How to pass user-defined gradients for anonymous non-linear constraints in fmincon

Hi to all,

I have a non-linear optimization problem where both non-linear constraints and objective functions have to be defined anonymously. I want to use fmincon as efficiently as possible by providing user-supplied gradients for both gradients and objective, and user-supplied Hessian for objective.

How can I pass these gradients for anonymously defined functions?

Thanks in advance

Ebru

[Matt J]

where both non-linear constraints and objective functions have to be defined anonymously.

Why is that?

[Ebru Angun]

This is the case when black-box simulation optimization problem is solved, where the mathematical forms of the functions are unknown, and they are to be approximated by some meta-models. These meta-models and possibly their gradients are to be evaluated at currently unknown x, which will be found iteratively by the fmincon.

[Matt J]

That still doesn't explain why you think the functions need to beanonymous, rather than non-anonymous functions. Why can't your blackbox be non-anonymous?

[Ebru Angun]

I have the idea that I am using the term anonymous function wrongly because I have the following problem.

I have the following command in the main program which provides the nonlinear constraint as a function of x:

nonlcon=@(x)non_linear_constraints(x,model_disservice,alpha,q,n,k,X_design);

The function non_linear_constraints is as follows. In this function, SKpredict_new is another function to evaluate y_hat_disservice and mspe_disservice at a currently unknown x.

c=zeros(q-1,1);

ceq=[];

z_a = norminv(1-(alpha/2));

B_predict=1;

[y_hat_disservice,mspe_disservice] = SKpredict_new(model_disservice,x,B_predict);

c(1)= y_hat_disservice-z_a * sqrt(mspe_disservice);

[Matt J]

You should just modify non_linear_constraints to (optionally) return the gradients:

function [c,ceq,gc,gceq]=non_linear_constraints(x,model_disservice,alpha,q,n,k,X_design);   

    c=zeros(q-1,1); 
    ceq=[];
    z_a = norminv(1-(alpha/2));
    B_predict=1;
    [y_hat_disservice,mspe_disservice] = SKpredict_new(model_disservice,x,B_predict);
    c(1)= y_hat_disservice-z_a * sqrt(mspe_disservice); 
   
   if nargout>3 %check if fmincon really asked for the gradients
   
       gc=...
       gceq=[];
       
   end
   
end

[Ebru Angun]

Yes, indeed. Thanks

Answer 1

[Matt J]

I suspect you've misled yourself into believing the functions must be defined anonymously. I don't know of any situation where that truly is the case. Nevertheless, you can accomplish what you ask as in the example below. The convergence of fmincon will benefit from providing the gradients, however, you will never be able to get the same efficiency as a non-anonymous function implemention. This is because an anonymous function must always compute the function gradients whenever it is invoked, even at steps where fmincon requires only the function values.

nonlcon=@(x)f(nargout,{x^2-1,tan(x)+3,2*x,sec(x)^2});

[c,ceq]=nonlcon(1)

c =

0

ceq =

4.5574

[c,ceq,gc,gceq]=nonlcon(1)

c =

0

ceq =

4.5574

gc =

2

gceq =

3.4255

function varargout=f(n,c)

 varargout=c(1:n);

end

[Ebru Angun]

I thought that fmincon requires a gradient to proceed (either user-supplied or computed through finite-difference by the function). Are there cases where fmincon uses only function evaluations?

Thanks

Ebru

[Matt J]

For example, some of the fmincon algorithms involve line search steps (e.g. SQP and, with certain settings, interior-point). Line searches will involve repeated evaluation of the function, but not its derivatives.

[Ebru Angun]

I see, thanks for the info. I will check interior-point since this is the algorithm I am planning to use.