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Why are the first order necessary conditions of optimality not satisfied for this problem?

Asked by Del on 30 Jan 2013

I am trying to find the optimal Lagrange multipliers for this problem:

min 100*(V4 - V2 + (V1 - V3)^2)^2 + (V3 - V1 + 1)^2
s.t 
[V5 - (V1 - V3)*(V2 - V4) + 1;  V3 - V1 + V6 - (V2 - V4)^2; V1 - V3 + V7 - 1/2]=0
[V1;V2;V3;V4;V5;V6;V7] >=0

The optimal minimizer that I am getting is:

V =
      0.5000
      2.0687
      0.0000
      0.0687
     -0.0000
      4.5000
      0.0000

MATLAB is giving me the Lagrange multipliers:

      lambda.eqnonlin=
1.0e+03 *
       -0.7000
        0.0000
        1.7510
    lambda.lower= 
1.0e+03 *
           0
           0
           0
           0
      0.7000
           0
      1.7510

However, when I take the gradient of the Lagrangian function at the optimal solution V, the answer is not 0!

Any idea why?

5 Comments

Matt J on 30 Jan 2013

What solver are you using? Are you sure that the optimization succeeded/completed? What exitflag did it return? Are you sure your gradient and Jacobian functions are correct (it would help to show them)? Did the optimization use them or did you use the default finite difference differentiation?

Del on 30 Jan 2013

I used fmincon, starting with the value V0=zeros(7,1), it returned the exitflag 4. I checked the gradient and jacobian functions by hand, and they are correct. I only used fmincon to find the minimizer V, and to find the optimal Lagrange multipliers. I just calculated the gradient and Jacobian so that I could use them in the gradient lagrangian function, and I was hoping to get 0.

Del on 30 Jan 2013

Here is what was written: "fmincon stopped because the size of the current search direction is less than twice the default value of the step size tolerance and constraints are satisfied to within the default value of the constraint tolerance."

and

grad_f_V'=

 2*V1 - 2*V3 + 200*(2*V1 - 2*V3)*(V4 - V2 + (V1 - V3)^2) - 2
                           200*V2 - 200*V4 - 200*(V1 - V3)^2
 2*V3 - 2*V1 - 200*(2*V1 - 2*V3)*(V4 - V2 + (V1 - V3)^2) + 2
                           200*V4 - 200*V2 + 200*(V1 - V3)^2
                                                           0
                                                           0
                                                           0

Jacobian_h_V =

[ V4 - V2,     V3 - V1, V2 - V4,     V1 - V3, 1, 0, 0]
[      -1, 2*V4 - 2*V2,       1, 2*V2 - 2*V4, 0, 1, 0]
[       1,           0,      -1,           0, 0, 0, 1]
Del

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1 Answer

Answer by Matt J on 30 Jan 2013

Here is what was written: "fmincon stopped because the size of the current search direction is less than twice the default value of the step size tolerance and constraints are satisfied to within the default value of the constraint tolerance."

which means you didn't converge with respect to the first order optimality measure. Your objective is a variant of Rosenbrock, so presumably it's supposed to be hard to converge to a proper solution. Try increasing MaxIter to something ridiculously large and make sure you get an exitflag=1.

3 Comments

Del on 30 Jan 2013

alright, how to change/Increase MaxIter ?

Del on 30 Jan 2013

Here is what I am typing:

options = optimset( 'MaxIter', Inf);
V0=zeros(7,1);lb=zeros(7,1);
[V,fval,exitflag,output,lambda,grad]  =  fmincon(@function_TEST2_V,V0,[],[],[],[],lb,[],@constraints_TEST2_V) 

I am not sure if that is what I am supposed to do, however, it is not making any difference, the exit flag is still 4

Matt J on 31 Jan 2013

Try also setting TolX=0 and MaxFunEvals=Inf.

Matt J

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