最快下降法

理解：

$F(x)^{'} = \lim_{\Delta x\rightarrow 0}\frac{F(x+\Delta x)-F(x)}{\Delta x}$

a*b = |a|*|b|*cos( $\theta$ )

最小二乘问题

通常的最小二乘问题都可以表示为：

GaussNewton

close all;
clc;a=1;b=2;c=1;               %待求解的系数x=(0:0.01:1)';
w=rand(length(x),1)*2-1;   %生成噪声
y_noise=exp(a*x.^2+b*x+c)+w;     %带噪声的模型 pre=rand(3,1);  %步骤1，待拟合的参数，初始值是随机数；for i=1:1000 f = exp(pre(1)*x.^2+pre(2)*x+pre(3));   % g = y_noise-f;                                % 步骤2，构建拟合的函数和真实函数的误差；视觉中，也是根据观测的值减去预测的值，然后优化其中的参数，使两者最小； p1 = exp(pre(1)*x.^2+pre(2)*x+pre(3)).*x.^2;    % 残差函数g对a求偏导，在ceres中也是有对应的偏导项的p2 = exp(pre(1)*x.^2+pre(2)*x+pre(3)).*x;       % 残差函数g对b求偏导；p3 = exp(pre(1)*x.^2+pre(2)*x+pre(3));          % 残差函数g对c求偏导；J = [p1 p2 p3];                                 % 步骤2中的雅克比矩阵delta = inv(J'*J)*J'* g;                        %步骤3，inv(J'*J)*J'为H的逆pcur = pre+delta;                               %步骤4if norm(delta) <1e-16break;endpre = pcur;
endyreal = exp(a*x.^2+b*x+c);
ypre=exp(pre(1)*x.^2+pre(2)*x+pre(3));  % 拟合的曲线figure(1)
plot(x,y_noise,'b')
hold on
plot(x,yreal,'r')
hold on
plot(x,ypre,'k');
grid on;
set(gca, 'GridLineStyle', '--');   % 设置为虚线
set(gca, 'GridAlpha', 1.0);        % 设置透明度
xlabel('X');
ylabel('Y');
legend('带有噪声的真实曲线','不带噪声的真实曲线','拟合曲线')figure(2)
plot(x,yreal - ypre,'b')
grid on;
set(gca, 'GridLineStyle', '--');   % 设置为虚线
set(gca, 'GridAlpha', 1.0);        % 设置透明度
xlabel('X');
ylabel('Y');
legend('真值和优化值之间的误差曲线')%比较一下
[a b c]
pre'

LM阻尼最小二乘法

Dog-Leg最小二乘法

举例

Gauss Newton代码：

double func(const VectorXd& input, const VectorXd& output, const VectorXd& params, double objIndex)
{// obj = A * sin(Bx) + C * cos(D*x) - Fdouble x1 = params(0);double x2 = params(1);double x3 = params(2);double x4 = params(3);double t = input(objIndex);double f = output(objIndex);return x1 * sin(x2 * t) + x3 * cos( x4 * t) - f;
}//return vector make up of func() element.
VectorXd objF(const VectorXd& input, const VectorXd& output, const VectorXd& params)
{VectorXd obj(input.rows());for(int i = 0; i < input.rows(); i++)obj(i) = func(input, output, params, i);return obj;
}//F = (f ^t * f)/2
double Func(const VectorXd& obj)
{return obj.squaredNorm()/2;
}double Deriv(const VectorXd& input, const VectorXd& output, int objIndex, const VectorXd& params,int paraIndex)
{VectorXd para1 = params;VectorXd para2 = params;para1(paraIndex) -= DERIV_STEP;para2(paraIndex) += DERIV_STEP;double obj1 = func(input, output, para1, objIndex);double obj2 = func(input, output, para2, objIndex);return (obj2 - obj1) / (2 * DERIV_STEP);
}MatrixXd Jacobin(const VectorXd& input, const VectorXd& output, const VectorXd& params)
{int rowNum = input.rows();int colNum = params.rows();MatrixXd Jac(rowNum, colNum);for (int i = 0; i < rowNum; i++){for (int j = 0; j < colNum; j++){Jac(i,j) = Deriv(input, output, i, params, j);}}return Jac;
}void gaussNewton(const VectorXd& input, const VectorXd& output, VectorXd& params)
{int errNum = input.rows();      //error  numint paraNum = params.rows();    //parameter  numVectorXd obj(errNum);double last_sum = 0;int iterCnt = 0;while (iterCnt < MAX_ITER){obj = objF(input, output, params);double sum = 0;sum = Func(obj);cout << "Iterator index: " << iterCnt << endl;cout << "parameter: " << endl << params << endl;cout << "error sum: " << endl << sum << endl << endl;if (fabs(sum - last_sum) <= 1e-12)break;last_sum = sum;MatrixXd Jac = Jacobin(input, output, params);VectorXd delta(paraNum);delta = (Jac.transpose() * Jac).inverse() * Jac.transpose() * obj;params -= delta;iterCnt++;}
}

LM代码：

double maxMatrixDiagonale(const MatrixXd& Hessian)
{int max = 0;for(int i = 0; i < Hessian.rows(); i++){if(Hessian(i,i) > max)max = Hessian(i,i);}return max;
}//L(h) = F(x) + h^t*J^t*f + h^t*J^t*J*h/2
//deltaL = h^t * (u * h - g)/2
double linerDeltaL(const VectorXd& step, const VectorXd& gradient, const double u)
{double L = step.transpose() * (u * step - gradient);return L/2;
}void levenMar(const VectorXd& input, const VectorXd& output, VectorXd& params)
{int errNum = input.rows();      //error numint paraNum = params.rows();    //parameter num//initial parameter VectorXd obj = objF(input,output,params);MatrixXd Jac = Jacobin(input, output, params);  //jacobinMatrixXd A = Jac.transpose() * Jac;             //HessianVectorXd gradient = Jac.transpose() * obj;      //gradient//initial parameter tao v epsilon1 epsilon2double tao = 1e-3;long long v = 2;double eps1 = 1e-12, eps2 = 1e-12;double u = tao * maxMatrixDiagonale(A);bool found = gradient.norm() <= eps1;if(found) return;double last_sum = 0;int iterCnt = 0;while (iterCnt < MAX_ITER){VectorXd obj = objF(input,output,params);MatrixXd Jac = Jacobin(input, output, params);  //jacobinMatrixXd A = Jac.transpose() * Jac;             //HessianVectorXd gradient = Jac.transpose() * obj;      //gradientif( gradient.norm() <= eps1 ){cout << "stop g(x) = 0 for a local minimizer optimizer." << endl;break;}cout << "A: " << endl << A << endl; VectorXd step = (A + u * MatrixXd::Identity(paraNum, paraNum)).inverse() * gradient; //negtive Hlm.cout << "step: " << endl << step << endl;if( step.norm() <= eps2*(params.norm() + eps2) ){cout << "stop because change in x is small" << endl;break;} VectorXd paramsNew(params.rows());paramsNew = params - step; //h_lm = -step;//compute f(x)obj = objF(input,output,params);//compute f(x_new)VectorXd obj_new = objF(input,output,paramsNew);double deltaF = Func(obj) - Func(obj_new);double deltaL = linerDeltaL(-1 * step, gradient, u);double roi = deltaF / deltaL;cout << "roi is : " << roi << endl;if(roi > 0){params = paramsNew;u *= max(1.0/3.0, 1-pow(2*roi-1, 3));v = 2;}else{u = u * v;v = v * 2;}cout << "u = " << u << " v = " << v << endl;iterCnt++;cout << "Iterator " << iterCnt << " times, result is :" << endl << endl;}
}

Dog-Leg代码：

void dogLeg(const VectorXd& input, const VectorXd& output, VectorXd& params)
{int errNum = input.rows();      //error numint paraNum = params.rows();    //parameter numVectorXd obj = objF(input, output, params);MatrixXd Jac = Jacobin(input, output, params);  //jacobinVectorXd gradient = Jac.transpose() * obj;      //gradient//initial parameter tao v epsilon1 epsilon2double eps1 = 1e-12, eps2 = 1e-12, eps3 = 1e-12;double radius = 1.0;bool found  = obj.norm() <= eps3 || gradient.norm() <= eps1;if(found) return;double last_sum = 0;int iterCnt = 0;while(iterCnt < MAX_ITER){VectorXd obj = objF(input, output, params);MatrixXd Jac = Jacobin(input, output, params);  //jacobinVectorXd gradient = Jac.transpose() * obj;      //gradientif( gradient.norm() <= eps1 ){cout << "stop F'(x) = g(x) = 0 for a global minimizer optimizer." << endl;break;}if(obj.norm() <= eps3){cout << "stop f(x) = 0 for f(x) is so small";break;}//compute how far go along stepest descent direction.double alpha = gradient.squaredNorm() / (Jac * gradient).squaredNorm();//compute gauss newton step and stepest descent step.VectorXd stepest_descent = -alpha * gradient;VectorXd gauss_newton = (Jac.transpose() * Jac).inverse() * Jac.transpose() * obj * (-1);double beta = 0;//compute dog-leg step.VectorXd dog_leg(params.rows());if(gauss_newton.norm() <= radius)dog_leg = gauss_newton;else if(alpha * stepest_descent.norm() >= radius)dog_leg = (radius / stepest_descent.norm()) * stepest_descent;else{VectorXd a = alpha * stepest_descent;VectorXd b = gauss_newton;double c = a.transpose() * (b - a);beta = (sqrt(c*c + (b-a).squaredNorm()*(radius*radius-a.squaredNorm()))-c)/(b-a).squaredNorm();dog_leg = alpha * stepest_descent + beta * (gauss_newton - alpha * stepest_descent);}cout << "dog-leg: " << endl << dog_leg << endl;if(dog_leg.norm() <= eps2 *(params.norm() + eps2)){cout << "stop because change in x is small" << endl;break;}VectorXd new_params(params.rows());new_params = params + dog_leg;cout << "new parameter is: " << endl << new_params << endl;//compute f(x)obj = objF(input,output,params);//compute f(x_new)VectorXd obj_new = objF(input,output,new_params);//compute delta F = F(x) - F(x_new)double deltaF = Func(obj) - Func(obj_new);//compute delat L =L(0)-L(dog_leg)double deltaL = 0;if(gauss_newton.norm() <= radius)deltaL = Func(obj);else if(alpha * stepest_descent.norm() >= radius)deltaL = radius*(2*alpha*gradient.norm() - radius)/(2.0*alpha);else{VectorXd a = alpha * stepest_descent;VectorXd b = gauss_newton;double c = a.transpose() * (b - a);beta = (sqrt(c*c + (b-a).squaredNorm()*(radius*radius-a.squaredNorm()))-c)/(b-a).squaredNorm();deltaL = alpha*(1-beta)*(1-beta)*gradient.squaredNorm()/2.0 + beta*(2.0-beta)*Func(obj);}double roi = deltaF / deltaL;if(roi > 0){params = new_params;}if(roi > 0.75){radius = max(radius, 3.0 * dog_leg.norm());}else if(roi < 0.25){radius = radius / 2.0;if(radius <= eps2*(params.norm()+eps2)){cout << "trust region radius is too small." << endl;break;}}cout << "roi: " << roi << " dog-leg norm: " << dog_leg.norm() << endl;cout << "radius: " << radius << endl;iterCnt++;cout << "Iterator " << iterCnt << " times" << endl << endl;}
}

main()

#include <eigen3/Eigen/Dense>
#include <eigen3/Eigen/Sparse>
#include <iostream>
#include <iomanip>
#include <math.h>using namespace std;
using namespace Eigen;const double DERIV_STEP = 1e-5;
const int MAX_ITER = 100;#define max(a,b) (((a)>(b))?(a):(b))int main(int argc, char* argv[])
{// obj = A * sin(Bx) + C * cos(D*x) - F//there are 4 parameter: A, B, C, D.int num_params = 4;//generate random data using these parameterint total_data = 100;VectorXd input(total_data);VectorXd output(total_data);double A = 5, B= 1, C = 10, D = 2;//load observation datafor (int i = 0; i < total_data; i++){//generate a random variable [-10 10]double x = 20.0 * ((random() % 1000) / 1000.0) - 10.0;double deltaY = 2.0 * (random() % 1000) /1000.0;double y = A*sin(B*x)+C*cos(D*x) + deltaY;input(i) = x;output(i) = y;}//gauss the parametersVectorXd params_gaussNewton(num_params);//init gaussparams_gaussNewton << 1.6, 1.4, 6.2, 1.7;VectorXd params_levenMar = params_gaussNewton;VectorXd params_dogLeg   = params_gaussNewton;gaussNewton(input, output, params_gaussNewton);levenMar(input, output, params_levenMar);dogLeg(input, output, params_dogLeg);cout << "gauss newton parameter: " << endl << params_gaussNewton << endl << endl << endl;cout << "Levenberg-Marquardt parameter: " << endl << params_levenMar << endl << endl << endl;cout << "dog-leg parameter: " << endl << params_dogLeg << endl << endl << endl;
}

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非线性最小二乘法之Gauss Newton、L-M、Dog-Leg9

最优化理论·非线性最小二乘

最快下降法

最小二乘问题

GaussNewton

LM阻尼最小二乘法

Dog-Leg最小二乘法

举例

Gauss Newton代码：

LM代码：

Dog-Leg代码：

main()

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