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Commit 4d0b2035 authored by Davis King's avatar Davis King
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Just moved the try block to reduce the indentation level.

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examples/optimization_ex.cpp
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......@@ -113,213 +113,208 @@ public:
// ----------------------------------------------------------------------------------------
int main()
int main() try
{
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_max_global() routine. Like
// find_max_bobyqa(), this is a technique specially designed to maximize
// 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_max_global()
// won't. find_max_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
// maximizer. 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 the bounds constraints. This makes it the method
// of choice for derivative free optimization in the presence of 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)
{
// 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_max_global() routine. Like
// find_max_bobyqa(), this is a technique specially designed to maximize
// 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_max_global()
// won't. find_max_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
// maximizer. 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 the bounds constraints. This makes it the method
// of choice for derivative free optimization in the presence of 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)
{
// 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_max_global()
auto result = find_max_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_max_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;
}
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_max_global()
auto result = find_max_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_max_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;
}
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