    // Copyright (C) 2008  Davis E. King (davis@dlib.net)
    // License: Boost Software License   See LICENSE.txt for the full license.

    #include "optimization_test_functions.h"
    #include <dlib/optimization.h>
    #include <dlib/statistics.h>
    #include <sstream>
    #include <string>
    #include <cstdlib>
    #include <ctime>
    #include <vector>
    #include "../stl_checked.h"
    #include "../array.h"
    #include "../rand.h"

    #include "tester.h"

    namespace
    {

        using namespace test;
        using namespace dlib;
        using namespace std;

        logger dlog("test.optimization");

    // ----------------------------------------------------------------------------------------

        bool approx_equal (
            double a,
            double b
        )
        {
            return std::abs(a - b) < 100*std::numeric_limits<double>::epsilon();
        }

    // ----------------------------------------------------------------------------------------

        long total_count = 0;

        template <typename T>
        double apq ( const T& x)
        {
            DLIB_ASSERT(x.nr() > 1 && x.nc() == 1,"");
            COMPILE_TIME_ASSERT(is_matrix<T>::value);
            double temp = 0;
            for (long r = 0; r < x.nr(); ++r)
            {
                temp += (r+1)*x(r)*x(r);
            }

            ++total_count;

            return temp + 1/100.0*(x(0) + x(x.nr()-1))*(x(0) + x(x.nr()-1));
        }

        template <typename T>
        T der_apq ( const T& x)
        {
            DLIB_ASSERT(x.nr() > 1 && x.nc() == 1,"");
            COMPILE_TIME_ASSERT(is_matrix<T>::value);
            T temp(x.nr());
            for (long r = 0; r < x.nr(); ++r)
            {
                temp(r) = 2*(r+1)*x(r) ;
            }

            temp(0) += 1/50.0*(x(0) + x(x.nr()-1));
            temp(x.nr()-1) += 1/50.0*(x(0) + x(x.nr()-1));

            ++total_count;

            return temp;
        }

    // ----------------------------------------------------------------------------------------

        // Rosenbrock's function.  minimum at (1,1)
        double rosen ( const matrix<double,2,1>& x)
        {
            ++total_count;
            return 100*pow(x(1) - x(0)*x(0),2) + pow(1 - x(0),2);
        }

        matrix<double,2,1> der_rosen ( const matrix<double,2,1>& x)
        {
            ++total_count;
            matrix<double,2,1> res;
            res(0) = -400*x(0)*(x(1)-x(0)*x(0)) - 2*(1-x(0));
            res(1) = 200*(x(1)-x(0)*x(0));
            return res;
        }

    // ----------------------------------------------------------------------------------------

        // negative of Rosenbrock's function.  minimum at (1,1)
        double neg_rosen ( const matrix<double,2,1>& x)
        {
            ++total_count;
            return -(100*pow(x(1) - x(0)*x(0),2) + pow(1 - x(0),2));
        }

        matrix<double,2,1> der_neg_rosen ( const matrix<double,2,1>& x)
        {
            ++total_count;
            matrix<double,2,1> res;
            res(0) = -400*x(0)*(x(1)-x(0)*x(0)) - 2*(1-x(0));
            res(1) = 200*(x(1)-x(0)*x(0));
            return -res;
        }

    // ----------------------------------------------------------------------------------------

        double simple ( const matrix<double,2,1>& x)
        {
            ++total_count;
            return 10*x(0)*x(0) + x(1)*x(1);
        }

        matrix<double,2,1> der_simple ( const matrix<double,2,1>& x)
        {
            ++total_count;
            matrix<double,2,1> res;
            res(0) = 20*x(0);
            res(1) = 2*x(1);
            return res;
        }

    // ----------------------------------------------------------------------------------------

        double powell ( const matrix<double,4,1>& x)
        {
            ++total_count;
            return pow(x(0) + 10*x(1),2) +
                pow(std::sqrt(5.0)*(x(2) - x(3)),2) +
                pow((x(1) - 2*x(2))*(x(1) - 2*x(2)),2) +
                pow(std::sqrt(10.0)*(x(0) - x(3))*(x(0) - x(3)),2);
        }

    // ----------------------------------------------------------------------------------------

    // a simple function with a minimum at zero
        double single_variable_function ( double x)
        {
            ++total_count;
            return 3*x*x + 5;
        }

    // ----------------------------------------------------------------------------------------
    // ----------------------------------------------------------------------------------------
    // ----------------------------------------------------------------------------------------

        void test_apq (
            const matrix<double,0,1> p
        )
        {
            typedef matrix<double,0,1> T;
            const double eps = 1e-12;
            const double minf = -10;
            matrix<double,0,1> x(p.nr()), opt(p.nr());
            set_all_elements(opt, 0);
            double val = 0;

            if (p.size() < 20)
                dlog << LINFO << "testing with apq and the start point: " << trans(p);
            else
                dlog << LINFO << "testing with apq and a big vector with " << p.size() << " components.";

            // don't use bfgs on really large vectors
            if (p.size() < 20)
            {
                total_count = 0;
                x = p;
                val = find_min(bfgs_search_strategy(),
                         objective_delta_stop_strategy(eps),
                         wrap_function(apq<T>), wrap_function(der_apq<T>), x, minf);
                DLIB_TEST_MSG(dlib::equal(x,opt, 1e-5),opt-x);
                DLIB_TEST(approx_equal(val , apq(x)));
                dlog << LINFO << "find_min() bgfs: got apq in " << total_count;

                total_count = 0;
                x = p;
                find_min(bfgs_search_strategy(),
                         gradient_norm_stop_strategy(),
                         wrap_function(apq<T>), wrap_function(der_apq<T>), x, minf);
                DLIB_TEST_MSG(dlib::equal(x,opt, 1e-5),opt-x);
                dlog << LINFO << "find_min() bgfs(gn): got apq in " << total_count;
            }

            if (p.size() < 100)
            {
                total_count = 0;
                x = p;
                val=find_min_bobyqa(wrap_function(apq<T>), x, 2*x.size()+1,
                                uniform_matrix<double>(x.size(),1,-1e100),
                                uniform_matrix<double>(x.size(),1,1e100),
                                (max(abs(x))+1)/10,
                                1e-6,
                                10000);
                DLIB_TEST_MSG(dlib::equal(x,opt, 1e-5),opt-x);
                DLIB_TEST(approx_equal(val , apq(x)));
                dlog << LINFO << "find_min_bobyqa(): got apq in " << total_count;
            }

            total_count = 0;
            x = p;
            val=find_min(lbfgs_search_strategy(10),
                     objective_delta_stop_strategy(eps),
                     wrap_function(apq<T>), wrap_function(der_apq<T>), x, minf);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-5),opt-x);
            DLIB_TEST(approx_equal(val , apq(x)));
            dlog << LINFO << "find_min() lbgfs-10: got apq in " << total_count;

            total_count = 0;
            x = p;
            val=find_min(lbfgs_search_strategy(1),
                     objective_delta_stop_strategy(eps),
                     wrap_function(apq<T>), wrap_function(der_apq<T>), x, minf);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-5),opt-x);
            DLIB_TEST(approx_equal(val , apq(x)));
            dlog << LINFO << "find_min() lbgfs-1: got apq in " << total_count;

            total_count = 0;
            x = p;
            val=find_min(cg_search_strategy(),
                     objective_delta_stop_strategy(eps),
                     wrap_function(apq<T>), wrap_function(der_apq<T>), x, minf);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-5),opt-x);
            DLIB_TEST(approx_equal(val , apq(x)));
            dlog << LINFO << "find_min() cg: got apq in " << total_count;

            // don't do approximate derivative tests if the input point is really long
            if (p.size() < 20)
            {
                total_count = 0;
                x = p;
                val=find_min(bfgs_search_strategy(),
                         objective_delta_stop_strategy(eps),
                         wrap_function(apq<T>), derivative(wrap_function(apq<T>)), x, minf);
                DLIB_TEST_MSG(dlib::equal(x,opt, 1e-5),opt-x);
                DLIB_TEST(approx_equal(val , apq(x)));
                dlog << LINFO << "find_min() bfgs: got apq/noder in " << total_count;

                total_count = 0;
                x = p;
                val=find_min(cg_search_strategy(),
                         objective_delta_stop_strategy(eps),
                         wrap_function(apq<T>), derivative(wrap_function(apq<T>)), x, minf);
                DLIB_TEST_MSG(dlib::equal(x,opt, 1e-5),opt-x);
                DLIB_TEST(approx_equal(val , apq(x)));
                dlog << LINFO << "find_min() cg: got apq/noder in " << total_count;

                total_count = 0;
                x = p;
                val=find_min_using_approximate_derivatives(bfgs_search_strategy(),
                                                       objective_delta_stop_strategy(eps),
                                                       wrap_function(apq<T>), x, minf);
                DLIB_TEST_MSG(dlib::equal(x,opt, 1e-5),opt-x);
                DLIB_TEST(approx_equal(val , apq(x)));
                dlog << LINFO << "find_min() bfgs: got apq/noder2 in " << total_count;

                total_count = 0;
                x = p;
                val=find_min_using_approximate_derivatives(lbfgs_search_strategy(10),
                                                       objective_delta_stop_strategy(eps),
                                                       wrap_function(apq<T>), x, minf);
                DLIB_TEST_MSG(dlib::equal(x,opt, 1e-5),opt-x);
                dlog << LINFO << "find_min() lbfgs-10: got apq/noder2 in " << total_count;

                total_count = 0;
                x = p;
                val=find_min_using_approximate_derivatives(cg_search_strategy(),
                                                       objective_delta_stop_strategy(eps),
                                                       wrap_function(apq<T>), x, minf);
                DLIB_TEST_MSG(dlib::equal(x,opt, 1e-5),opt-x);
                DLIB_TEST(approx_equal(val , apq(x)));
                dlog << LINFO << "find_min() cg: got apq/noder2 in " << total_count;
            }
        }

        void test_powell (
            const matrix<double,4,1> p
        )
        {
            const double eps = 1e-15;
            const double minf = -1;
            matrix<double,4,1> x, opt;
            opt(0) = 0;
            opt(1) = 0;
            opt(2) = 0;
            opt(3) = 0;

            double val = 0;

            dlog << LINFO << "testing with powell and the start point: " << trans(p);

            /*
            total_count = 0;
            x = p;
            val=find_min(bfgs_search_strategy(),
                     objective_delta_stop_strategy(eps),
                     powell, derivative(powell,1e-8), x, minf);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-2),opt-x);
            DLIB_TEST(approx_equal(val , powell(x)));
            dlog << LINFO << "find_min() bfgs: got powell/noder in " << total_count;

            total_count = 0;
            x = p;
            val=find_min(cg_search_strategy(),
                     objective_delta_stop_strategy(eps),
                     powell, derivative(powell,1e-9), x, minf);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-2),opt-x);
            DLIB_TEST(approx_equal(val , powell(x)));
            dlog << LINFO << "find_min() cg: got powell/noder in " << total_count;
            */

            total_count = 0;
            x = p;
            val=find_min_using_approximate_derivatives(bfgs_search_strategy(),
                                                   objective_delta_stop_strategy(eps),
                                                   powell, x, minf, 1e-10);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-1),opt-x);
            DLIB_TEST(approx_equal(val , powell(x)));
            dlog << LINFO << "find_min() bfgs: got powell/noder2 in " << total_count;

            total_count = 0;
            x = p;
            val=find_min_using_approximate_derivatives(lbfgs_search_strategy(4),
                                                   objective_delta_stop_strategy(eps),
                                                   powell, x, minf, 1e-10);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-1),opt-x);
            DLIB_TEST(approx_equal(val , powell(x)));
            dlog << LINFO << "find_min() lbfgs-4: got powell/noder2 in " << total_count;

            total_count = 0;
            x = p;
            val=find_min_using_approximate_derivatives(lbfgs_search_strategy(4),
                                                   gradient_norm_stop_strategy(),
                                                   powell, x, minf, 1e-10);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-1),opt-x);
            DLIB_TEST(approx_equal(val , powell(x)));
            dlog << LINFO << "find_min() lbfgs-4(gn): got powell/noder2 in " << total_count;

            total_count = 0;
            x = p;
            val=find_min_using_approximate_derivatives(cg_search_strategy(),
                                                   objective_delta_stop_strategy(eps),
                                                   powell, x, minf, 1e-10);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-1),opt-x);
            DLIB_TEST(approx_equal(val , powell(x)));
            dlog << LINFO << "find_min() cg: got powell/noder2 in " << total_count;

            total_count = 0;
            x = p;
            val=find_min_bobyqa(powell, x, 2*x.size()+1,
                            uniform_matrix<double>(x.size(),1,-1e100),
                            uniform_matrix<double>(x.size(),1,1e100),
                            (max(abs(x))+1)/10,
                            1e-7,
                            10000);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-3),opt-x);
            DLIB_TEST(approx_equal(val , powell(x)));
            dlog << LINFO << "find_min_bobyqa(): got powell in " << total_count;

        }

        void test_simple (
            const matrix<double,2,1> p
        )
        {
            const double eps = 1e-12;
            const double minf = -10000;
            matrix<double,2,1> x, opt;
            opt(0) = 0;
            opt(1) = 0;
            double val = 0;

            dlog << LINFO << "testing with simple and the start point: " << trans(p);

            total_count = 0;
            x = p;
            val=find_min(bfgs_search_strategy(),
                     objective_delta_stop_strategy(eps),
                     simple, der_simple, x, minf);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-5),opt-x);
            DLIB_TEST(approx_equal(val , simple(x)));
            dlog << LINFO << "find_min() bfgs: got simple in " << total_count;

            total_count = 0;
            x = p;
            val=find_min(bfgs_search_strategy(),
                     gradient_norm_stop_strategy(),
                     simple, der_simple, x, minf);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-5),opt-x);
            DLIB_TEST(approx_equal(val , simple(x)));
            dlog << LINFO << "find_min() bfgs(gn): got simple in " << total_count;

            total_count = 0;
            x = p;
            val=find_min(lbfgs_search_strategy(3),
                     objective_delta_stop_strategy(eps),
                     simple, der_simple, x, minf);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-5),opt-x);
            DLIB_TEST(approx_equal(val , simple(x)));
            dlog << LINFO << "find_min() lbfgs-3: got simple in " << total_count;

            total_count = 0;
            x = p;
            val=find_min(cg_search_strategy(),
                     objective_delta_stop_strategy(eps),
                     simple, der_simple, x, minf);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-5),opt-x);
            DLIB_TEST(approx_equal(val , simple(x)));
            dlog << LINFO << "find_min() cg: got simple in " << total_count;

            total_count = 0;
            x = p;
            val=find_min(bfgs_search_strategy(),
                     objective_delta_stop_strategy(eps),
                     simple, derivative(simple), x, minf);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-5),opt-x);
            DLIB_TEST(approx_equal(val , simple(x)));
            dlog << LINFO << "find_min() bfgs: got simple/noder in " << total_count;

            total_count = 0;
            x = p;
            val=find_min(lbfgs_search_strategy(8),
                     objective_delta_stop_strategy(eps),
                     simple, derivative(simple), x, minf);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-5),opt-x);
            DLIB_TEST(approx_equal(val , simple(x)));
            dlog << LINFO << "find_min() lbfgs-8: got simple/noder in " << total_count;

            total_count = 0;
            x = p;
            val=find_min(cg_search_strategy(),
                     objective_delta_stop_strategy(eps),
                     simple, derivative(simple), x, minf);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-5),opt-x);
            DLIB_TEST(approx_equal(val , simple(x)));
            dlog << LINFO << "find_min() cg: got simple/noder in " << total_count;

            total_count = 0;
            x = p;
            val=find_min_using_approximate_derivatives(bfgs_search_strategy(),
                                                   objective_delta_stop_strategy(eps),
                                                   simple, x, minf);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-5),opt-x);
            DLIB_TEST(approx_equal(val , simple(x)));
            dlog << LINFO << "find_min() bfgs: got simple/noder2 in " << total_count;

            total_count = 0;
            x = p;
            val=find_min_using_approximate_derivatives(lbfgs_search_strategy(6),
                                                   objective_delta_stop_strategy(eps),
                                                   simple, x, minf);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-5),opt-x);
            DLIB_TEST(approx_equal(val , simple(x)));
            dlog << LINFO << "find_min() lbfgs-6: got simple/noder2 in " << total_count;

            total_count = 0;
            x = p;
            val=find_min_using_approximate_derivatives(cg_search_strategy(),
                                                   objective_delta_stop_strategy(eps),
                                                   simple, x, minf);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-5),opt-x);
            DLIB_TEST(approx_equal(val , simple(x)));
            dlog << LINFO << "find_min() cg: got simple/noder2 in " << total_count;

            total_count = 0;
            x = p;
            val=find_min_bobyqa(simple, x, 2*x.size()+1,
                            uniform_matrix<double>(x.size(),1,-1e100),
                            uniform_matrix<double>(x.size(),1,1e100),
                            (max(abs(x))+1)/10,
                            1e-6,
                            10000);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-5),opt-x);
            DLIB_TEST(approx_equal(val , simple(x)));
            dlog << LINFO << "find_min_bobyqa(): got simple in " << total_count;

        }

        void test_rosen (
            const matrix<double,2,1> p
        )
        {
            const double eps = 1e-15;
            const double minf = -10;
            matrix<double,2,1> x, opt;
            opt(0) = 1;
            opt(1) = 1;

            double val = 0;

            dlog << LINFO << "testing with rosen and the start point: " << trans(p);

            total_count = 0;
            x = p;
            val=find_min(bfgs_search_strategy(),
                     objective_delta_stop_strategy(eps),
                     rosen, der_rosen, x, minf);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-7),opt-x);
            DLIB_TEST(approx_equal(val , rosen(x)));
            dlog << LINFO << "find_min() bfgs: got rosen in " << total_count;

            total_count = 0;
            x = p;
            val=find_min(bfgs_search_strategy(),
                     gradient_norm_stop_strategy(),
                     rosen, der_rosen, x, minf);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-7),opt-x);
            DLIB_TEST(approx_equal(val , rosen(x)));
            dlog << LINFO << "find_min() bfgs(gn): got rosen in " << total_count;

            total_count = 0;
            x = p;
            val=find_min(lbfgs_search_strategy(20),
                     objective_delta_stop_strategy(eps),
                     rosen, der_rosen, x, minf);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-7),opt-x);
            DLIB_TEST(approx_equal(val , rosen(x)));
            dlog << LINFO << "find_min() lbfgs-20: got rosen in " << total_count;

            total_count = 0;
            x = p;
            val=find_min(cg_search_strategy(),
                     objective_delta_stop_strategy(eps),
                     rosen, der_rosen, x, minf);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-7),opt-x);
            DLIB_TEST(approx_equal(val , rosen(x)));
            dlog << LINFO << "find_min() cg: got rosen in " << total_count;

            total_count = 0;
            x = p;
            val=find_min(bfgs_search_strategy(),
                     objective_delta_stop_strategy(eps),
                     rosen, derivative(rosen,1e-5), x, minf);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-4),opt-x);
            DLIB_TEST(approx_equal(val , rosen(x)));
            dlog << LINFO << "find_min() bfgs: got rosen/noder in " << total_count;

            total_count = 0;
            x = p;
            val=find_min(lbfgs_search_strategy(5),
                     objective_delta_stop_strategy(eps),
                     rosen, derivative(rosen,1e-5), x, minf);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-4),opt-x);
            DLIB_TEST(approx_equal(val , rosen(x)));
            dlog << LINFO << "find_min() lbfgs-5: got rosen/noder in " << total_count;

            total_count = 0;
            x = p;
            val=find_min(cg_search_strategy(),
                     objective_delta_stop_strategy(eps),
                     rosen, derivative(rosen,1e-5), x, minf);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-4),opt-x);
            DLIB_TEST(approx_equal(val , rosen(x)));
            dlog << LINFO << "find_min() cg: got rosen/noder in " << total_count;

            total_count = 0;
            x = p;
            val=find_min_using_approximate_derivatives(cg_search_strategy(),
                                                   objective_delta_stop_strategy(eps),
                                                   rosen, x, minf);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-4),opt-x);
            DLIB_TEST(approx_equal(val , rosen(x)));
            dlog << LINFO << "find_min() cg: got rosen/noder2 in " << total_count;

            if (max(abs(p)) < 1000)
            {
                total_count = 0;
                x = p;
                val=find_min_bobyqa(rosen, x, 2*x.size()+1,
                                uniform_matrix<double>(x.size(),1,-1e100),
                                uniform_matrix<double>(x.size(),1,1e100),
                                (max(abs(x))+1)/10,
                                1e-6,
                                10000);
                DLIB_TEST_MSG(dlib::equal(x,opt, 1e-5),opt-x);
                DLIB_TEST(approx_equal(val , rosen(x)));
                dlog << LINFO << "find_min_bobyqa(): got rosen in " << total_count;
            }
        }

        void test_neg_rosen (
            const matrix<double,2,1> p
        )
        {
            const double eps = 1e-15;
            const double maxf = 10;
            matrix<double,2,1> x, opt;
            opt(0) = 1;
            opt(1) = 1;

            double val = 0;

            dlog << LINFO << "testing with neg_rosen and the start point: " << trans(p);

            total_count = 0;
            x = p;
            val=find_max(
                bfgs_search_strategy(),
                objective_delta_stop_strategy(eps), neg_rosen, der_neg_rosen, x, maxf);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-7),opt-x);
            DLIB_TEST(approx_equal(val , neg_rosen(x)));
            dlog << LINFO << "find_max() bfgs: got neg_rosen in " << total_count;

            total_count = 0;
            x = p;
            val=find_max(
                lbfgs_search_strategy(5),
                objective_delta_stop_strategy(eps), neg_rosen, der_neg_rosen, x, maxf);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-7),opt-x);
            DLIB_TEST(approx_equal(val , neg_rosen(x)));
            dlog << LINFO << "find_max() lbfgs-5: got neg_rosen in " << total_count;

            total_count = 0;
            x = p;
            val=find_max(
                lbfgs_search_strategy(5),
                objective_delta_stop_strategy(eps), neg_rosen, derivative(neg_rosen), x, maxf);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-7),opt-x);
            DLIB_TEST(approx_equal(val , neg_rosen(x)));
            dlog << LINFO << "find_max() lbfgs-5: got neg_rosen/noder in " << total_count;

            total_count = 0;
            x = p;
            val=find_max_using_approximate_derivatives(
                cg_search_strategy(),
                objective_delta_stop_strategy(eps), neg_rosen, x, maxf);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-7),opt-x);
            DLIB_TEST(approx_equal(val , neg_rosen(x)));
            dlog << LINFO << "find_max() cg: got neg_rosen/noder2 in " << total_count;

            total_count = 0;
            x = p;
            val=find_max_bobyqa(neg_rosen, x, 2*x.size()+1,
                            uniform_matrix<double>(x.size(),1,-1e100),
                            uniform_matrix<double>(x.size(),1,1e100),
                            (max(abs(x))+1)/10,
                            1e-6,
                            10000);
            DLIB_TEST_MSG(dlib::equal(x,opt, 1e-5),opt-x);
            DLIB_TEST(approx_equal(val , neg_rosen(x)));
            dlog << LINFO << "find_max_bobyqa(): got neg_rosen in " << total_count;
        }

    // ----------------------------------------------------------------------------------------

        void test_single_variable_function (
            const double p
        )
        {
            const double eps = 1e-7;

            dlog << LINFO << "testing with single_variable_function and the start point: " << p;
            double out, x;

            total_count = 0;
            x = p;
            out = find_min_single_variable(single_variable_function, x, -1e100, 1e100, eps, 1000);
            DLIB_TEST_MSG(std::abs(out-5) < 1e-6, out-5);
            DLIB_TEST_MSG(std::abs(x) < 1e-6, x);
            dlog << LINFO << "find_min_single_variable(): got single_variable_function in " << total_count;

            total_count = 0;
            x = p;
            out = -find_max_single_variable(negate_function(single_variable_function), x, -1e100, 1e100, eps, 1000);
            DLIB_TEST_MSG(std::abs(out-5) < 1e-6, out-5);
            DLIB_TEST_MSG(std::abs(x) < 1e-6, x);
            dlog << LINFO << "find_max_single_variable(): got single_variable_function in " << total_count;

            if (p > 0)
            {
                total_count = 0;
                x = p;
                out = find_min_single_variable(single_variable_function, x, -1e-4, 1e100, eps, 1000);
                DLIB_TEST_MSG(std::abs(out-5) < 1e-6, out-5);
                DLIB_TEST_MSG(std::abs(x) < 1e-6, x);
                dlog << LINFO << "find_min_single_variable(): got single_variable_function in " << total_count;

                if (p > 3)
                {
                    total_count = 0;
                    x = p;
                    out = -find_max_single_variable(negate_function(single_variable_function), x, 3, 1e100, eps, 1000);
                    DLIB_TEST_MSG(std::abs(out - (3*3*3+5)) < 1e-6, out-(3*3*3+5));
                    DLIB_TEST_MSG(std::abs(x-3) < 1e-6, x);
                    dlog << LINFO << "find_max_single_variable(): got single_variable_function in " << total_count;
                }
            }

            if (p < 0)
            {
                total_count = 0;
                x = p;
                out = find_min_single_variable(single_variable_function, x, -1e100, 1e-4, eps, 1000);
                DLIB_TEST_MSG(std::abs(out-5) < 1e-6, out-5);
                DLIB_TEST_MSG(std::abs(x) < 1e-6, x);
                dlog << LINFO << "find_min_single_variable(): got single_variable_function in " << total_count;

                if (p < -3)
                {
                    total_count = 0;
                    x = p;
                    out = find_min_single_variable(single_variable_function, x, -1e100, -3, eps, 1000);
                    DLIB_TEST_MSG(std::abs(out - (3*3*3+5)) < 1e-6, out-(3*3*3+5));
                    DLIB_TEST_MSG(std::abs(x+3) < 1e-6, x);
                    dlog << LINFO << "find_min_single_variable(): got single_variable_function in " << total_count;
                }
            }

        }

    // ----------------------------------------------------------------------------------------

        void optimization_test (
        )
        /*!
            ensures
                - runs tests on the optimization stuff compliance with the specs
        !*/
        {
            matrix<double,0,1> p;

            print_spinner();

            p.set_size(2);

            // test with single_variable_function
            test_single_variable_function(0);
            test_single_variable_function(1);
            test_single_variable_function(-10);
            test_single_variable_function(-100);
            test_single_variable_function(900.53);

            // test with the rosen function
            p(0) = 9;
            p(1) = -4.9;
            test_rosen(p);
            test_neg_rosen(p);

            p(0) = 0;
            p(1) = 0;
            test_rosen(p);

            p(0) = 5323;
            p(1) = 98248;
            test_rosen(p);

            // test with the simple function
            p(0) = 1;
            p(1) = 1;
            test_simple(p);

            p(0) = 0.5;
            p(1) = -9;
            test_simple(p);

            p(0) = 645;
            p(1) = 839485;
            test_simple(p);

            print_spinner();

            // test with the apq function
            p.set_size(5);

            p(0) = 1;
            p(1) = 1;
            p(2) = 1;
            p(3) = 1;
            p(4) = 1;
            test_apq(p);

            p(0) = 1;
            p(1) = 2;
            p(2) = 3;
            p(3) = 4;
            p(4) = 5;
            test_apq(p);

            p(0) = 1;
            p(1) = 2;
            p(2) = -3;
            p(3) = 4;
            p(4) = 5;
            test_apq(p);

            print_spinner();

            p(0) = 1;
            p(1) = 2324;
            p(2) = -3;
            p(3) = 4;
            p(4) = 534534;
            test_apq(p);

            p.set_size(10);
            p(0) = 1;
            p(1) = 2;
            p(2) = -3;
            p(3) = 4;
            p(4) = 5;
            p(5) = 1;
            p(6) = 2;
            p(7) = -3;
            p(8) = 4;
            p(9) = 5;
            test_apq(p);

            // test apq with a big vector
            p.set_size(500);
            dlib::rand rnd;
            for (long i = 0; i < p.size(); ++i)
            {
                p(i) = rnd.get_random_double()*20 - 10;
            }
            test_apq(p);

            print_spinner();

            // test with the powell function
            p.set_size(4);

            p(0) = 3;
            p(1) = -1;
            p(2) = 0;
            p(3) = 1;
            test_powell(p);

            {
                matrix<double,2,1> m;
                m(0) = -0.43;
                m(1) = 0.919;
                DLIB_TEST(dlib::equal(der_rosen(m) , derivative(rosen)(m),1e-5));

                DLIB_TEST_MSG(std::abs(derivative(make_line_search_function(rosen,m,m))(0) -
                                      make_line_search_function(derivative(rosen),m,m)(0)) < 1e-5,"");
                DLIB_TEST_MSG(std::abs(derivative(make_line_search_function(rosen,m,m))(1) -
                                      make_line_search_function(derivative(rosen),m,m)(1)) < 1e-5,"");

                DLIB_TEST_MSG(std::abs(derivative(make_line_search_function(rosen,m,m))(0) -
                                      make_line_search_function(der_rosen,m,m)(0)) < 1e-5,"");
                DLIB_TEST_MSG(std::abs(derivative(make_line_search_function(rosen,m,m))(1) -
                                      make_line_search_function(der_rosen,m,m)(1)) < 1e-5,"");
            }
            {
                matrix<double,2,1> m;
                m(0) = 1;
                m(1) = 2;
                DLIB_TEST(dlib::equal(der_rosen(m) , derivative(rosen)(m),1e-5));

                DLIB_TEST_MSG(std::abs(derivative(make_line_search_function(rosen,m,m))(0) -
                                      make_line_search_function(derivative(rosen),m,m)(0)) < 1e-5,"");
                DLIB_TEST_MSG(std::abs(derivative(make_line_search_function(rosen,m,m))(1) -
                                      make_line_search_function(derivative(rosen),m,m)(1)) < 1e-5,"");

                DLIB_TEST_MSG(std::abs(derivative(make_line_search_function(rosen,m,m))(0) -
                                      make_line_search_function(der_rosen,m,m)(0)) < 1e-5,"");
                DLIB_TEST_MSG(std::abs(derivative(make_line_search_function(rosen,m,m))(1) -
                                      make_line_search_function(der_rosen,m,m)(1)) < 1e-5,"");
            }

            {
                matrix<double,2,1> m;
                m = 1,2;
                DLIB_TEST(std::abs(neg_rosen(m) - negate_function(rosen)(m) ) < 1e-16);
            }

        }

        template <typename der_funct, typename T>
        double unconstrained_gradient_magnitude (
            const der_funct& grad,
            const T& x,
            const T& lower,
            const T& upper
        )
        {
            T g = grad(x);

            double unorm = 0;

            for (long i = 0; i < g.size(); ++i)
            {
                if (lower(i) < x(i) && x(i) < upper(i))
                    unorm += g(i)*g(i);
                else if (x(i) == lower(i) && g(i) < 0)
                    unorm += g(i)*g(i);
                else if (x(i) == upper(i) && g(i) > 0)
                    unorm += g(i)*g(i);
            }

            return unorm;
        }

        template <typename der_funct, typename T>
        double unconstrained_gradient_magnitude_neg_funct (
            const der_funct& grad,
            const T& x,
            const T& lower,
            const T& upper
        )
        {
            T g = grad(x);

            double unorm = 0;

            for (long i = 0; i < g.size(); ++i)
            {
                if (lower(i) < x(i) && x(i) < upper(i))
                    unorm += g(i)*g(i);
                else if (x(i) == lower(i) && g(i) > 0)
                    unorm += g(i)*g(i);
                else if (x(i) == upper(i) && g(i) < 0)
                    unorm += g(i)*g(i);
            }

            return unorm;
        }

        template <typename search_strategy_type>
        double test_bound_solver_neg_rosen (dlib::rand& rnd, search_strategy_type search_strategy)
        {
            using namespace dlib::test_functions;
            print_spinner();
            matrix<double,2,1> starting_point, lower, upper, x;

            // pick random bounds
            lower = rnd.get_random_gaussian()+1, rnd.get_random_gaussian()+1;
            upper = rnd.get_random_gaussian()+1, rnd.get_random_gaussian()+1;
            while (upper(0) < lower(0)) upper(0) = rnd.get_random_gaussian()+1;
            while (upper(1) < lower(1)) upper(1) = rnd.get_random_gaussian()+1;

            starting_point = rnd.get_random_double()*(upper(0)-lower(0))+lower(0),
                           rnd.get_random_double()*(upper(1)-lower(1))+lower(1);

            dlog << LINFO << "lower: "<< trans(lower);
            dlog << LINFO << "upper: "<< trans(upper);
            dlog << LINFO << "starting: "<< trans(starting_point);

            x = starting_point;
            double val = find_max_box_constrained(
                search_strategy,
                objective_delta_stop_strategy(1e-16, 500),
                neg_rosen, der_neg_rosen, x,
                lower,
                upper
            );

            DLIB_TEST_MSG(std::abs(val - neg_rosen(x)) < 1e-11, std::abs(val - neg_rosen(x)));
            dlog << LINFO << "neg_rosen solution:\n" << x;

            dlog << LINFO << "neg_rosen gradient: "<< trans(der_neg_rosen(x));
            const double gradient_residual = unconstrained_gradient_magnitude_neg_funct(der_neg_rosen, x, lower, upper);
            dlog << LINFO << "gradient_residual: "<< gradient_residual;

            return gradient_residual;
        }

        template <typename search_strategy_type>
        double test_bound_solver_rosen (dlib::rand& rnd, search_strategy_type search_strategy)
        {
            using namespace dlib::test_functions;
            print_spinner();
            matrix<double,2,1> starting_point, lower, upper, x;

            // pick random bounds and sometimes put the upper bound at zero so we can have
            // a test where the optimal value has a bound active at 0 so make sure this case
            // works properly.
            if (rnd.get_random_double() > 0.2)
            {
                lower = rnd.get_random_gaussian()+1, rnd.get_random_gaussian()+1;
                upper = rnd.get_random_gaussian()+1, rnd.get_random_gaussian()+1;
                while (upper(0) < lower(0)) upper(0) = rnd.get_random_gaussian()+1;
                while (upper(1) < lower(1)) upper(1) = rnd.get_random_gaussian()+1;
            }
            else
            {
                upper = 0,0;
                if (rnd.get_random_double() > 0.5)
                    upper(0) = -rnd.get_random_double();
                if (rnd.get_random_double() > 0.5)
                    upper(1) = -rnd.get_random_double();

                lower = rnd.get_random_double()+1, rnd.get_random_double()+1;
                lower = upper - lower;
            }
            const bool pick_uniform_bounds = rnd.get_random_double() > 0.9;
            if (pick_uniform_bounds)
            {
                double x = rnd.get_random_gaussian()*2;
                double y = rnd.get_random_gaussian()*2;
                lower = min(x,y);
                upper = max(x,y);
            }

            starting_point = rnd.get_random_double()*(upper(0)-lower(0))+lower(0),
                           rnd.get_random_double()*(upper(1)-lower(1))+lower(1);

            dlog << LINFO << "lower: "<< trans(lower);
            dlog << LINFO << "upper: "<< trans(upper);
            dlog << LINFO << "starting: "<< trans(starting_point);

            x = starting_point;
            double val;
            if (!pick_uniform_bounds)
            {
                val = find_min_box_constrained(
                    search_strategy,
                    objective_delta_stop_strategy(1e-16, 500),
                    rosen, der_rosen, x,
                    lower,
                    upper
                );
            }
            else
            {
                val = find_min_box_constrained(
                    search_strategy,
                    objective_delta_stop_strategy(1e-16, 500),
                    rosen, der_rosen, x,
                    lower(0),
                    upper(0)
                );
            }

            DLIB_TEST_MSG(std::abs(val - rosen(x)) < 1e-11, std::abs(val - rosen(x)));
            dlog << LINFO << "rosen solution:\n" << x;

            dlog << LINFO << "rosen gradient: "<< trans(der_rosen(x));
            const double gradient_residual = unconstrained_gradient_magnitude(der_rosen, x, lower, upper);
            dlog << LINFO << "gradient_residual: "<< gradient_residual;

            return gradient_residual;
        }

        template <typename search_strategy_type>
        double test_bound_solver_brown (dlib::rand& rnd, search_strategy_type search_strategy)
        {
            using namespace dlib::test_functions;
            print_spinner();
            matrix<double,4,1> starting_point(4), lower(4), upper(4), x;

            const matrix<double,0,1> solution = brown_solution();

            // pick random bounds
            lower = rnd.get_random_gaussian(), rnd.get_random_gaussian(), rnd.get_random_gaussian(), rnd.get_random_gaussian();
            lower = lower*10 + solution;
            upper = rnd.get_random_gaussian(), rnd.get_random_gaussian(), rnd.get_random_gaussian(), rnd.get_random_gaussian();
            upper = upper*10 + solution;
            for (int i = 0; i < lower.size(); ++i)
            {
                if (upper(i) < lower(i))
                    swap(upper(i),lower(i));
            }

            starting_point = rnd.get_random_double()*(upper(0)-lower(0))+lower(0),
                           rnd.get_random_double()*(upper(1)-lower(1))+lower(1),
                           rnd.get_random_double()*(upper(2)-lower(2))+lower(2),
                           rnd.get_random_double()*(upper(3)-lower(3))+lower(3);

            dlog << LINFO << "lower: "<< trans(lower);
            dlog << LINFO << "upper: "<< trans(upper);
            dlog << LINFO << "starting: "<< trans(starting_point);

            x = starting_point;
            double val = find_min_box_constrained(
                search_strategy,
                objective_delta_stop_strategy(1e-16, 500),
                brown, brown_derivative, x,
                lower,
                upper
            );

            DLIB_TEST(std::abs(val - brown(x)) < 1e-14);
            dlog << LINFO << "brown solution:\n" << x;
            return unconstrained_gradient_magnitude(brown_derivative, x, lower, upper);
        }

        template <typename search_strategy_type>
        void test_box_constrained_optimizers(search_strategy_type search_strategy)
        {
            dlib::rand rnd;
            running_stats<double> rs;

            dlog << LINFO << "test find_min_box_constrained() on rosen";
            for (int i = 0; i < 10000; ++i)
                rs.add(test_bound_solver_rosen(rnd, search_strategy));
            dlog << LINFO << "mean rosen gradient: " << rs.mean();
            dlog << LINFO << "max rosen gradient:  " << rs.max();
            DLIB_TEST(rs.mean() < 1e-12);
            DLIB_TEST(rs.max() < 1e-9);

            dlog << LINFO << "test find_min_box_constrained() on brown";
            rs.clear();
            for (int i = 0; i < 1000; ++i)
                rs.add(test_bound_solver_brown(rnd, search_strategy));
            dlog << LINFO << "mean brown gradient: " << rs.mean();
            dlog << LINFO << "max brown gradient:  " << rs.max();
            dlog << LINFO << "min brown gradient:  " << rs.min();
            DLIB_TEST(rs.mean() < 1e-5);
            DLIB_TEST(rs.max() < 1e-2);
            DLIB_TEST(rs.min() < 1e-10);

            dlog << LINFO << "test find_max_box_constrained() on neg_rosen";
            rs.clear();
            for (int i = 0; i < 1000; ++i)
                rs.add(test_bound_solver_neg_rosen(rnd, search_strategy));
            dlog << LINFO << "mean neg_rosen gradient: " << rs.mean();
            dlog << LINFO << "max neg_rosen gradient:  " << rs.max();
            DLIB_TEST(rs.mean() < 1e-12);
            DLIB_TEST(rs.max() < 1e-9);

        }

        void test_poly_min_extract_2nd()
        {
            double off;

            off = 0.0; DLIB_TEST(std::abs( poly_min_extrap(off*off, -2*off, (1-off)*(1-off)) - off) < 1e-13);
            off = 0.1; DLIB_TEST(std::abs( poly_min_extrap(off*off, -2*off, (1-off)*(1-off)) - off) < 1e-13);
            off = 0.2; DLIB_TEST(std::abs( poly_min_extrap(off*off, -2*off, (1-off)*(1-off)) - off) < 1e-13);
            off = 0.3; DLIB_TEST(std::abs( poly_min_extrap(off*off, -2*off, (1-off)*(1-off)) - off) < 1e-13);
            off = 0.4; DLIB_TEST(std::abs( poly_min_extrap(off*off, -2*off, (1-off)*(1-off)) - off) < 1e-13);
            off = 0.5; DLIB_TEST(std::abs( poly_min_extrap(off*off, -2*off, (1-off)*(1-off)) - off) < 1e-13);
            off = 0.6; DLIB_TEST(std::abs( poly_min_extrap(off*off, -2*off, (1-off)*(1-off)) - off) < 1e-13);
            off = 0.8; DLIB_TEST(std::abs( poly_min_extrap(off*off, -2*off, (1-off)*(1-off)) - off) < 1e-13);
            off = 0.9; DLIB_TEST(std::abs( poly_min_extrap(off*off, -2*off, (1-off)*(1-off)) - off) < 1e-13);
            off = 1.0; DLIB_TEST(std::abs( poly_min_extrap(off*off, -2*off, (1-off)*(1-off)) - off) < 1e-13);
        }

        class optimization_tester : public tester
        {
        public:
            optimization_tester (
            ) :
                tester ("test_optimization",
                        "Runs tests on the optimization component.")
            {}

            void perform_test (
            )
            {
                dlog << LINFO << "test_box_constrained_optimizers(bfgs_search_strategy())";
                test_box_constrained_optimizers(bfgs_search_strategy());
                dlog << LINFO << "test_box_constrained_optimizers(lbfgs_search_strategy(5))";
                test_box_constrained_optimizers(lbfgs_search_strategy(5));
                test_poly_min_extract_2nd();
                optimization_test();
            }
        } a;

    }