you please explain in more detail? Why are some of the runs generating initial values, when I have already defined initial value x0 in my code? How can I solve this problem?

[Matt J]

That's the way the GlobalSearch algorithm works. It explores the surface of your cost function for the global minimum by testing many initial points. There is nothing wrong that needs solving.

[Debapriya Sengupta]

Okay. Thank you so much.

[Matt J]

Your welcome, but please Accpet-click this answer.

Answer 2

[Matt J]

That might happen if you have additional, tighter box bounds encoded into your linear inequality constraint matrices A,b..For example, if one of the rows of your A,b data were

A(i,:)=[0,0,-1]; b(i)=0;  %equivalent to x(3)>=0

then fmincon preprocessing would delete this row and convert lb(3) to zero. Once the input data has been simplified in this way, x0(3) would then violate the bounds.

[Debapriya Sengupta]

Mine is an unconstrained optimization problem. I do not have inequality constraints defined in my code.