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Stochastic search techniques are used when the structure of a space is not well understood or is not smooth, so that techniques like Newton's method (which requires calculating Jacobian derivative matrices) cannot be used. In particular, these techniques are frequently used to solve combinatorial optimization problems, such as the traveling salesman problem.
The goal is to find a point in the space at which a real valued energy function (or cost function) is minimized. Simulated annealing is a minimization technique which has given good results in avoiding local minima; it is based on the idea of taking a random walk through the space at successively lower temperatures, where the probability of taking a step is given by a Boltzmann distribution.
The functions described in this chapter are declared in the header file 'gsl_siman.h'.
The simulated annealing algorithm takes random walks through the problem space, looking for points with low energies; in these random walks, the probability of taking a step is determined by the Boltzmann distribution,
p = e^{-(E_{i+1} - E_i)/(kT)}
if E_{i+1} > E_i, and p = 1 when E_{i+1} <= E_i.
In other words, a step will occur if the new energy is lower. If the new energy is higher, the transition can still occur, and its likelihood is proportional to the temperature T and inversely proportional to the energy difference E_{i+1} - E_i.
The temperature T is initially set to a high value, and a random walk is carried out at that temperature. Then the temperature is lowered very slightly according to a cooling schedule, for example: T -> T/mu_T where \mu_T is slightly greater than 1.
The slight probability of taking a step that gives higher energy is what allows simulated annealing to frequently get out of local minima.
This function performs a simulated annealing search through a given space. The space is specified by providing the functions Ef and distance. The simulated annealing steps are generated using the random number generator r and the function take_step.
The starting configuration of the system should be given by x0_p.
The routine offers two modes for updating configurations, a fixed-size
mode and a variable-size mode. In the fixed-size mode the configuration
is stored as a single block of memory of size element_size.
Copies of this configuration are created, copied and destroyed
internally using the standard library functions malloc,
memcpy and free. The function pointers copyfunc,
copy_constructor and destructor should be null pointers in
fixed-size mode. In the variable-size mode the functions
copyfunc, copy_constructor and destructor are used to
create, copy and destroy configurations internally. The variable
element_size should be zero in the variable-size mode.
The params structure (described below) controls the run by providing the temperature schedule and other tunable parameters to the algorithm.
On exit the best result achieved during the search is placed in
*x0_p. If the annealing process has been successful this
should be a good approximation to the optimal point in the space.
If the function pointer print_position is not null, a debugging
log will be printed to stdout with the following columns:
number_of_iterations temperature x x-(*x0_p) Ef(x)
and the output of the function print_position itself. If print_position is null then no information is printed.
double (*gsl_siman_Efunc_t) (void *xp)
void (*gsl_siman_step_t) (const gsl_rng *r, void *xp,
double step_size)
double (*gsl_siman_metric_t) (void *xp, void *yp)
void (*gsl_siman_print_t) (void *xp)
void (*gsl_siman_copy_t) (void *source, void *dest)
void * (*gsl_siman_copy_construct_t) (void *xp)
void (*gsl_siman_destroy_t) (void *xp)
gsl_siman_solve.
This structure contains all the information needed to control the
search, beyond the energy function, the step function and the initial
guess.
int n_tries
int iters_fixed_T
double step_size
double k, t_initial, mu_t, t_min
The simulated Annealing package is clumsy, and it has to be because it is written in C, for C callers, and tries to be polymorphic at the same time. But here we provide some examples which can be pasted into your application with little change and should make things easier.
The first example, in one dimensional cartesian space, sets up an energy function which is a damped sine wave; this has many local minima, but only one global minimum, somewhere between 1.0 and 1.5. The initial guess given is 15.5, which is several local minima away from the global minimum.
#include <math.h>
#include <stdlib.h>
#include <gsl/gsl_siman.h>
/* set up parameters for this simulated annealing run */
/* how many points do we try before stepping */
#define N_TRIES 200
/* how many iterations for each T? */
#define ITERS_FIXED_T 10
/* max step size in random walk */
#define STEP_SIZE 10
/* Boltzmann constant */
#define K 1.0
/* initial temperature */
#define T_INITIAL 0.002
/* damping factor for temperature */
#define MU_T 1.005
#define T_MIN 2.0e-6
gsl_siman_params_t params
= {N_TRIES, ITERS_FIXED_T, STEP_SIZE,
K, T_INITIAL, MU_T, T_MIN};
/* now some functions to test in one dimension */
double E1(void *xp)
{
double x = * ((double *) xp);
return exp(-pow((x-1.0),2.0))*sin(8*x);
}
double M1(void *xp, void *yp)
{
double x = *((double *) xp);
double y = *((double *) yp);
return fabs(x - y);
}
void S1(const gsl_rng * r, void *xp, double step_size)
{
double old_x = *((double *) xp);
double new_x;
double u = gsl_rng_uniform(r);
new_x = u * 2 * step_size - step_size + old_x;
memcpy(xp, &new_x, sizeof(new_x));
}
void P1(void *xp)
{
printf ("%12g", *((double *) xp));
}
int
main(int argc, char *argv[])
{
const gsl_rng_type * T;
gsl_rng * r;
double x_initial = 15.5;
gsl_rng_env_setup();
T = gsl_rng_default;
r = gsl_rng_alloc(T);
gsl_siman_solve(r, &x_initial, E1, S1, M1, P1,
NULL, NULL, NULL,
sizeof(double), params);
return 0;
}
Here are a couple of plots that are generated by running
siman_test in the following way:
./siman_test | grep -v "^#" | xyplot -xyil -y -0.88 -0.83 -d "x...y" | xyps -d > siman-test.eps ./siman_test | grep -v "^#" | xyplot -xyil -xl "generation" -yl "energy" -d "x..y" | xyps -d > siman-energy.eps
The TSP (Traveling Salesman Problem) is the classic combinatorial optimization problem. I have provided a very simple version of it, based on the coordinates of twelve cities in the southwestern United States. This should maybe be called the Flying Salesman Problem, since I am using the great-circle distance between cities, rather than the driving distance. Also: I assume the earth is a sphere, so I don't use geoid distances.
The gsl_siman_solve() routine finds a route which is 3490.62
Kilometers long; this is confirmed by an exhaustive search of all
possible routes with the same initial city.
The full code can be found in 'siman/siman_tsp.c', but I include here some plots generated with in the following way:
./siman_tsp > tsp.output
grep -v "^#" tsp.output
| xyplot -xyil -d "x................y"
-lx "generation" -ly "distance"
-lt "TSP -- 12 southwest cities"
| xyps -d > 12-cities.eps
grep initial_city_coord tsp.output
| awk '{print $2, $3, $4, $5}'
| xyplot -xyil -lb0 -cs 0.8
-lx "longitude (- means west)"
-ly "latitude"
-lt "TSP -- initial-order"
| xyps -d > initial-route.eps
grep final_city_coord tsp.output
| awk '{print $2, $3, $4, $5}'
| xyplot -xyil -lb0 -cs 0.8
-lx "longitude (- means west)"
-ly "latitude"
-lt "TSP -- final-order"
| xyps -d > final-route.eps
This is the output showing the initial order of the cities; longitude is negative, since it is west and I want the plot to look like a map.
# initial coordinates of cities (longitude and latitude) ###initial_city_coord: -105.95 35.68 Santa Fe ###initial_city_coord: -112.07 33.54 Phoenix ###initial_city_coord: -106.62 35.12 Albuquerque ###initial_city_coord: -103.2 34.41 Clovis ###initial_city_coord: -107.87 37.29 Durango ###initial_city_coord: -96.77 32.79 Dallas ###initial_city_coord: -105.92 35.77 Tesuque ###initial_city_coord: -107.84 35.15 Grants ###initial_city_coord: -106.28 35.89 Los Alamos ###initial_city_coord: -106.76 32.34 Las Cruces ###initial_city_coord: -108.58 37.35 Cortez ###initial_city_coord: -108.74 35.52 Gallup ###initial_city_coord: -105.95 35.68 Santa Fe
The optimal route turns out to be:
# final coordinates of cities (longitude and latitude) ###final_city_coord: -105.95 35.68 Santa Fe ###final_city_coord: -106.28 35.89 Los Alamos ###final_city_coord: -106.62 35.12 Albuquerque ###final_city_coord: -107.84 35.15 Grants ###final_city_coord: -107.87 37.29 Durango ###final_city_coord: -108.58 37.35 Cortez ###final_city_coord: -108.74 35.52 Gallup ###final_city_coord: -112.07 33.54 Phoenix ###final_city_coord: -106.76 32.34 Las Cruces ###final_city_coord: -96.77 32.79 Dallas ###final_city_coord: -103.2 34.41 Clovis ###final_city_coord: -105.92 35.77 Tesuque ###final_city_coord: -105.95 35.68 Santa Fe
Here's a plot of the cost function (energy) versus generation (point in the calculation at which a new temperature is set) for this problem:
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