Optimising multi-dimensional functions with Ceres - c++

Question

I want to use Ceres to optimise a function of n parameters. How to find the gradient of this function is unclear, although I do have a well defined cost. I have been using GSL so far, with numeric differentiation, but I thought I would try to use Ceres with automatic diff.
I have had a look at the toy example helloworld_analytic_diff.cc in which they use AutoDiff to minimise the function f(x) = 0.5 (10 - x)^2 and read the tutorial, so I thought I would try and extend this to the two dimensional function f(x,y) = (10-x)^2 +(20- y)^2, which has a global minimum at x, y = 10, 20. But I am getting a little stuck on this:
#include "ceres/ceres.h"
#include "glog/logging.h"
using ceres::AutoDiffCostFunction;
using ceres::CostFunction;
using ceres::Problem;
using ceres::Solver;
using ceres::Solve;
struct CostFunctor {
template <typename T> bool operator()(const T* const x, T* residual) const {
const T x1 = x[0];
const T y1 = x[1];
residual[0] = (10.0-x[0]) + (20.0-x[1]);
return true;
}
};
int main(int argc, char** argv) {
google::InitGoogleLogging(argv[0]);
double x[2] = {0.5, -3.0};
const double initial_x[2] = {0.5, -3.0};
Problem problem;
CostFunction* cost_function = new AutoDiffCostFunction<CostFunctor, 1, 2>(new CostFunctor);
problem.AddResidualBlock(cost_function, NULL, &x[0]);
// Run the solver!
Solver::Options options;
options.minimizer_progress_to_stdout = true;
Solver::Summary summary;
Solve(options, &problem, &summary);
std::cout << summary.BriefReport() << "\n";
std::cout << "x : " << initial_x[0] << ", " << initial_x[0]
<< " -> " << x[0] << ", " << x[1]<< "\n";
return 0;
}
If I run this however, it winds up converging to something incorrect, depending on the initial guess:
Ceres Solver Report: Iterations: 3, Initial cost: 5.281250e+02, Final cost: 3.667046e-16, Termination: CONVERGENCE
x : 0.5, 0.5 -> 16.75, 13.25
Any ideas on what I've done wrong here?
Many thanks!

Answer 1

Your cost functor is wrong.
the optimization problem you are solving is
[(10.0-x) + (20.0-y)]^2
and not
(10-x)^2 +(20- y)^2,
what you should do here is to have two cost functors, something like as follows:
struct CostFunctor1 {
template <typename T> bool operator()(const T* const x, T* residual) const {
residual[0] = 10.0-x[0];
return true;
}
};
struct CostFunctor2 {
template <typename T> bool operator()(const T* const x, T* residual) const {
residual[0] = 20.0-x[1];
return true;
}
};
int main(int argc, char** argv) {
google::InitGoogleLogging(argv[0]);
double x[2] = {0.5, -3.0};
const double initial_x[2] = {0.5, -3.0};
Problem problem;
problem.AddResidualBlock(new AutoDiffCostFunction<CostFunctor1, 1, 2>(new
CostFunctor1), NULL, &x[0]);
problem.AddResidualBlock(new AutoDiffCostFunction<CostFunctor2, 1, 2>(new
CostFunctor2), NULL, &x[0]);
// Run the solver!
Solver::Options options;
options.minimizer_progress_to_stdout = true;
Solver::Summary summary;
Solve(options, &problem, &summary);
std::cout << summary.BriefReport() << "\n";
std::cout << "x : " << initial_x[0] << ", " << initial_x[0]
<< " -> " << x[0] << ", " << x[1]<< "\n";
return 0;
}