// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2015 Google Inc. All rights reserved.
// http://ceres-solver.org/
//
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are met:
//
// * Redistributions of source code must retain the above copyright notice,
// this list of conditions and the following disclaimer.
// * Redistributions in binary form must reproduce the above copyright notice,
// this list of conditions and the following disclaimer in the documentation
// and/or other materials provided with the distribution.
// * Neither the name of Google Inc. nor the names of its contributors may be
// used to endorse or promote products derived from this software without
// specific prior written permission.
//
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
// POSSIBILITY OF SUCH DAMAGE.
//
// Author: sameeragarwal@google.com (Sameer Agarwal)
//
// The National Institute of Standards and Technology has released a
// set of problems to test non-linear least squares solvers.
//
// More information about the background on these problems and
// suggested evaluation methodology can be found at:
//
// http://www.itl.nist.gov/div898/strd/nls/nls_info.shtml
//
// The problem data themselves can be found at
//
// http://www.itl.nist.gov/div898/strd/nls/nls_main.shtml
//
// The problems are divided into three levels of difficulty, Easy,
// Medium and Hard. For each problem there are two starting guesses,
// the first one far away from the global minimum and the second
// closer to it.
//
// A problem is considered successfully solved, if every components of
// the solution matches the globally optimal solution in at least 4
// digits or more.
//
// This dataset was used for an evaluation of Non-linear least squares
// solvers:
//
// P. F. Mondragon & B. Borchers, A Comparison of Nonlinear Regression
// Codes, Journal of Modern Applied Statistical Methods, 4(1):343-351,
// 2005.
//
// The results from Mondragon & Borchers can be summarized as
// Excel Gnuplot GaussFit HBN MinPack
// Average LRE 2.3 4.3 4.0 6.8 4.4
// Winner 1 5 12 29 12
//
// Where the row Winner counts, the number of problems for which the
// solver had the highest LRE.
// In this file, we implement the same evaluation methodology using
// Ceres. Currently using Levenberg-Marquardt with DENSE_QR, we get
//
// Excel Gnuplot GaussFit HBN MinPack Ceres
// Average LRE 2.3 4.3 4.0 6.8 4.4 9.4
// Winner 0 0 5 11 2 41
#include <iostream>
#include <iterator>
#include <fstream>
#include "ceres/ceres.h"
#include "gflags/gflags.h"
#include "glog/logging.h"
#include "Eigen/Core"
DEFINE_string(nist_data_dir, "", "Directory containing the NIST non-linear"
"regression examples");
DEFINE_string(minimizer, "trust_region",
"Minimizer type to use, choices are: line_search & trust_region");
DEFINE_string(trust_region_strategy, "levenberg_marquardt",
"Options are: levenberg_marquardt, dogleg");
DEFINE_string(dogleg, "traditional_dogleg",
"Options are: traditional_dogleg, subspace_dogleg");
DEFINE_string(linear_solver, "dense_qr", "Options are: "
"sparse_cholesky, dense_qr, dense_normal_cholesky and"
"cgnr");
DEFINE_string(preconditioner, "jacobi", "Options are: "
"identity, jacobi");
DEFINE_string(line_search, "wolfe",
"Line search algorithm to use, choices are: armijo and wolfe.");
DEFINE_string(line_search_direction, "lbfgs",
"Line search direction algorithm to use, choices: lbfgs, bfgs");
DEFINE_int32(max_line_search_iterations, 20,
"Maximum number of iterations for each line search.");
DEFINE_int32(max_line_search_restarts, 10,
"Maximum number of restarts of line search direction algorithm.");
DEFINE_string(line_search_interpolation, "cubic",
"Degree of polynomial aproximation in line search, "
"choices are: bisection, quadratic & cubic.");
DEFINE_int32(lbfgs_rank, 20,
"Rank of L-BFGS inverse Hessian approximation in line search.");
DEFINE_bool(approximate_eigenvalue_bfgs_scaling, false,
"Use approximate eigenvalue scaling in (L)BFGS line search.");
DEFINE_double(sufficient_decrease, 1.0e-4,
"Line search Armijo sufficient (function) decrease factor.");
DEFINE_double(sufficient_curvature_decrease, 0.9,
"Line search Wolfe sufficient curvature decrease factor.");
DEFINE_int32(num_iterations, 10000, "Number of iterations");
DEFINE_bool(nonmonotonic_steps, false, "Trust region algorithm can use"
" nonmonotic steps");
DEFINE_double(initial_trust_region_radius, 1e4, "Initial trust region radius");
DEFINE_bool(use_numeric_diff, false,
"Use numeric differentiation instead of automatic "
"differentiation.");
DEFINE_string(numeric_diff_method, "ridders", "When using numeric "
"differentiation, selects algorithm. Options are: central, "
"forward, ridders.");
DEFINE_double(ridders_step_size, 1e-9, "Initial step size for Ridders "
"numeric differentiation.");
DEFINE_int32(ridders_extrapolations, 3, "Maximal number of Ridders "
"extrapolations.");
namespace ceres {
namespace examples {
using Eigen::Dynamic;
using Eigen::RowMajor;
typedef Eigen::Matrix<double, Dynamic, 1> Vector;
typedef Eigen::Matrix<double, Dynamic, Dynamic, RowMajor> Matrix;
using std::atof;
using std::atoi;
using std::cout;
using std::ifstream;
using std::string;
using std::vector;
void SplitStringUsingChar(const string& full,
const char delim,
vector<string>* result) {
std::back_insert_iterator< vector<string> > it(*result);
const char* p = full.data();
const char* end = p + full.size();
while (p != end) {
if (*p == delim) {
++p;
} else {
const char* start = p;
while (++p != end && *p != delim) {
// Skip to the next occurence of the delimiter.
}
*it++ = string(start, p - start);
}
}
}
bool GetAndSplitLine(ifstream& ifs, vector<string>* pieces) {
pieces->clear();
char buf[256];
ifs.getline(buf, 256);
SplitStringUsingChar(string(buf), ' ', pieces);
return true;
}
void SkipLines(ifstream& ifs, int num_lines) {
char buf[256];
for (int i = 0; i < num_lines; ++i) {
ifs.getline(buf, 256);
}
}
class NISTProblem {
public:
explicit NISTProblem(const string& filename) {
ifstream ifs(filename.c_str(), ifstream::in);
CHECK(ifs) << "Unable to open : " << filename;
vector<string> pieces;
SkipLines(ifs, 24);
GetAndSplitLine(ifs, &pieces);
const int kNumResponses = atoi(pieces[1].c_str());
GetAndSplitLine(ifs, &pieces);
const int kNumPredictors = atoi(pieces[0].c_str());
GetAndSplitLine(ifs, &pieces);
const int kNumObservations = atoi(pieces[0].c_str());
SkipLines(ifs, 4);
GetAndSplitLine(ifs, &pieces);
const int kNumParameters = atoi(pieces[0].c_str());
SkipLines(ifs, 8);
// Get the first line of initial and final parameter values to
// determine the number of tries.
GetAndSplitLine(ifs, &pieces);
const int kNumTries = pieces.size() - 4;
predictor_.resize(kNumObservations, kNumPredictors);
response_.resize(kNumObservations, kNumResponses);
initial_parameters_.resize(kNumTries, kNumParameters);
final_parameters_.resize(1, kNumParameters);
// Parse the line for parameter b1.
int parameter_id = 0;
for (int i = 0; i < kNumTries; ++i) {
initial_parameters_(i, parameter_id) = atof(pieces[i + 2].c_str());
}
final_parameters_(0, parameter_id) = atof(pieces[2 + kNumTries].c_str());
// Parse the remaining parameter lines.
for (int parameter_id = 1; parameter_id < kNumParameters; ++parameter_id) {
GetAndSplitLine(ifs, &pieces);
// b2, b3, ....
for (int i = 0; i < kNumTries; ++i) {
initial_parameters_(i, parameter_id) = atof(pieces[i + 2].c_str());
}
final_parameters_(0, parameter_id) = atof(pieces[2 + kNumTries].c_str());
}
// Certfied cost
SkipLines(ifs, 1);
GetAndSplitLine(ifs, &pieces);
certified_cost_ = atof(pieces[4].c_str()) / 2.0;
// Read the observations.
SkipLines(ifs, 18 - kNumParameters);
for (int i = 0; i < kNumObservations; ++i) {
GetAndSplitLine(ifs, &pieces);
// Response.
for (int j = 0; j < kNumResponses; ++j) {
response_(i, j) = atof(pieces[j].c_str());
}
// Predictor variables.
for (int j = 0; j < kNumPredictors; ++j) {
predictor_(i, j) = atof(pieces[j + kNumResponses].c_str());
}
}
}
Matrix initial_parameters(int start) const { return initial_parameters_.row(start); } // NOLINT
Matrix final_parameters() const { return final_parameters_; }
Matrix predictor() const { return predictor_; }
Matrix response() const { return response_; }
int predictor_size() const { return predictor_.cols(); }
int num_observations() const { return predictor_.rows(); }
int response_size() const { return response_.cols(); }
int num_parameters() const { return initial_parameters_.cols(); }
int num_starts() const { return initial_parameters_.rows(); }
double certified_cost() const { return certified_cost_; }
private:
Matrix predictor_;
Matrix response_;
Matrix initial_parameters_;
Matrix final_parameters_;
double certified_cost_;
};
#define NIST_BEGIN(CostFunctionName) \
struct CostFunctionName { \
CostFunctionName(const double* const x, \
const double* const y) \
: x_(*x), y_(*y) {} \
double x_; \
double y_; \
template <typename T> \
bool operator()(const T* const b, T* residual) const { \
const T y(y_); \
const T x(x_); \
residual[0] = y - (
#define NIST_END ); return true; }};
// y = b1 * (b2+x)**(-1/b3) + e
NIST_BEGIN(Bennet5)
b[0] * pow(b[1] + x, -1.0 / b[2])
NIST_END
// y = b1*(1-exp[-b2*x]) + e
NIST_BEGIN(BoxBOD)
b[0] * (1.0 - exp(-b[1] * x))
NIST_END
// y = exp[-b1*x]/(b2+b3*x) + e
NIST_BEGIN(Chwirut)
exp(-b[0] * x) / (b[1] + b[2] * x)
NIST_END
// y = b1*x**b2 + e
NIST_BEGIN(DanWood)
b[0] * pow(x, b[1])
NIST_END
// y = b1*exp( -b2*x ) + b3*exp( -(x-b4)**2 / b5**2 )
// + b6*exp( -(x-b7)**2 / b8**2 ) + e
NIST_BEGIN(Gauss)
b[0] * exp(-b[1] * x) +
b[2] * exp(-pow((x - b[3])/b[4], 2)) +
b[5] * exp(-pow((x - b[6])/b[7], 2))
NIST_END
// y = b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x) + e
NIST_BEGIN(Lanczos)
b[0] * exp(-b[1] * x) + b[2] * exp(-b[3] * x) + b[4] * exp(-b[5] * x)
NIST_END
// y = (b1+b2*x+b3*x**2+b4*x**3) /
// (1+b5*x+b6*x**2+b7*x**3) + e
NIST_BEGIN(Hahn1)
(b[0] + b[1] * x + b[2] * x * x + b[3] * x * x * x) /
(1.0 + b[4] * x + b[5] * x * x + b[6] * x * x * x)
NIST_END
// y = (b1 + b2*x + b3*x**2) /
// (1 + b4*x + b5*x**2) + e
NIST_BEGIN(Kirby2)
(b[0] + b[1] * x + b[2] * x * x) /
(1.0 + b[3] * x + b[4] * x * x)
NIST_END
// y = b1*(x**2+x*b2) / (x**2+x*b3+b4) + e
NIST_BEGIN(MGH09)
b[0] * (x * x + x * b[1]) / (x * x + x * b[2] + b[3])
NIST_END
// y = b1 * exp[b2/(x+b3)] + e
NIST_BEGIN(MGH10)
b[0] * exp(b[1] / (x + b[2]))
NIST_END
// y = b1 + b2*exp[-x*b4] + b3*exp[-x*b5]
NIST_BEGIN(MGH17)
b[0] + b[1] * exp(-x * b[3]) + b[2] * exp(-x * b[4])
NIST_END
// y = b1*(1-exp[-b2*x]) + e
NIST_BEGIN(Misra1a)
b[0] * (1.0 - exp(-b[1] * x))
NIST_END
// y = b1 * (1-(1+b2*x/2)**(-2)) + e
NIST_BEGIN(Misra1b)
b[0] * (1.0 - 1.0/ ((1.0 + b[1] * x / 2.0) * (1.0 + b[1] * x / 2.0))) // NOLINT
NIST_END
// y = b1 * (1-(1+2*b2*x)**(-.5)) + e
NIST_BEGIN(Misra1c)
b[0] * (1.0 - pow(1.0 + 2.0 * b[1] * x, -0.5))
NIST_END
// y = b1*b2*x*((1+b2*x)**(-1)) + e
NIST_BEGIN(Misra1d)
b[0] * b[1] * x / (1.0 + b[1] * x)
NIST_END
const double kPi = 3.141592653589793238462643383279;
// pi = 3.141592653589793238462643383279E0
// y = b1 - b2*x - arctan[b3/(x-b4)]/pi + e
NIST_BEGIN(Roszman1)
b[0] - b[1] * x - atan2(b[2], (x - b[3])) / kPi
NIST_END
// y = b1 / (1+exp[b2-b3*x]) + e
NIST_BEGIN(Rat42)
b[0] / (1.0 + exp(b[1] - b[2] * x))
NIST_END
// y = b1 / ((1+exp[b2-b3*x])**(1/b4)) + e
NIST_BEGIN(Rat43)
b[0] / pow(1.0 + exp(b[1] - b[2] * x), 1.0 / b[3])
NIST_END
// y = (b1 + b2*x + b3*x**2 + b4*x**3) /
// (1 + b5*x + b6*x**2 + b7*x**3) + e
NIST_BEGIN(Thurber)
(b[0] + b[1] * x + b[2] * x * x + b[3] * x * x * x) /
(1.0 + b[4] * x + b[5] * x * x + b[6] * x * x * x)
NIST_END
// y = b1 + b2*cos( 2*pi*x/12 ) + b3*sin( 2*pi*x/12 )
// + b5*cos( 2*pi*x/b4 ) + b6*sin( 2*pi*x/b4 )
// + b8*cos( 2*pi*x/b7 ) + b9*sin( 2*pi*x/b7 ) + e
NIST_BEGIN(ENSO)
b[0] + b[1] * cos(2.0 * kPi * x / 12.0) +
b[2] * sin(2.0 * kPi * x / 12.0) +
b[4] * cos(2.0 * kPi * x / b[3]) +
b[5] * sin(2.0 * kPi * x / b[3]) +
b[7] * cos(2.0 * kPi * x / b[6]) +
b[8] * sin(2.0 * kPi * x / b[6])
NIST_END
// y = (b1/b2) * exp[-0.5*((x-b3)/b2)**2] + e
NIST_BEGIN(Eckerle4)
b[0] / b[1] * exp(-0.5 * pow((x - b[2])/b[1], 2))
NIST_END
struct Nelson {
public:
Nelson(const double* const x, const double* const y)
: x1_(x[0]), x2_(x[1]), y_(y[0]) {}
template <typename T>
bool operator()(const T* const b, T* residual) const {
// log[y] = b1 - b2*x1 * exp[-b3*x2] + e
residual[0] = log(y_) - (b[0] - b[1] * x1_ * exp(-b[2] * x2_));
return true;
}
private:
double x1_;
double x2_;
double y_;
};
static void SetNumericDiffOptions(ceres::NumericDiffOptions* options) {
options->max_num_ridders_extrapolations = FLAGS_ridders_extrapolations;
options->ridders_relative_initial_step_size = FLAGS_ridders_step_size;
}
string JoinPath(const string& dirname, const string& basename) {
#ifdef _WIN32
static const char separator = '\\';
#else
static const char separator = '/';
#endif // _WIN32
if ((!basename.empty() && basename[0] == separator) || dirname.empty()) {
return basename;
} else if (dirname[dirname.size() - 1] == separator) {
return dirname + basename;
} else {
return dirname + string(&separator, 1) + basename;
}
}
template <typename Model, int num_residuals, int num_parameters>
int RegressionDriver(const string& filename,
const ceres::Solver::Options& options) {
NISTProblem nist_problem(JoinPath(FLAGS_nist_data_dir, filename));
CHECK_EQ(num_residuals, nist_problem.response_size());
CHECK_EQ(num_parameters, nist_problem.num_parameters());
Matrix predictor = nist_problem.predictor();
Matrix response = nist_problem.response();
Matrix final_parameters = nist_problem.final_parameters();
printf("%s\n", filename.c_str());
// Each NIST problem comes with multiple starting points, so we
// construct the problem from scratch for each case and solve it.
int num_success = 0;
for (int start = 0; start < nist_problem.num_starts(); ++start) {
Matrix initial_parameters = nist_problem.initial_parameters(start);
ceres::Problem problem;
for (int i = 0; i < nist_problem.num_observations(); ++i) {
Model* model = new Model(
predictor.data() + nist_problem.predictor_size() * i,
response.data() + nist_problem.response_size() * i);
ceres::CostFunction* cost_function = NULL;
if (FLAGS_use_numeric_diff) {
ceres::NumericDiffOptions options;
SetNumericDiffOptions(&options);
if (FLAGS_numeric_diff_method == "central") {
cost_function = new NumericDiffCostFunction<Model,
ceres::CENTRAL,
num_residuals,
num_parameters>(
model, ceres::TAKE_OWNERSHIP, num_residuals, options);
} else if (FLAGS_numeric_diff_method == "forward") {
cost_function = new NumericDiffCostFunction<Model,
ceres::FORWARD,
num_residuals,
num_parameters>(
model, ceres::TAKE_OWNERSHIP, num_residuals, options);
} else if (FLAGS_numeric_diff_method == "ridders") {
cost_function = new NumericDiffCostFunction<Model,
ceres::RIDDERS,
num_residuals,
num_parameters>(
model, ceres::TAKE_OWNERSHIP, num_residuals, options);
} else {
LOG(ERROR) << "Invalid numeric diff method specified";
return 0;
}
} else {
cost_function =
new ceres::AutoDiffCostFunction<Model,
num_residuals,
num_parameters>(model);
}
problem.AddResidualBlock(cost_function,
NULL,
initial_parameters.data());
}
ceres::Solver::Summary summary;
Solve(options, &problem, &summary);
// Compute the LRE by comparing each component of the solution
// with the ground truth, and taking the minimum.
Matrix final_parameters = nist_problem.final_parameters();
const double kMaxNumSignificantDigits = 11;
double log_relative_error = kMaxNumSignificantDigits + 1;
for (int i = 0; i < num_parameters; ++i) {
const double tmp_lre =
-std::log10(std::fabs(final_parameters(i) - initial_parameters(i)) /
std::fabs(final_parameters(i)));
// The maximum LRE is capped at 11 - the precision at which the
// ground truth is known.
//
// The minimum LRE is capped at 0 - no digits match between the
// computed solution and the ground truth.
log_relative_error =
std::min(log_relative_error,
std::max(0.0, std::min(kMaxNumSignificantDigits, tmp_lre)));
}
const int kMinNumMatchingDigits = 4;
if (log_relative_error > kMinNumMatchingDigits) {
++num_success;
}
printf("start: %d status: %s lre: %4.1f initial cost: %e final cost:%e "
"certified cost: %e total iterations: %d\n",
start + 1,
log_relative_error < kMinNumMatchingDigits ? "FAILURE" : "SUCCESS",
log_relative_error,
summary.initial_cost,
summary.final_cost,
nist_problem.certified_cost(),
(summary.num_successful_steps + summary.num_unsuccessful_steps));
}
return num_success;
}
void SetMinimizerOptions(ceres::Solver::Options* options) {
CHECK(ceres::StringToMinimizerType(FLAGS_minimizer,
&options->minimizer_type));
CHECK(ceres::StringToLinearSolverType(FLAGS_linear_solver,
&options->linear_solver_type));
CHECK(ceres::StringToPreconditionerType(FLAGS_preconditioner,
&options->preconditioner_type));
CHECK(ceres::StringToTrustRegionStrategyType(
FLAGS_trust_region_strategy,
&options->trust_region_strategy_type));
CHECK(ceres::StringToDoglegType(FLAGS_dogleg, &options->dogleg_type));
CHECK(ceres::StringToLineSearchDirectionType(
FLAGS_line_search_direction,
&options->line_search_direction_type));
CHECK(ceres::StringToLineSearchType(FLAGS_line_search,
&options->line_search_type));
CHECK(ceres::StringToLineSearchInterpolationType(
FLAGS_line_search_interpolation,
&options->line_search_interpolation_type));
options->max_num_iterations = FLAGS_num_iterations;
options->use_nonmonotonic_steps = FLAGS_nonmonotonic_steps;
options->initial_trust_region_radius = FLAGS_initial_trust_region_radius;
options->max_lbfgs_rank = FLAGS_lbfgs_rank;
options->line_search_sufficient_function_decrease = FLAGS_sufficient_decrease;
options->line_search_sufficient_curvature_decrease =
FLAGS_sufficient_curvature_decrease;
options->max_num_line_search_step_size_iterations =
FLAGS_max_line_search_iterations;
options->max_num_line_search_direction_restarts =
FLAGS_max_line_search_restarts;
options->use_approximate_eigenvalue_bfgs_scaling =
FLAGS_approximate_eigenvalue_bfgs_scaling;
options->function_tolerance = std::numeric_limits<double>::epsilon();
options->gradient_tolerance = std::numeric_limits<double>::epsilon();
options->parameter_tolerance = std::numeric_limits<double>::epsilon();
}
void SolveNISTProblems() {
if (FLAGS_nist_data_dir.empty()) {
LOG(FATAL) << "Must specify the directory containing the NIST problems";
}
ceres::Solver::Options options;
SetMinimizerOptions(&options);
cout << "Lower Difficulty\n";
int easy_success = 0;
easy_success += RegressionDriver<Misra1a, 1, 2>("Misra1a.dat", options);
easy_success += RegressionDriver<Chwirut, 1, 3>("Chwirut1.dat", options);
easy_success += RegressionDriver<Chwirut, 1, 3>("Chwirut2.dat", options);
easy_success += RegressionDriver<Lanczos, 1, 6>("Lanczos3.dat", options);
easy_success += RegressionDriver<Gauss, 1, 8>("Gauss1.dat", options);
easy_success += RegressionDriver<Gauss, 1, 8>("Gauss2.dat", options);
easy_success += RegressionDriver<DanWood, 1, 2>("DanWood.dat", options);
easy_success += RegressionDriver<Misra1b, 1, 2>("Misra1b.dat", options);
cout << "\nMedium Difficulty\n";
int medium_success = 0;
medium_success += RegressionDriver<Kirby2, 1, 5>("Kirby2.dat", options);
medium_success += RegressionDriver<Hahn1, 1, 7>("Hahn1.dat", options);
medium_success += RegressionDriver<Nelson, 1, 3>("Nelson.dat", options);
medium_success += RegressionDriver<MGH17, 1, 5>("MGH17.dat", options);
medium_success += RegressionDriver<Lanczos, 1, 6>("Lanczos1.dat", options);
medium_success += RegressionDriver<Lanczos, 1, 6>("Lanczos2.dat", options);
medium_success += RegressionDriver<Gauss, 1, 8>("Gauss3.dat", options);
medium_success += RegressionDriver<Misra1c, 1, 2>("Misra1c.dat", options);
medium_success += RegressionDriver<Misra1d, 1, 2>("Misra1d.dat", options);
medium_success += RegressionDriver<Roszman1, 1, 4>("Roszman1.dat", options);
medium_success += RegressionDriver<ENSO, 1, 9>("ENSO.dat", options);
cout << "\nHigher Difficulty\n";
int hard_success = 0;
hard_success += RegressionDriver<MGH09, 1, 4>("MGH09.dat", options);
hard_success += RegressionDriver<Thurber, 1, 7>("Thurber.dat", options);
hard_success += RegressionDriver<BoxBOD, 1, 2>("BoxBOD.dat", options);
hard_success += RegressionDriver<Rat42, 1, 3>("Rat42.dat", options);
hard_success += RegressionDriver<MGH10, 1, 3>("MGH10.dat", options);
hard_success += RegressionDriver<Eckerle4, 1, 3>("Eckerle4.dat", options);
hard_success += RegressionDriver<Rat43, 1, 4>("Rat43.dat", options);
hard_success += RegressionDriver<Bennet5, 1, 3>("Bennett5.dat", options);
cout << "\n";
cout << "Easy : " << easy_success << "/16\n";
cout << "Medium : " << medium_success << "/22\n";
cout << "Hard : " << hard_success << "/16\n";
cout << "Total : "
<< easy_success + medium_success + hard_success << "/54\n";
}
} // namespace examples
} // namespace ceres
int main(int argc, char** argv) {
CERES_GFLAGS_NAMESPACE::ParseCommandLineFlags(&argc, &argv, true);
google::InitGoogleLogging(argv[0]);
ceres::examples::SolveNISTProblems();
return 0;
}