Write the f(c) as an anonymous function

I have to test the following problem in an optimization course.

The following function f(c) is used to fit a set of data Dn.

$$

f(c)=\sum_{i=1}^{n}|c_1\ x_i+c_2+c_3 e^{(-(x_i-c_4)^2/c_5)}-y_i\ |^2\

\\c=\left(\begin{matrix}c_1\\\begin{matrix}c_2\\\begin{matrix}c_3\\\begin{matrix}c_4\\c_5\\\end{matrix}\\\end{matrix}\\\end{matrix}\\\end{matrix}\right)\in\mathbb{R}^5\\

D_n=\left\{\left(x_i,\ y_i\right)\ \right|\ i=1,\ \ldots,\ n)

$$

I wish to minimize this function, using the steepest descent algorithm, shown below:

function x=steep_descent_non_quad(f,nablaf,x0,tol,kmax)
% f is a scalar function on R^n
% nablaf is a vector function on R^n
x=x0;r=nablaf(x0);RelErr=1;k=1;n0=norm(x0);
while (k<=kmax)&&(RelErr>tol)
% Compute the optimal step
alpha=optim(f,d,x,tol/2);
% The function optim seeks an approximation to the exact optimal alpha_k
% Using either bisection or golden section
x=x-alpha*r;
RelErr=abs(alpha)*norm(d)/n0;
% Compute new remainder
r=nablaf(x);
k=k+1;
end

I want to write the function f(c) in a way that fits its use in the algorithm.

Let's say I write it this way

c = ones(5,1);
f = @(c)(sum(abs(c(1)*x + c(2) + c(3)*exp(-(x-c(4)^2 / c(5) - y)^2))));

1- Is this correct?

Will this function provide the sum from i to n?

Or should I write it otherwise?

2- can MATLAB directly give me the gradient vector of f or should I program it myself?

Thank you for your help!

Answer 1

[Matt J]

Is this correct?

There is no need for abs(). Also, your division and exponentiation operations need to be element-wise,

f = @(c) sum( ( c(1)*x + c(2) + c(3)*exp(-(x-c(4)).^2 ./ c(5) - y).^2 );

can MATLAB directly give me the gradient vector of f or should I program it myself?

If you want an analytical gradient calculation, you must do it yourself. However, there are file exchange tools that will implement a finite difference derivative calculation, if you're willing to settle for that, e.g.,

https://www.mathworks.com/matlabcentral/fileexchange/13490-adaptive-robust-numerical-differentiation

[Charbel]

@Matt J Thank you for the answer. Should I define the vector c beforehand though?

[Matt J]

Not for the purpose of creating the anonymous function. Obviously, you need an initial guess c0.

[Charbel]

@Matt J thanks a lot!

[Matt J]

You're quite welcome, but please Accept-click the answer to indicate that your quesiton is resolved.

[Charbel]

One more question, when I run the code with that function f(c), will it do the sum based on all the (x,y) pairs?

[Matt J]

It should, but you can easily check it.

[Charbel]

ok thank you.

[Walter Roberson]

With the c being in R, the abs() would be needed only if x or y are in C... we are not actually told that they are in R.

[Torsten]

I think a bracket is missing:

f = @(c) sum( ( c(1)*x + c(2) + c(3)*exp(-(x-c(4)).^2 ./ c(5) ) - y).^2 );

instead of

f = @(c) sum( ( c(1)*x + c(2) + c(3)*exp(-(x-c(4)).^2 ./ c(5) - y).^2 );

And maybe you have to assume c(5) > 0 in the optimizer.