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340 lines
15 KiB
C++
340 lines
15 KiB
C++
// The contents of this file are in the public domain. See LICENSE_FOR_EXAMPLE_PROGRAMS.txt
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/*
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This is an example illustrating the use the general purpose non-linear
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optimization routines from the dlib C++ Library.
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The library provides implementations of the conjugate gradient, BFGS,
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L-BFGS, and BOBYQA optimization algorithms. These algorithms allow you to
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find the minimum of a function of many input variables. This example walks
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though a few of the ways you might put these routines to use.
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*/
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#include <dlib/optimization.h>
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#include <iostream>
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using namespace std;
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using namespace dlib;
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// ----------------------------------------------------------------------------------------
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// In dlib, the general purpose solvers optimize functions that take a column
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// vector as input and return a double. So here we make a typedef for a
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// variable length column vector of doubles. This is the type we will use to
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// represent the input to our objective functions which we will be minimizing.
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typedef matrix<double,0,1> column_vector;
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// ----------------------------------------------------------------------------------------
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// Below we create a few functions. When you get down into main() you will see that
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// we can use the optimization algorithms to find the minimums of these functions.
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// ----------------------------------------------------------------------------------------
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double rosen (const column_vector& m)
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/*
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This function computes what is known as Rosenbrock's function. It is
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a function of two input variables and has a global minimum at (1,1).
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So when we use this function to test out the optimization algorithms
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we will see that the minimum found is indeed at the point (1,1).
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*/
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{
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const double x = m(0);
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const double y = m(1);
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// compute Rosenbrock's function and return the result
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return 100.0*pow(y - x*x,2) + pow(1 - x,2);
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}
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// This is a helper function used while optimizing the rosen() function.
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const column_vector rosen_derivative (const column_vector& m)
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/*!
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ensures
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- returns the gradient vector for the rosen function
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!*/
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{
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const double x = m(0);
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const double y = m(1);
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// make us a column vector of length 2
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column_vector res(2);
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// now compute the gradient vector
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res(0) = -400*x*(y-x*x) - 2*(1-x); // derivative of rosen() with respect to x
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res(1) = 200*(y-x*x); // derivative of rosen() with respect to y
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return res;
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}
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// This function computes the Hessian matrix for the rosen() fuction. This is
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// the matrix of second derivatives.
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matrix<double> rosen_hessian (const column_vector& m)
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{
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const double x = m(0);
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const double y = m(1);
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matrix<double> res(2,2);
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// now compute the second derivatives
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res(0,0) = 1200*x*x - 400*y + 2; // second derivative with respect to x
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res(1,0) = res(0,1) = -400*x; // derivative with respect to x and y
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res(1,1) = 200; // second derivative with respect to y
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return res;
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}
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// ----------------------------------------------------------------------------------------
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class test_function
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{
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/*
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This object is an example of what is known as a "function object" in C++.
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It is simply an object with an overloaded operator(). This means it can
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be used in a way that is similar to a normal C function. The interesting
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thing about this sort of function is that it can have state.
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In this example, our test_function object contains a column_vector
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as its state and it computes the mean squared error between this
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stored column_vector and the arguments to its operator() function.
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This is a very simple function, however, in general you could compute
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any function you wanted here. An example of a typical use would be
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to find the parameters of some regression function that minimized
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the mean squared error on a set of data. In this case the arguments
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to the operator() function would be the parameters of your regression
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function. You would loop over all your data samples and compute the output
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of the regression function for each data sample given the parameters and
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return a measure of the total error. The dlib optimization functions
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could then be used to find the parameters that minimized the error.
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*/
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public:
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test_function (
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const column_vector& input
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)
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{
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target = input;
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}
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double operator() ( const column_vector& arg) const
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{
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// return the mean squared error between the target vector and the input vector
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return mean(squared(target-arg));
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}
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private:
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column_vector target;
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};
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// ----------------------------------------------------------------------------------------
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class rosen_model
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{
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/*!
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This object is a "function model" which can be used with the
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find_min_trust_region() routine.
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!*/
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public:
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typedef ::column_vector column_vector;
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typedef matrix<double> general_matrix;
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double operator() (
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const column_vector& x
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) const { return rosen(x); }
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void get_derivative_and_hessian (
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const column_vector& x,
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column_vector& der,
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general_matrix& hess
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) const
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{
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der = rosen_derivative(x);
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hess = rosen_hessian(x);
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}
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};
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// ----------------------------------------------------------------------------------------
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int main()
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{
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try
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{
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// make a column vector of length 2
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column_vector starting_point(2);
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// Set the starting point to (4,8). This is the point the optimization algorithm
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// will start out from and it will move it closer and closer to the function's
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// minimum point. So generally you want to try and compute a good guess that is
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// somewhat near the actual optimum value.
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starting_point = 4, 8;
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// The first example below finds the minimum of the rosen() function and uses the
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// analytical derivative computed by rosen_derivative(). Since it is very easy to
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// make a mistake while coding a function like rosen_derivative() it is a good idea
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// to compare your derivative function against a numerical approximation and see if
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// the results are similar. If they are very different then you probably made a
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// mistake. So the first thing we do is compare the results at a test point:
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cout << "Difference between analytic derivative and numerical approximation of derivative: "
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<< length(derivative(rosen)(starting_point) - rosen_derivative(starting_point)) << endl;
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cout << "Find the minimum of the rosen function()" << endl;
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// Now we use the find_min() function to find the minimum point. The first argument
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// to this routine is the search strategy we want to use. The second argument is the
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// stopping strategy. Below I'm using the objective_delta_stop_strategy which just
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// says that the search should stop when the change in the function being optimized
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// is small enough.
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// The other arguments to find_min() are the function to be minimized, its derivative,
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// then the starting point, and the last is an acceptable minimum value of the rosen()
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// function. That is, if the algorithm finds any inputs to rosen() that gives an output
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// value <= -1 then it will stop immediately. Usually you supply a number smaller than
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// the actual global minimum. So since the smallest output of the rosen function is 0
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// we just put -1 here which effectively causes this last argument to be disregarded.
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find_min(bfgs_search_strategy(), // Use BFGS search algorithm
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objective_delta_stop_strategy(1e-7), // Stop when the change in rosen() is less than 1e-7
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rosen, rosen_derivative, starting_point, -1);
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// Once the function ends the starting_point vector will contain the optimum point
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// of (1,1).
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cout << "rosen solution:\n" << starting_point << endl;
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// Now let's try doing it again with a different starting point and the version
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// of find_min() that doesn't require you to supply a derivative function.
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// This version will compute a numerical approximation of the derivative since
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// we didn't supply one to it.
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starting_point = -94, 5.2;
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find_min_using_approximate_derivatives(bfgs_search_strategy(),
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objective_delta_stop_strategy(1e-7),
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rosen, starting_point, -1);
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// Again the correct minimum point is found and stored in starting_point
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cout << "rosen solution:\n" << starting_point << endl;
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// Here we repeat the same thing as above but this time using the L-BFGS
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// algorithm. L-BFGS is very similar to the BFGS algorithm, however, BFGS
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// uses O(N^2) memory where N is the size of the starting_point vector.
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// The L-BFGS algorithm however uses only O(N) memory. So if you have a
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// function of a huge number of variables the L-BFGS algorithm is probably
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// a better choice.
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starting_point = 0.8, 1.3;
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find_min(lbfgs_search_strategy(10), // The 10 here is basically a measure of how much memory L-BFGS will use.
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objective_delta_stop_strategy(1e-7).be_verbose(), // Adding be_verbose() causes a message to be
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// printed for each iteration of optimization.
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rosen, rosen_derivative, starting_point, -1);
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cout << endl << "rosen solution: \n" << starting_point << endl;
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starting_point = -94, 5.2;
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find_min_using_approximate_derivatives(lbfgs_search_strategy(10),
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objective_delta_stop_strategy(1e-7),
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rosen, starting_point, -1);
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cout << "rosen solution: \n"<< starting_point << endl;
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// dlib also supports solving functions subject to bounds constraints on
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// the variables. So for example, if you wanted to find the minimizer
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// of the rosen function where both input variables were in the range
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// 0.1 to 0.8 you would do it like this:
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starting_point = 0.1, 0.1; // Start with a valid point inside the constraint box.
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find_min_box_constrained(lbfgs_search_strategy(10),
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objective_delta_stop_strategy(1e-9),
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rosen, rosen_derivative, starting_point, 0.1, 0.8);
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// Here we put the same [0.1 0.8] range constraint on each variable, however, you
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// can put different bounds on each variable by passing in column vectors of
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// constraints for the last two arguments rather than scalars.
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cout << endl << "constrained rosen solution: \n" << starting_point << endl;
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// You can also use an approximate derivative like so:
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starting_point = 0.1, 0.1;
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find_min_box_constrained(bfgs_search_strategy(),
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objective_delta_stop_strategy(1e-9),
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rosen, derivative(rosen), starting_point, 0.1, 0.8);
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cout << endl << "constrained rosen solution: \n" << starting_point << endl;
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// In many cases, it is useful if we also provide second derivative information
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// to the optimizers. Two examples of how we can do that are shown below.
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starting_point = 0.8, 1.3;
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find_min(newton_search_strategy(rosen_hessian),
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objective_delta_stop_strategy(1e-7),
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rosen,
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rosen_derivative,
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starting_point,
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-1);
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cout << "rosen solution: \n"<< starting_point << endl;
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// We can also use find_min_trust_region(), which is also a method which uses
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// second derivatives. For some kinds of non-convex function it may be more
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// reliable than using a newton_search_strategy with find_min().
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starting_point = 0.8, 1.3;
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find_min_trust_region(objective_delta_stop_strategy(1e-7),
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rosen_model(),
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starting_point,
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10 // initial trust region radius
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);
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cout << "rosen solution: \n"<< starting_point << endl;
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// Now let's look at using the test_function object with the optimization
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// functions.
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cout << "\nFind the minimum of the test_function" << endl;
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column_vector target(4);
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starting_point.set_size(4);
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// This variable will be used as the target of the test_function. So,
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// our simple test_function object will have a global minimum at the
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// point given by the target. We will then use the optimization
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// routines to find this minimum value.
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target = 3, 5, 1, 7;
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// set the starting point far from the global minimum
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starting_point = 1,2,3,4;
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find_min_using_approximate_derivatives(bfgs_search_strategy(),
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objective_delta_stop_strategy(1e-7),
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test_function(target), starting_point, -1);
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// At this point the correct value of (3,5,1,7) should be found and stored in starting_point
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cout << "test_function solution:\n" << starting_point << endl;
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// Now let's try it again with the conjugate gradient algorithm.
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starting_point = -4,5,99,3;
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find_min_using_approximate_derivatives(cg_search_strategy(),
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objective_delta_stop_strategy(1e-7),
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test_function(target), starting_point, -1);
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cout << "test_function solution:\n" << starting_point << endl;
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// Finally, let's try the BOBYQA algorithm. This is a technique specially
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// designed to minimize a function in the absence of derivative information.
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// Generally speaking, it is the method of choice if derivatives are not available.
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starting_point = -4,5,99,3;
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find_min_bobyqa(test_function(target),
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starting_point,
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9, // number of interpolation points
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uniform_matrix<double>(4,1, -1e100), // lower bound constraint
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uniform_matrix<double>(4,1, 1e100), // upper bound constraint
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10, // initial trust region radius
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1e-6, // stopping trust region radius
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100 // max number of objective function evaluations
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);
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cout << "test_function solution:\n" << starting_point << endl;
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}
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catch (std::exception& e)
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{
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cout << e.what() << endl;
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}
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}
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