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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.

__Function:__void**gsl_siman_solve***(const gsl_rng **`r`, void *`x0_p`, gsl_siman_Efunc_t`Ef`, gsl_siman_step_t`take_step`, gsl_siman_metric_t`distance`, gsl_siman_print_t`print_position`, gsl_siman_copy_t`copyfunc`, gsl_siman_copy_construct_t`copy_constructor`, gsl_siman_destroy_t`destructor`, size_t`element_size`, gsl_siman_params_t`params`)-
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

`*`

. If the annealing process has been successful this should be a good approximation to the optimal point in the space.`x0_p`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.

__Data Type:__**gsl_siman_Efunc_t**-
This function type should return the energy of a configuration
`xp`.double (*gsl_siman_Efunc_t) (void *xp)

__Data Type:__**gsl_siman_step_t**-
This function type should modify the configuration
`xp`using a random step taken from the generator`r`, up to a maximum distance of`step_size`.void (*gsl_siman_step_t) (const gsl_rng *r, void *xp, double step_size)

__Data Type:__**gsl_siman_metric_t**-
This function type should return the distance between two configurations
`xp`and`yp`.double (*gsl_siman_metric_t) (void *xp, void *yp)

__Data Type:__**gsl_siman_print_t**-
This function type should print the contents of the configuration
`xp`.void (*gsl_siman_print_t) (void *xp)

__Data Type:__**gsl_siman_copy_t**-
This function type should copy the configuration
`dest`into`source`.void (*gsl_siman_copy_t) (void *source, void *dest)

__Data Type:__**gsl_siman_copy_construct_t**-
This function type should create a new copy of the configuration
`xp`.void * (*gsl_siman_copy_construct_t) (void *xp)

__Data Type:__**gsl_siman_destroy_t**-
This function type should destroy the configuration
`xp`, freeing its memory.void (*gsl_siman_destroy_t) (void *xp)

__Data Type:__**gsl_siman_params_t**-
These are the parameters that control a run of
`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`

- The number of points to try for each step
`int iters_fixed_T`

- The number of iterations at each temperature
`double step_size`

- The maximum step size in the random walk
`double k, t_initial, mu_t, t_min`

- The parameters of the Boltzmann distribution and cooling schedule

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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