dlib C++ Library - optimization_ex.cpp

// 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() function. 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
 std::chrono::milliseconds(500) // run this long
 );
 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;
}