Generalized Simulated Annealing Function

Description

This function searches for global minimum of a very complex non-linear objective function with a very large number of optima.

Usage

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    GenSA(par, fn, lower, upper, control=list(), ...)

Arguments

par

Vector. Initial values for the components to be optimized. Default is NULL, in which case, default values will be generated automatically.

fn

A function to be minimized, with first argument the vector of parameters over which minimization is to take place. It should return a scalar result.

lower

Vector with length of par. Lower bounds for components.

upper

Vector with length of par. Upper bounds for components.

...

allows the user to pass additional arguments to the function fn.

control

The argument is a list that can be used to control the behavior of the algorithm:

maxit

Integer. Maximum number of iterations of the algorithm.

threshold.stop

Numeric. The program will stop when the expected objective function value threshold.stop is reached. Default value is NULL

nb.stop.improvement

Integer. The program will stop when there is no any improvement in nb.stop.improvement steps.

smooth

Logical.TRUE when the objective function is smooth, or differentiable almost everywhere in the region of par, FALSE otherwise. Default value is TRUE.

max.call

Integer. Maximum number of call of the objective function. Default is set to 1e7.

max.time

Numeric. Maximum running time in seconds.

temperature

Numeric. Initial value for temperature.

visiting.param

Numeric. Parameter for visiting distribution.

acceptance.param

Numeric. Parameter for acceptance distribution.

verbose

Logical. TRUE means that messages from the algorithm are shown. Default is FALSE.

simple.function

Logical. FALSE means that the objective function has only a few local minima. Default is FALSE which means that the objective function is complicated with many local minima.

trace.mat

Logical. Default is TRUE which means that the trace matrix will be available in the returned value of GenSA call.

Details

The default values of the control components are set for a complex optimization problem. For usual optimization problem with medium complexity, GenSA can find a reasonable solution quickly so the user is recommended to let GenSA stop earlier by setting threshold.stop if threshold.stop is the expected function value, or by setting max.time if the user just want to run GenSA for max.time seconds, or by setting max.call if the user just want to run GenSA within max.call function calls. Please refer to the examples below. For very complex optimization problems, the user is recommended to increase maxit and temp.

Value

The returned value is a list with the following fields:

value:

Numeric. The value of fn corresponding to par.

par:

Vector. The best set of parameters found.

trace.mat:

A matrix which contains the history of the algorithm. (By columns: Step number, temperature, current objective function value, current minimal objective function value).

counts:

Integer. Total number of calls of the objective function.

Author(s)

Yang Xiang, Sylvain Gubian, Brian Suomela, Julia Hoeng, PMP SA.

References

Xiang Y, Gubian S, Suomela B, Hoeng (2012). "Generalized Simulated Annealing for Efficient Global Optimization: the GenSA Package for R". The R Journal, Forthcoming. http://journal.r-project.org/.

Tsallis C (1988). "Possible generalization of Boltzmann-Gibbs statistics." Journal of Statistical Physics, 52, 479–487.

Tsallis C, Stariolo DA (1996). "Generalized Simulated Annealing." Physica A, 233, 395–406.

Xiang Y, Sun DY, Fan W, Gong XG (1997). "Generalized Simulated Annealing Algorithm and Its Application to the Thomson Model." Physics Letters A, 233, 216–220.

Xiang Y, Gong XG (2000a). "Efficiency of Generalized Simulated Annealing." PHYSICAL REVIEW E, 62, 4473.

Xiang Y, Sun DY, Gong XG (2000). "Generalized Simulated Annealing Studies on Structures and Properties of Nin (n=2-55) Clusters." Journal of Physical Chemistry A, 104, 2746–2751.

See Also

optim

Examples

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    library(GenSA)
# Try Rastrgin function (The objective function value for global minimum
# is 0 with all components of par are 0.)
        Rastrigin <- function(x) {
            sum(x^2 - 10 * cos(2 * pi  * x)) + 10 * length(x)
        }
# Perform the search on a 30 dimensions rastrigin function. Rastrigin
# function with dimension 30 is known as the most
# difficult optimization problem according to "Yao X, Liu Y, Lin G (1999).
# \Evolutionary Programming Made Faster."
# IEEE Transactions on Evolutionary Computation, 3(2), 82-102.

# GenSA will stop after finding the targeted function value 0 with
# absolute tolerance 1e-13
    set.seed(1234) # The user can use any seed.
        dimension <- 30
        global.min <- 0
        tol <- 1e-13
        lower <- rep(-5.12, dimension)
        upper <- rep(5.12, dimension)
        out <- GenSA(lower = lower, upper = upper, fn = Rastrigin,
                control=list(threshold.stop=global.min+tol,verbose=TRUE))
        out[c("value","par","counts")]

# GenSA will stop after running for about 2 seconds
# Note: The time for solving this problem by GenSA may vary
# depending on the computer used. 
        set.seed(1234) # The user can use any seed.
        dimension <- 30
        global.min <- 0
        tol <- 1e-13
        lower <- rep(-5.12, dimension)
        upper <- rep(5.12, dimension)
        out <- GenSA(lower = lower, upper = upper, fn = Rastrigin,
                control=list(max.time=2))
        out[c("value","par","counts")]

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