Trying to minimise a function subject to nonlinear constraints

function= xz+y^(2)

constraints: xz=3

y^2+z^2=1

Code:

fun = @(x)x(1)*x(3)+x(2)^2;

options = optimoptions('fmincon','Display','iter','Algorithm','sqp');

nonlcon = @confuneq;

A = [];

b = [];

Aeq = [];

beq = [];

lb = [];

ub = [];

x0 = [0,0,0];

x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)

Separate file:

function [c,ceq] = confuneq(x)

c = x(2)^2 + x(3)^2 -1;

ceq = x(2)*x(3) -3;

end

Can't seem to get the correct answers of (3,0,1) and (-3,0,-1)

Any help is welcome

Answer 1

[John D'Errico]

Simple mistake(s) here. You have TWO equality constraints. But your constraint function is written to have ONE equality constraint, and ONE inequality constraint. And you also wrote the x*z=3 constraint wrong. You wrote it as y*z=3.

Instead, try this:

function [c,ceq] = confuneq(x)
    c = [];
    ceq = [x(2)^2+x(3)^2-1; x(1)*x(3)-3];
end

That should now be a correct way to write the constraints.

Of course, don't expect fmincon to provide both solutions in one call. You need to start the optimizer out in the correct basin of attraction to get the solution you might want to see.

x = fmincon(fun,[1 1 2],A,b,Aeq,beq,lb,ub,nonlcon,options)
 Iter  Func-count            Fval   Feasibility   Step Length       Norm of   First-order  
                                                                       step    optimality
    0           4    3.000000e+00     4.000e+00     1.000e+00     0.000e+00     2.000e+00  
    1           9    2.928889e+00     3.352e+00     7.000e-01     1.504e+00     9.644e-01  
    2          13    2.571265e+00     7.559e-01     1.000e+00     1.060e+00     3.470e-01  
    3          17    2.789068e+00     2.110e-01     1.000e+00     7.947e-01     3.581e-01  
    4          21    2.988095e+00     1.191e-02     1.000e+00     3.066e-01     4.893e-02  
    5          25    2.999988e+00     1.161e-05     1.000e+00     1.437e-02     9.121e-04  
    6          29    3.000000e+00     5.045e-09     1.000e+00     7.238e-05     1.451e-04  
    7          33    3.000000e+00     5.265e-09     1.000e+00     7.255e-05     1.646e-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.

<stopping criteria details>
x =
          2.99999999210178     -1.73125747607761e-08          1.00000000263274

          
>> x = fmincon(fun,[-1 1 0],A,b,Aeq,beq,lb,ub,nonlcon,options)
 Iter  Func-count            Fval   Feasibility   Step Length       Norm of   First-order  
                                                                       step    optimality
    0           4    1.000000e+00     3.000e+00     1.000e+00     0.000e+00     2.000e+00  
    1          12    1.720300e+00     2.280e+00     2.401e-01     7.203e-01     1.520e+00  
    2          19    1.688289e+00     1.455e+00     3.430e-01     9.684e-01     6.358e-01  
    3          23    2.587102e+00     4.133e-01     1.000e+00     1.342e+00     6.046e-01  
    4          27    2.929079e+00     7.341e-02     1.000e+00     6.240e-01     2.038e-01  
    5          31    3.000225e+00     6.050e-04     1.000e+00     9.457e-02     5.761e-02  
    6          35    3.000027e+00     5.576e-04     1.000e+00     2.362e-02     1.620e-02  
    7          39    3.000000e+00     2.761e-05     1.000e+00     5.308e-03     4.556e-04  
    8          43    3.000000e+00     2.435e-09     1.000e+00     6.453e-05     2.124e-05  
    9          47    3.000000e+00     0.000e+00     1.000e+00     4.398e-09     1.192e-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.

<stopping criteria details>
x =
         -3.00000000000053      5.92273862321821e-07        -0.999999999999825

[Adam Stevenson]

Thankyou kind sir.