error in fmincon problem

Hello!

I have to solve a system of non linear equalities and inequalities. First I used fsolve, but it only works with non linear equalities. So I found that to include also inequalities I should use fmincon.

My system depends on the unknown vector "a" (what I want to find out by solving the system) and a matrix "X" of random variables that I would like to generate in advance, and fix, so that the system is solved only with respect to the vector "a".

Now maybe I have a little bit of confusion about the objective function that I should indicate to fmincon. I mean I want to solve a system, so should the objective function be the system itself?

The main script is the following "Main Program"

    X = binornd(5,0.6,5,2);

    a0 = 100*randn(14,1);

    opts = optimset('Algorithm','interior-point','Display','off');

    x = fmincon(minconstr,a0,[],[],[],[],[],[],[],opts)

---------------------------------------------------------------------

where "minconstr" should be the objective function, therefore my system of non linear equalities and inequalities, is it correct? Here I use "global X" to try to fix the matrix X generated in the previous script. Am I right?

    function [c, ceq]=minconstr(a)

    global X;

    b=[1;1;1;1;1;1;1;0;0;0;0;0;0;0;0];

    ceq=confun(a);

    c=[-a(5)-b(1);-a(6)-b(1);-a(4)+a(5)-b(2);-a(4)+a(6)-b(2);-a(3)+a(4)-b(3);
        -a(2)+a(3)-b(4);-a(1)+a(2)-b(5);-a(7)-b(8);-a(8)-b(9);-a(9)-b(10);
        -a(10)-b(11);-a(11)-b(12);-a(12)-b(13);-a(13)-b(14);-a(14)-b(15)
        ];% Linear inequalities

--------------------------------------------------------------

where "confun" is a function that creates the vector of all the non linear equalities of my system. Here again I use "global X" to try to fix the matrix X generated in the previous script. The code of this function is quite long, but it is something like

    function F = confun(a)

    global X;

    ...
    ....
    ...

    % Nonlinear equality constraints 

    F = [dY(1); 
         dc(1); dc(2); dc(3); dc(4); dLambda;
         dmi(1);dmi(2);
         dmi2(1);dmi2(2);dmi2(3)];

------------------------------------------------------------------------

If I run my "Main Program" script, the error says

    ??? Input argument "a" is undefined.

    Error in ==> minconstr at 7
    ceq=confun(a);

    Error in ==> MainProgram at 9
    x = fmincon(minconstr,a0,[],[],[],[],[],[],[],opts)

"a" should be the unknown of my system of which I already provided an initial guess "a0". Does anyone know where could it be the problem?

Thank you all!

Answer 1

[Matt J]

Your objective function shouldn't be myconstr. Your objective should really be norm(confun(a)).^2

Also, you don't really have any nonlinear inequalities, or at least all of your inequalities looks quite linear to me. So you should use matrix data to express them and your call to fmincon should really look like,

   x = fmincon(@(a) norm(confun(a)).^2,a0, A,B,[],[],[],[],[],opts)

for an appropriate matrix A and vector B.

[Marty]

Thank you Matt for your answer.

Yes, I have non linear equalities that are collected in the vector F as output of the function "confun", and also linear inequalities included in "minconstr" (I fixed the post, because before I did a mistake in writing).

I know that for linear inequalities I could use the matrices A and B, but I simply wrote explicitly the conditions that would derive from this matrix product and collected them in the vector "c". Should it work anyway?

Could you please explain me why should norm(confun(a)).^2 be the objective function?

And why norm(confun(a)).^2 and not norm(mincontr(a)).^2?

I mean "confun" includes only the non linear equalities, while "minconstr" should represents the whole system (non linear equalities and linear inequalities).

I tried to use

 x = fmincon(@(a) norm(confun(a)).^2,a0, [],[],[],[],[],[],[],opts)

but the error says

 ??? Error using ==> fmincon at 217
 FMINCON is for constrained problems. Use FMINUNC for unconstrained  problems.

 Error in ==> MainProgram at 11
x = fmincon(@(a) norm(confun(a)).^2,a0, [],[],[],[],[],[],[],opts)

[Matt J]

I know that for linear inequalities I could use the matrices A and B, but I simply wrote explicitly the conditions that would derive from this matrix product and collected them in the vector "c". Should it work anyway?

It should work, but less efficiently. FMINCON can handle inequalities in an improved way when it knows that they are linear.

Could you please explain me why should norm(confun(a)).^2 be the objective function? And why norm(confun(a)).^2 and not norm(mincontr(a)).^2?

FMINCON expects the objective function to return a scalar value that you are trying to minimize. Since you said you were trying to generalize fsolve, I proposed the objective f(a)=norm(confun(a)).^2 so that the solver will look for a a least squares solution to your equalities. This is the most similar to what fsolve does in the unconstrained case.

However, I should point out that my proposal puts more priority on satisfying the inequalities than the equalities. It will try to satsify the inequalities exactly, but will give you only a least squares solution to the equalities if that's the best that satisfying the inequalities will allow. I don't know if this is acceptable to you, but it is similar to what lsqnonlin does when solving equalities with bound constraints.

I mean "confun" includes only the non linear equalities, while "minconstr" should represents the whole system (non linear equalities and linear inequalities).

I did not tell you to ignore the inequalities. I advised you to treat them as constraints and to handle them through matrix data A,B. Because you did not do this and instead made all 7 input arguments for the constraints [],[],[],[],[],[],[] empty, you received the error message

 FMINCON is for constrained problems. Use FMINUNC for unconstrained  problems.

Since you did not specify any constraints, fmincon wonders why you wouldn't just use fminunc.

[Marty]

I see, but when I tried to express the linear inequalities through the matrices A and b, matlab repeated infinite times "Warning: Matrix is singular to working precision." before giving the result.

No idea why. It worked fine for me. I obviously used a "dummy" objective function (norm(a).^2), because I don't know your confun().

Another thing you should note is that many of your inequalities are just simple lower bounds on a(5:14). For example

     c(1)=-a(5)-b(1)

is really just the lower bound a(5)>=-b(1). You should use the lb, ub input arguments for simple bounds like those. Again, this is so that fmincon can know that they are simple bounds and take advantage of that.

I tried another way ... since I don't have any objective function, but I simply want the solution of a system of non linear equalities and inequalities. Do you think this way is similar or tends to give a similar result as the one you proposed?

They should both give a solution to all of your (in)equalities as long as a solution satisfying the entire system does indeed exist. And as long as in both cases you choose a good enough initial guess that fmincon can find that solution.

If such a solution does not exist, however, you have to start prioritizing which of the (in)equalities you want satisifed rigorously and which you can tolerate least squares solutions for.