How to use the Eigen unsupported levenberg marquardt implementation?

Question

I'm trying to minimize a following sample function:

F(x) = f[0]^2(x[0],...,x[n-1]) + ... + f[m-1]^2(x[0],...,x[n-1])

A normal way to minimize such a funct could be the Levenberg-Marquardt algorithm. I would like to perform this minimization in c++ and have done some initial tests with Eigen that resulted in the expected solution.

My question is the following: I'm used to optimization in python using i.e. scipy.optimize.fmin_powell. Here the input function parameters are (func, x0, args=(), xtol=0.0001, ftol=0.0001, maxiter=None, maxfun=None, full_output=0, disp=1, retall=0, callback=None, direc=None). So I can define a func(x0), give the x0 vector and start optimizing. If needed I can change the optimization parameters.

Now the Eigen Lev-Marq algorithm works in a different way. I need to define a function vector (why?) Furthermore I can't manage to set the optimization parameters. According to:
http://eigen.tuxfamily.org/dox/unsupported/classEigen_1_1LevenbergMarquardt.html
I should be able to use the setEpsilon() and other set functions.

But when I have the following code:

my_functor functor;
Eigen::NumericalDiff<my_functor> numDiff(functor);
Eigen::LevenbergMarquardt<Eigen::NumericalDiff<my_functor>,double> lm(numDiff);
lm.setEpsilon(); //doesn't exist!

So I have 2 questions:

1. Why is a function vector needed and why wouldn't a function scalar be enough?
   References where I've searched for an answer:
   http://www.ultimatepp.org/reference$Eigen_demo$en-us.html
   http://www.alglib.net/optimization/levenbergmarquardt.php
   

2. How do I set the optimization parameters using the set functions?

Answer 1

So I believe I've found the answers.

1) The function is able to work as a function vector and as a function scalar.
If there are m solveable parameters, a Jacobian matrix of m x m needs to be created or numerically calculated. In order to do a Matrix-Vector multiplication J(x[m]).transpose*f(x[m]) the function vector f(x) should have m items. This can be the m different functions, but we can also give f1 the complete function and make the other items 0.

2) The parameters can be set and read using lm.parameters.maxfev = 2000;

Both answers have been tested in the following example code:

#include <iostream>
#include <Eigen/Dense>

#include <unsupported/Eigen/NonLinearOptimization>
#include <unsupported/Eigen/NumericalDiff>

// Generic functor
template<typename _Scalar, int NX = Eigen::Dynamic, int NY = Eigen::Dynamic>
struct Functor
{
typedef _Scalar Scalar;
enum {
    InputsAtCompileTime = NX,
    ValuesAtCompileTime = NY
};
typedef Eigen::Matrix<Scalar,InputsAtCompileTime,1> InputType;
typedef Eigen::Matrix<Scalar,ValuesAtCompileTime,1> ValueType;
typedef Eigen::Matrix<Scalar,ValuesAtCompileTime,InputsAtCompileTime> JacobianType;

int m_inputs, m_values;

Functor() : m_inputs(InputsAtCompileTime), m_values(ValuesAtCompileTime) {}
Functor(int inputs, int values) : m_inputs(inputs), m_values(values) {}

int inputs() const { return m_inputs; }
int values() const { return m_values; }

};

struct my_functor : Functor<double>
{
my_functor(void): Functor<double>(2,2) {}
int operator()(const Eigen::VectorXd &x, Eigen::VectorXd &fvec) const
{
    // Implement y = 10*(x0+3)^2 + (x1-5)^2
    fvec(0) = 10.0*pow(x(0)+3.0,2) +  pow(x(1)-5.0,2);
    fvec(1) = 0;

    return 0;
}
};


int main(int argc, char *argv[])
{
Eigen::VectorXd x(2);
x(0) = 2.0;
x(1) = 3.0;
std::cout << "x: " << x << std::endl;

my_functor functor;
Eigen::NumericalDiff<my_functor> numDiff(functor);
Eigen::LevenbergMarquardt<Eigen::NumericalDiff<my_functor>,double> lm(numDiff);
lm.parameters.maxfev = 2000;
lm.parameters.xtol = 1.0e-10;
std::cout << lm.parameters.maxfev << std::endl;

int ret = lm.minimize(x);
std::cout << lm.iter << std::endl;
std::cout << ret << std::endl;

std::cout << "x that minimizes the function: " << x << std::endl;

std::cout << "press [ENTER] to continue " << std::endl;
std::cin.get();
return 0;
}