What is this strange difference between c++ eigen Levenberg-Marquardt and Java L-M optimizer implementation?

Question

Related background and interconnected problem posted on math stackexchange: https://math.stackexchange.com/questions/3972648/non-linear-optimization-levenberg-marquardt-gives-different-results-using-diffe/3972674

Posting this question here as it is not (only) math related.

I'm using L-M algorithm to minimize (x*,y*) = SUM(x-x0)^2+(y-y0)^2-r^2

Forms of function to minimize (non-squared as both c++ and java libs squares it for us apparently):

1. sqrt((x-x0)^2+(y-y0)^2) - r

2. 1 - sqrt((x-x0)^2+(y-y0)^2)/r

Using java L-M algorithm both forms give good result (almost identical, very small difference, I wonder what is the reason for the diff.)

BUT using Eigen's L-M algorithm only the 2nd form is working properly. Somehow the 1st form has huge error.

What do you think is the problem here? I'm wondering if there is a fundamental difference in the JAVA/Eigen c++ L-M implementation? Would be great to clear this up.

I tried eigen's L-M with automatic and "handwritten" derivatives for both forms. In Java I only used "handwritten" derivatives.

This is how I use eigen:

int operator()(const Eigen::VectorXd &b, Eigen::VectorXd &fvec) const {
for(int i = 0; i < matrix.rows(); i++) {
  fvec[i] = 1 - sqrt(pow(matrix(i, 0) - b[0], 2) + pow(matrix(i, 1) - b[1], 2)) / matrix(i, 2);
  //fvec[i] = sqrt(pow(matrix(i, 0) - b[0], 2) + pow(matrix(i, 1) - b[1], 2)) - matrix(i, 2);

  /*
    * Important: LevenbergMarquardt is designed to work with objective functions that are a sum
    * of squared terms. The algorithm takes this into account: do not do it yourself.
    * In other words: objFun = sum(fvec(i)^2)
    */
}

return 0;}

Answer 1

等待大神答复