How to combine multiple objective functions for lsqnonlin?

Hello all,

I'm wondering how can I combine multiple objective functions into one so as to use "lsqnonlin" to optimize the 4 unknown parameters existing throughout each of the objective functions.

I'm currently able to optimize the parameters from a single objective function, which is comprised of "model prediction - experimental data", with the below expression:

function rss = objfnc1(params)
  a = params(1);
  b = params(2);
  c = params(3);
  d = params(4);
  rss = (pred(:) - expt(:))./(expt(:))

[paramsopt,resnorm,residual,exitflag,output] = lsqnonlin(@objfnc1,params_0,lb,ub,options);

But I'd like to combine other objective functions from different experiments under different time/temperature, for instance if there are two more objective functions "objfnc2" and "objfnc3", how can I combine them with "objfunc1" to optimize the 4 known parameters?

Can I directly concatenate the objective functions together like the following? What would be the best way to do it? I think I will be dealing with around 5-10 objective functions

objfnc = [objfnc1; objfnc2; objfnc3];

[paramsopt,resnorm,residual,exitflag,output] = lsqnonlin(@objfnc,params_0,lb,ub,options);

Thank you in advance for your great help!

Answer 1

[Abel Babu]

Hi David,

One way to go about the question is to define all the objective functions in a MATLAB function that accepts the parameters, the input data and the output data and return the objective functions as a vector. For example :

function [ rssOutput ] = objFunctions( params,X,Y) 

    a = params(1); % parameter 1
    b = params(2); % parameter 2

    rss1 = a.*exp(b.*X) - Y; % objective function 1 
    rss2 = a.*sin(b.*X) - Y; % objective function 2 

    rssOutput = [rss1; rss2]; 
  end

Now, initialize the parameters defined in the above function:

 x0 = [1,1]'; % initial value of a and b

Then get the input data ('X') and output data ('Y') and then call the ' lsqnonlin ' in the following manner:

x=lsqnonlin(@objFunctions,x0,[],[],[],X,Y);

I have left lower bound , upper bound and options parameters empty for the sake of simplicity right now, but please have a look at the documentation of lsqnonlin , for details more on the same:

http://in.mathworks.com/help/optim/ug/lsqnonlin.html?s_tid=srchtitle

This should optimize the parameters over multiple objective functions.

Regards,

Abel Paul Babu

MathWorks.

[Carlos Ortega]

Dear Abel,

I didn´t ask the original question, but your answer did help me with my problem.

You gave a very clear and easy-to-follow example. In my opinion the 'lsqnonlin' documentation could benefit a lot by adding this example you shared. Mostly, for user that doesn't have a very strong background in mathematics.

Something I'd like to highlight is taht you are passing variables X and Y to the lsqnonlin function. I also did it and it work, but Matlab documentation doesn't show this possibility (https://nl.mathworks.com/help/optim/ug/lsqnonlin.html#buuhch7-3). When you look at the input arguments it is not possible to add anything after the 'options' field

Thanks again!

Carlos

[Alan Weiss]

For the documented, supported ways of passing extra parameters, see Passing Extra Parameters.

Alan Weiss

MATLAB mathematical toolbox documentation