```// The contents of this file are in the public domain. See LICENSE_FOR_EXAMPLE_PROGRAMS.txt
/*

This is an example illustrating the use the general purpose non-linear
optimization routines from the dlib C++ Library.

The library provides implementations of many popular algorithms such as L-BFGS
and BOBYQA.  These algorithms allow you to find the minimum or maximum of a
function of many input variables.  This example walks though a few of the ways
you might put these routines to use.

*/

#include <dlib/optimization.h>
#include <dlib/global_optimization.h>
#include <iostream>

using namespace std;
using namespace dlib;

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

// In dlib, most of the general purpose solvers optimize functions that take a
// column vector as input and return a double.  So here we make a typedef for a
// variable length column vector of doubles.  This is the type we will use to
// represent the input to our objective functions which we will be minimizing.
typedef matrix<double,0,1> column_vector;

// ----------------------------------------------------------------------------------------
// Below we create a few functions.  When you get down into main() you will see that
// we can use the optimization algorithms to find the minimums of these functions.
// ----------------------------------------------------------------------------------------

double rosen (const column_vector& m)
/*
This function computes what is known as Rosenbrock's function.  It is
a function of two input variables and has a global minimum at (1,1).
So when we use this function to test out the optimization algorithms
we will see that the minimum found is indeed at the point (1,1).
*/
{
const double x = m(0);
const double y = m(1);

// compute Rosenbrock's function and return the result
return 100.0*pow(y - x*x,2) + pow(1 - x,2);
}

// This is a helper function used while optimizing the rosen() function.
const column_vector rosen_derivative (const column_vector& m)
/*!
ensures
- returns the gradient vector for the rosen function
!*/
{
const double x = m(0);
const double y = m(1);

// make us a column vector of length 2
column_vector res(2);

// now compute the gradient vector
res(0) = -400*x*(y-x*x) - 2*(1-x); // derivative of rosen() with respect to x
res(1) = 200*(y-x*x);              // derivative of rosen() with respect to y
return res;
}

// This function computes the Hessian matrix for the rosen() fuction.  This is
// the matrix of second derivatives.
matrix<double> rosen_hessian (const column_vector& m)
{
const double x = m(0);
const double y = m(1);

matrix<double> res(2,2);

// now compute the second derivatives
res(0,0) = 1200*x*x - 400*y + 2; // second derivative with respect to x
res(1,0) = res(0,1) = -400*x;   // derivative with respect to x and y
res(1,1) = 200;                 // second derivative with respect to y
return res;
}

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

class rosen_model
{
/*!
This object is a "function model" which can be used with the
find_min_trust_region() routine.
!*/

public:
typedef ::column_vector column_vector;
typedef matrix<double> general_matrix;

double operator() (
const column_vector& x
) const { return rosen(x); }

void get_derivative_and_hessian (
const column_vector& x,
column_vector& der,
general_matrix& hess
) const
{
der = rosen_derivative(x);
hess = rosen_hessian(x);
}
};

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

int main() try
{
// Set the starting point to (4,8).  This is the point the optimization algorithm
// will start out from and it will move it closer and closer to the function's
// minimum point.   So generally you want to try and compute a good guess that is
// somewhat near the actual optimum value.
column_vector starting_point = {4, 8};

// The first example below finds the minimum of the rosen() function and uses the
// analytical derivative computed by rosen_derivative().  Since it is very easy to
// make a mistake while coding a function like rosen_derivative() it is a good idea
// to compare your derivative function against a numerical approximation and see if
// the results are similar.  If they are very different then you probably made a
// mistake.  So the first thing we do is compare the results at a test point:
cout << "Difference between analytic derivative and numerical approximation of derivative: "
<< length(derivative(rosen)(starting_point) - rosen_derivative(starting_point)) << endl;

cout << "Find the minimum of the rosen function()" << endl;
// Now we use the find_min() function to find the minimum point.  The first argument
// to this routine is the search strategy we want to use.  The second argument is the
// stopping strategy.  Below I'm using the objective_delta_stop_strategy which just
// says that the search should stop when the change in the function being optimized
// is small enough.

// The other arguments to find_min() are the function to be minimized, its derivative,
// then the starting point, and the last is an acceptable minimum value of the rosen()
// function.  That is, if the algorithm finds any inputs to rosen() that gives an output
// value <= -1 then it will stop immediately.  Usually you supply a number smaller than
// the actual global minimum.  So since the smallest output of the rosen function is 0
// we just put -1 here which effectively causes this last argument to be disregarded.

find_min(bfgs_search_strategy(),  // Use BFGS search algorithm
objective_delta_stop_strategy(1e-7), // Stop when the change in rosen() is less than 1e-7
rosen, rosen_derivative, starting_point, -1);
// Once the function ends the starting_point vector will contain the optimum point
// of (1,1).
cout << "rosen solution:\n" << starting_point << endl;

// Now let's try doing it again with a different starting point and the version
// of find_min() that doesn't require you to supply a derivative function.
// This version will compute a numerical approximation of the derivative since
// we didn't supply one to it.
starting_point = {-94, 5.2};
find_min_using_approximate_derivatives(bfgs_search_strategy(),
objective_delta_stop_strategy(1e-7),
rosen, starting_point, -1);
// Again the correct minimum point is found and stored in starting_point
cout << "rosen solution:\n" << starting_point << endl;

// Here we repeat the same thing as above but this time using the L-BFGS
// algorithm.  L-BFGS is very similar to the BFGS algorithm, however, BFGS
// uses O(N^2) memory where N is the size of the starting_point vector.
// The L-BFGS algorithm however uses only O(N) memory.  So if you have a
// function of a huge number of variables the L-BFGS algorithm is probably
// a better choice.
starting_point = {0.8, 1.3};
find_min(lbfgs_search_strategy(10),  // The 10 here is basically a measure of how much memory L-BFGS will use.
objective_delta_stop_strategy(1e-7).be_verbose(),  // Adding be_verbose() causes a message to be
// printed for each iteration of optimization.
rosen, rosen_derivative, starting_point, -1);

cout << endl << "rosen solution: \n" << starting_point << endl;

starting_point = {-94, 5.2};
find_min_using_approximate_derivatives(lbfgs_search_strategy(10),
objective_delta_stop_strategy(1e-7),
rosen, starting_point, -1);
cout << "rosen solution: \n"<< starting_point << endl;

// dlib also supports solving functions subject to bounds constraints on
// the variables.  So for example, if you wanted to find the minimizer
// of the rosen function where both input variables were in the range
// 0.1 to 0.8 you would do it like this:
starting_point = {0.1, 0.1}; // Start with a valid point inside the constraint box.
find_min_box_constrained(lbfgs_search_strategy(10),
objective_delta_stop_strategy(1e-9),
rosen, rosen_derivative, starting_point, 0.1, 0.8);
// Here we put the same [0.1 0.8] range constraint on each variable, however, you
// can put different bounds on each variable by passing in column vectors of
// constraints for the last two arguments rather than scalars.

cout << endl << "constrained rosen solution: \n" << starting_point << endl;

// You can also use an approximate derivative like so:
starting_point = {0.1, 0.1};
find_min_box_constrained(bfgs_search_strategy(),
objective_delta_stop_strategy(1e-9),
rosen, derivative(rosen), starting_point, 0.1, 0.8);
cout << endl << "constrained rosen solution: \n" << starting_point << endl;

// In many cases, it is useful if we also provide second derivative information
// to the optimizers.  Two examples of how we can do that are shown below.
starting_point = {0.8, 1.3};
find_min(newton_search_strategy(rosen_hessian),
objective_delta_stop_strategy(1e-7),
rosen,
rosen_derivative,
starting_point,
-1);
cout << "rosen solution: \n"<< starting_point << endl;

// We can also use find_min_trust_region(), which is also a method which uses
// second derivatives.  For some kinds of non-convex function it may be more
// reliable than using a newton_search_strategy with find_min().
starting_point = {0.8, 1.3};
find_min_trust_region(objective_delta_stop_strategy(1e-7),
rosen_model(),
starting_point,
10 // initial trust region radius
);
cout << "rosen solution: \n"<< starting_point << endl;

// Next, let's try the BOBYQA algorithm.  This is a technique specially
// designed to minimize a function in the absence of derivative information.
// Generally speaking, it is the method of choice if derivatives are not available
// and the function you are optimizing is smooth and has only one local optima.  As
// an example, consider the be_like_target function defined below:
column_vector target = {3, 5, 1, 7};
auto be_like_target = [&](const column_vector& x) {
return mean(squared(x-target));
};
starting_point = {-4,5,99,3};
find_min_bobyqa(be_like_target,
starting_point,
9,    // number of interpolation points
uniform_matrix<double>(4,1, -1e100),  // lower bound constraint
uniform_matrix<double>(4,1, 1e100),   // upper bound constraint
10,    // initial trust region radius
1e-6,  // stopping trust region radius
100    // max number of objective function evaluations
);
cout << "be_like_target solution:\n" << starting_point << endl;

// Finally, let's try the find_min_global() routine.  Like find_min_bobyqa(),
// this technique is specially designed to minimize a function in the absence
// of derivative information.  However, it is also designed to handle
// functions with many local optima.  Where BOBYQA would get stuck at the
// nearest local optima, find_min_global() won't.  find_min_global() uses a
// global optimization method based on a combination of non-parametric global
// function modeling and BOBYQA style quadratic trust region modeling to
// efficiently find a global minimizer.  It usually does a good job with a
// relatively small number of calls to the function being optimized.
//
// You also don't have to give it a starting point or set any parameters,
// other than defining bounds constraints.  This makes it the method of
// choice for derivative free optimization in the presence of multiple local
// optima.  Its API also allows you to define functions that take a
// column_vector as shown above or to explicitly use named doubles as
// arguments, which we do here.
auto complex_holder_table = [](double x0, double x1)
{
// This function is a version of the well known Holder table test
// function, which is a function containing a bunch of local optima.
// Here we make it even more difficult by adding more local optima
// and also a bunch of discontinuities.

// add discontinuities
double sign = 1;
for (double j = -4; j < 9; j += 0.5)
{
if (j < x0 && x0 < j+0.5)
x0 += sign*0.25;
sign *= -1;
}
// Holder table function tilted towards 10,10 and with additional
// high frequency terms to add more local optima.
return -( std::abs(sin(x0)*cos(x1)*exp(std::abs(1-std::sqrt(x0*x0+x1*x1)/pi))) -(x0+x1)/10 - sin(x0*10)*cos(x1*10));
};

// To optimize this difficult function all we need to do is call
// find_min_global()
auto result = find_min_global(complex_holder_table,
{-10,-10}, // lower bounds
{10,10}, // upper bounds
max_function_calls(300));

cout.precision(9);
// These cout statements will show that find_min_global() found the
// globally optimal solution to 9 digits of precision:
cout << "complex holder table function solution y (should be -21.9210397): " << result.y << endl;
cout << "complex holder table function solution x:\n" << result.x << endl;
}
catch (std::exception& e)
{
cout << e.what() << endl;
}

```