Provide gradient for fmincon when the function to minimize uses ODE

I am trying to fit a model to experimental points by using the function fmincon. The function I need to minimize is defined in a file called objfunction and is as following:

mi=zeros(n_exp,n_comp); 

for i=1: n_exp
    mi(i,:)=(pts_exp(i,:)-pts_mod(i,:)).^2;
end

Summin=sum(sum(mi));

Where pts_exp is a matrix obtained by solving differential equations using ode15s. I set the constraints (inequalities and bounds) and use fmincon:

[para,fval]=fmincon(@(para)objfunction(pts_exp,para,t_T_exp,n0,ordre,n_exp,n_comp,reactif,n_react),param0,A,b,Aeq,beq,lb,ub,nonlcon,options);

The vector para has 864 decision variables, all of them with lower and upper bounds and it takes around 10 minutes to optimize my problem. I know that providing the gradient and the hessian can make the code run faster but I do not know how to do it since my objective function uses a matrix given by ode15s. What I mean is that I do not have an analytical expression of my function as in f(x,y)=x+y or something like that. Is there any way of providing the gradient or making the code run faster?

Any help is greatly appreciated!

Answer 1

[Torsten]

Look up "sensitivity equations for ordinary differential equations".

The solution of these equations give you the derivative of the ODE solution with respect to the parameters.

Best wishes

Torsten.

[Alan Weiss]

The technique that Torsten refers to is outlined in the documentation topic Use a Gradient Evaluation Function. To do this involves solving a larger set of ODEs than the original set (you would need 864 new variables in your ODE). So it is not clear to me whether there would be a worthwhile speedup. But, if it is important to you, then you can try and see.

Good luck,

Alan Weiss

MATLAB mathematical toolbox documentation

[FM]

Thank you Torsten and Alan,

My main goal is to reduce computational time, I will take a look at the documentation. I know that computational time does not only depend on the number of variables but other factors as well, but I was wondering if the computational time I get for my number of variables, boundaries and constraints and the fact that the code involves solving ODE is considered "normal". It is my first time using fmincon so I do not know if it is possible to give an estimate on computational time.

I would like to know because in the future I might have more variables and I'd like to know if I should look at alternatives to optimize my function or if by improving my code fmincon will be a good tool to use.

Thank you

[Torsten]

More than 864 parameters to be fitted ? You are kidding, aren't you ?

[FM]

I have a set of 27 chemical reactions and 30 components and I am trying to optimize the stoichiometric coefficients as well as kinetic parameters, not all components intervene in all reactions though and I set those coefficients to zero in the vector containing the decision variables and lower and upper bounds for the optimization to zero as well, (maybe I am going the wrong way about that) so yes, I might have more than 864 parameters