optim_sa {optimization}

Flexible Optimization with Simulated Annealing

Description

Random search optimization method with systematic component that searches the global optimum. The loss function is allowed to be non-linear, non-differentiable and multimodal. Undefined responses are allowed as well.

Usage

optim_sa(fun, start, maximization = FALSE, trace = FALSE,
         lower, upper, control = list())

Arguments

fun

Loss function to be optimized. It must return a scalar value. The variables must be assigned as a vector. See 'details'.

start

Vector of initial values for the function variables. Must be of same length as the variables vector of the loss function. The response of the initial variables combination must be defined (NA or NaN responses are not allowed).

maximization

Logical. Default is FALSE.

trace

Logical. If TRUE, interim results are stored. Necessary for the plot function. Default is FALSE.

lower

Vector of lower boundaries for the function variables. Must be of same length as the variables vector of the function.

upper

Vector of upper boundaries for the function variables. Must be of same length as the variables vector of the function.

control

List with optional further arguments to modify the optimization specifically to the loss function:

vf

Function that determines the variation of the function variables for the next iteration. The variation function is allowed to depend on the vector of variables of the current iteration, the vector of random factors rf and the temperature of the current iteration. Default is a uniform distributed random number with relative range rf.

rf

Numeric vector. Random factor vector that determines the variation of the random number of vf in relation to the dimension of the function variables for the following iteration. Default is 1. If dyn_rf is enabled, the rf change dynamically over time.

dyn_rf

Logical. rf change dynamically over time to ensure increasing precision with increasing number of iterations. Default is TRUE, see 'details'.

t0

Numeric. Initial temperature. Default is 1000.

nlimit

Integer. Maximum number of iterations of the inner loop. Default is 100.

r

Numeric. Temperature reduction in the outer loop. Default is 0.6.

k

Numeric. Constant for the Metropolis function. Default is 1.

t_min

Numeric. Temperature where outer loop stops. Default is 0.1.

maxgood

Integer. Break criterion to improve the algorithm performance. Maximum number of loss function improvements in the inner loop. Breaks the inner loop. Default is 100.

stopac

Integer. Break criterion to improve the algorithm performance. Maximum number of repetitions where the loss improvement is lower than ac_acc. Breaks the inner loop. Default is 30.

ac_acc

Numeric. Accuracy of the stopac break criterion in relation to the response. Default is 1/10000 of the function value at initial variables combination.

Details

Simulated Annealing is an optimization algorithm for solving complex functions that may have several optima. The method is composed of a random and a systematic component. Basically, it randomly modifies the variables combination n_limit times to compare their response values. Depending on the temperature and the constant k, there is also a likelihood of choosing variables combinations with worse response. There is thus a time-decreasing likelihood of leaving local optima. The Simulated Annealing Optimization method is therefore advantageous for multimodal functions. Undefined response values (NA) are allowed as well. This can be useful for loss functions with variables restrictions. The high number of parameters allows a very flexible parameterization. optim_sa is able to solve mathematical formulas as well as complex rule sets.

The performance therefore highly depends on the settings. It is indispensable to parameterize the algorithm carefully. The control list is pre-parameterized for loss functions of medium complexity. To improve the performance, the settings should be changed when solving relatively simple functions (e. g. three dimensional multimodal functions). For complex functions the settings should be changed to improve the accuracy. Most important parameters are nlimit, r and t0.

The dynamic rf adjustment depends on the number of loss function calls which are out of the variables boundaries as well as the temperature of the current iteration. The obligatory decreasing rf ensures a relatively wide search grid at the beginning of the optimization process that shrinks over time. It thus automatically adjusts for the trade-off between range of the search grid and accuracy. See Pronzato (1984) for more details. It is sometimes useful to disable the dynamic rf changing when the most performant rf are known. As dyn_rf usually improves the performance as well as the accuracy, the default is TRUE.

Value

The output is a nmsa_optim list object with following entries:

par

Function variables after optimization.

function_value

Loss function response after optimization.

trace

Matrix with interim results. NULL if trace was not activated.

fun

The loss function.

start

The initial function variables.

lower

The lower boundaries of the function variables.

upper

The upper boundaries of the function variables.

control

Control arguments, see 'details'.

Author(s)

Kai Husmann

References

Corana, A., Marchesi, M., Martini, C. and Ridella, S. (1987), Minimizing Multimodal Functions of Continuous Variables with the 'Simulated Annealing' Algorithm. ACM Transactions on Mathematical Software, 13(3):262-280.

Kirkpatrick, S., Gelatt, C. D. and Vecchi, M. P. (1983). Optimization by Simulated Annealing. Science, 220(4598):671-680.

Pronzato, L., Walter, E., Venot, A. and Lebruchec, J.-F. (1984). A general-purpose global optimizer: Implementation and applications. Mathematics and Computers in Simulation, 26(5):412-422.

See Also

optim_nm, optim, plot.optim_nmsa

Examples

##### Rosenbrock function
# minimum at f(1,1) = 0
ro <- function(x){
  100*(x[2]-x[1]^2)^2+(1-x[1])^2
}

# Random start values. Example arguments for the relatively simple Rosenbrock function.
ro_sa <- optim_sa(fun = ro,
                  start = c(runif(2, min = -1, max = 1)),
                  lower = c(-5, -5),
                  upper = c(5, 5),
                  trace = TRUE,
                  control = list(t0 = 100,
                            nlimit = 550,
                            t_min = 0.1,
                            dyn_rf = FALSE,
                            rf = 1,
                            r = 0.7
                  )
         )


# Visual inspection.
plot(ro_sa)
plot(ro_sa, type = "contour")


##### Holder table function

# 4 minima at
  #f(8.055, 9.665) = -19.2085
  #f(-8.055, 9.665) = -19.2085
  #f(8.055, -9.665) = -19.2085
  #f(-8.055, -9.665) = -19.2085

ho <- function(x){
  x1 <- x[1]
  x2 <- x[2]

  fact1 <- sin(x1) * cos(x2)
  fact2 <- exp(abs(1 - sqrt(x1^2 + x2^2) / pi))
  y <- -abs(fact1 * fact2)
}

# Random start values. Example arguments for the relatively complex Holder table function.
optim_sa(fun = ho,
         start = c(1, 1),
         lower = c(-10, -10),
         upper = c(10, 10),
         trace = TRUE,
         control = list(dyn_rf = FALSE,
                        rf = 1.6,
                        t0 = 10,
                        nlimit = 200,
                        r = 0.6,
                        t_min = 0.1
         )
)