如何使用Eigen不支持的levenberg marquardt实现?

问题描述

我正在尝试最小化以下示例函数:

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

使这种功能最小化的正常方法是Levenberg-Marquardt算法.我想在C ++中执行此最小化，并做了一些初步测试与Eigen一起产生了预期的解决方案.

我的问题如下:我习惯使用scipy.optimize.fmin_powell在python中进行优化.这里输入功能参数为(func, x0, args=(), xtol=0.0001, ftol=0.0001, maxiter=None, maxfun=None, full_output=0, disp=1, retall=0, callback=None, direc=None).因此，我可以定义一个func(x0)，给出一个x0向量并开始优化.如果需要，我可以改变优化参数.

现在，Eigen Lev-Marq算法以不同的方式工作.我需要定义一个功能向量(为什么?)此外，我无法设法设置优化参数.根据:
http://eigen.tuxfamily.org/dox/unsupported/classEigen_1_1LevenbergMarquardt.html
我应该能够使用setEpsilon()和其他设置功能.

但是当我有以下代码时:

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

所以我有2个问题:

1. 为什么需要函数向量，为什么函数标量不够?
   我在其中寻找答案的参考文献:
   http://www.ultimatepp.org/reference$Eigen_demo$en-us.html
   http://www.alglib.net/optimization/levenbergmarquardt.php

2. 如何使用设置的功能设置优化参数?

解决方案

所以我相信已经找到了答案.

1)该函数能够用作函数向量和函数标量.
如果存在m个可求解的参数，则需要创建m x m的雅可比矩阵或对其进行数值计算.为了执行矩阵向量乘法J(x[m]).transpose*f(x[m])，功能向量f(x)应该具有m项.这可能是m的不同功能，但我们也可以为f1提供完整的功能，并使其他项目为0.

2)可以使用lm.parameters.maxfev = 2000;

设置和读取参数

两个答案均已在下面的示例代码中进行了测试:

#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;
}

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 testswith 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. Herethe 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 changethe optimization parameters.

Now the Eigen Lev-Marq algorithm works in a different way. I need to define a functionvector (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?

解决方案

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;
}

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