Problem of erf when using fmincon

Hello, I am using fmincon to solve a problem whose objective function involves erf and sqrt. Although I have added constraints, I still got the error message

"Error using erf

Input must be real and full."

The toy code is as follows.

clear all

x2=rand(2,1);

x1=x2+rand(2,1);

x0=[x1;x2];

fun = @(x)sum( erf(sqrt(-log(x(1:2)-x(3:4)))));

A0 = [eye(2) -eye(2)];

% the constraint below makes sure -log(x(1:2)-x(3:4)) is always a nonnegative vector.

A=[-A0; A0];

b = [zeros(2,1);ones(2,1)];

Aeq=[];

beq=[];

lb=zeros(4,1);

ub=[];

options = optimoptions('fmincon','Display','iter','Algorithm','sqp','TolFun',1e-10);

xsol= fmincon(fun,x0,A,b,Aeq,beq,lb,ub,[],options);

If I remove erf from the objective function, i.e., fun = @(x)sum( (sqrt(-log(x(1:2)-x(3:4))))), I don't have this error.

Since erf complains, it maens sqrt gives complex values, but why is this possible given the constraint? Could anyone help? Thank you.

Answer 1

[Are Mjaavatten]

Your problem has infinitely many solutions. Whenver x(1)-x(3) = 1 and x(2)-x(4) = 1, fun = 0, which is the minimum value possible.

That said, fmincon obviously searches beyond the constraints in your case. I wrote your 'fun' function as an ordinary function in order to see the offending x vector in the debugger, and you are absolutely right. The program crashes when:

K>> x
x =
   1.762231198224400
   0.964095464916107
   0.762231171965109
                   0
K>> A*x-b
ans =
  -1.000000026259291
  -0.964095464916107
   0.000000026259291   % Should be <= 0!!
  -0.035904535083893      

This may have to do with the lack of a unique solution, but anyway I think the Matworks staff should look into this.

[Max Wang]

Thank you for your reply. It is a toy version of the problem and I just used it to test erf in the objective funtion, so there may be many trivial solutions (but fmincon cannot find any).

I assume fmincon is not expected to search outside the feasible region? I tried several algorithms and all throw this kind of error.