SAopt {NMOF}

Optimisation with Simulated Annealing

Description

The function implements a Simulated-Annealing algorithm.

Usage

SAopt(OF, algo = list(), ...)

Arguments

OF

The objective function, to be minimised. Its first argument needs to be a solution x; it will be called as OF(x, ...).

algo

A list of settings for the algorithm. See Details.

...

other variables passed to OF and algo$neighbour. See Details.

Details

Simulated Annealing (SA) changes an initial solution iteratively; the algorithm stops after a fixed number of iterations. Conceptually, SA consists of a loop than runs for a number of iterations. In each iteration, a current solution xc is changed through a function algo$neighbour. If this new (or neighbour) solution xn is not worse than xc, ie, if OF(xn,...) <= OF(xc,...), then xn replaces xc. If xn is worse, it still replaces xc, but only with a certain probability. This probability is a function of the degree of the deterioration (the greater, the less likely the new solution is accepted) and the current iteration (the longer the algorithm has already run, the less likely the new solution is accepted).

The list algo contains the following items.

nS

The number of steps per temperature. The default is 1000; but this setting depends very much on the problem.

nT

The number of temperatures. Default is 10.

nI

Total number of iterations, with default NULL. If specified, it will override nS with ceiling(nI/nT). Using this option makes it easier to compare and switch between functions LSopt, TAopt and SAopt.

nD

The number of random steps to calibrate the temperature. Defaults to 2000.

initT

Initial temperature. Defaults to NULL, in which case it is automatically chosen so that initProb is achieved.

finalT

Final temperature. Defaults to 0.

alpha

The cooling constant. The current temperature is multiplied by this value. Default is 0.9.

mStep

Step multiplier. The default is 1, which implies constant number of steps per temperature. If greater than 1, the step number nS is increased to m*nS (and rounded).

x0

The initial solution. If this is a function, it will be called once without arguments to compute an initial solution, ie, x0 <- algo$x0(). This can be useful when the routine is called in a loop of restarts, and each restart is to have its own starting value.

neighbour

The neighbourhood function, called as neighbour(x, ...). Its first argument must be a solution x; it must return a changed solution.

printDetail

If TRUE (the default), information is printed. If an integer i greater then one, information is printed at very ith iteration.

printBar

If TRUE (default is FALSE), a txtProgressBar (from package utils) is printed. The progress bar is not shown if printDetail is an integer greater than 1.

storeF

if TRUE (the default), the objective function values for every solution in every generation are stored and returned as matrix Fmat.

storeSolutions

Default is FALSE. If TRUE, the solutions (ie, decision variables) in every generation are stored and returned in list xlist (see Value section below). To check, for instance, the current solution at the end of the ith generation, retrieve xlist[[c(2L, i)]].

classify

Logical; default is FALSE. If TRUE, the result will have a class attribute SAopt attached.

OF.target

Numeric; when specified, the algorithm will stop when an objective-function value as low as OF.target (or lower) is achieved. This is useful when an optimal objective-function value is known: the algorithm will then stop and not waste time searching for a better solution.

At the minimum, algo needs to contain an initial solution x0 and a neighbour function.

The total number of iterations equals algo$nT times algo$nS (plus possibly algo$nD).

Value

SAopt returns a list with five components:

xbest

the solution

OFvalue

objective function value of the solution, ie, OF(xbest, ...)

Fmat

if algo$storeF is TRUE, a matrix with one row for each iteration (excluding the initial algo$nD steps) and two columns. The first column contains the objective function values of the neighbour solution at a given iteration; the second column contains the value of the current solution. Since SA can walk away from locally-optimal solutions, the best solution can be monitored through cummin(Fmat[ ,2L]).

xlist

if algo$storeSolutions is TRUE, a list; else NA. Contains the neighbour solutions at a given iteration (xn) and the current solutions (xc). Example: Fmat[i, 2L] is the objective function value associated with xlist[[c(2L, i)]].

initial.state

the value of .Random.seed when the function was called.

If algo$classify was set to TRUE, the resulting list will have a class attribute TAopt.

Note

If the ... argument is used, then all the objects passed with ... need to go into the objective function and the neighbourhood function. It is recommended to collect all information in a list myList and then write OF and neighbour so that they are called as OF(x, myList) and neighbour(x, myList). Note that x need not be a vector but can be any data structure (eg, a matrix or a list).

Using an initial and final temperature of zero means that SA will be equivalent to a Local Search. The function LSopt may be preferred then because of smaller overhead.

Author(s)

Enrico Schumann

References

Gilli, M., Maringer, D. and Schumann, E. (2019) Numerical Methods and Optimization in Finance. 2nd edition. Elsevier. doi:10.1016/C2017-0-01621-X

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

Schumann, E. (2023) Financial Optimisation with R (NMOF Manual). http://enricoschumann.net/NMOF.htm#NMOFmanual

See Also

LSopt, TAopt, restartOpt

Examples

## Aim: given a matrix x with n rows and 2 columns,
##      divide the rows of x into two subsets such that
##      in one subset the columns are highly correlated,
##      and in the other lowly (negatively) correlated.
##      constraint: a single subset should have at least 40 rows

## create data with specified correlation
n <- 100L
rho <- 0.7
C <- matrix(rho, 2L, 2L); diag(C) <- 1
x <- matrix(rnorm(n * 2L), n, 2L) %*% chol(C)

## collect data
data <- list(x = x, n = n, nmin = 40L)

## a random initial solution
x0 <- runif(n) > 0.5

## a neighbourhood function
neighbour <- function(xc, data) {
    xn <- xc
    p <- sample.int(data$n, size = 1L)
    xn[p] <- abs(xn[p] - 1L)
    # reject infeasible solution
    c1 <- sum(xn) >= data$nmin
    c2 <- sum(xn) <= (data$n - data$nmin)
    if (c1 && c2) res <- xn else res <- xc
    as.logical(res)
}

## check (should be 1 FALSE and n-1 TRUE)
x0 == neighbour(x0, data)

## objective function
OF <- function(xc, data)
    -abs(cor(data$x[xc, ])[1L, 2L] - cor(data$x[!xc, ])[1L, 2L])

## check
OF(x0, data)
## check
OF(neighbour(x0, data), data)

## plot data
par(mfrow = c(1,3), bty = "n")
plot(data$x,
     xlim = c(-3,3), ylim = c(-3,3),
     main = "all data", col = "darkgreen")

## *Local Search*
algo <- list(nS = 3000L,
             neighbour = neighbour,
             x0 = x0,
             printBar = FALSE)
sol1 <- LSopt(OF, algo = algo, data=data)
sol1$OFvalue

## *Simulated Annealing*
algo$nT <- 10L
algo$nS <- ceiling(algo$nS/algo$nT)
sol <- SAopt(OF, algo = algo, data = data)
sol$OFvalue

c1 <- cor(data$x[ sol$xbest, ])[1L, 2L]
c2 <- cor(data$x[!sol$xbest, ])[1L, 2L]

lines(data$x[ sol$xbest, ], type = "p", col = "blue")

plot(data$x[ sol$xbest, ], col = "blue",
     xlim = c(-3, 3), ylim = c(-3, 3),
     main = paste("subset 1, corr.", format(c1, digits = 3)))

plot(data$x[!sol$xbest, ], col = "darkgreen",
     xlim = c(-3,3), ylim = c(-3,3),
     main = paste("subset 2, corr.", format(c2, digits = 3)))

## compare LS/SA
par(mfrow = c(1, 1), bty = "n")
plot(sol1$Fmat[ , 2L],type = "l", ylim=c(-1.5, 0.5),
     ylab = "OF", xlab = "Iterations")
lines(sol$Fmat[ , 2L],type = "l", col = "blue")
legend(x = "topright", legend = c("LS", "SA"),
       lty = 1, lwd = 2, col = c("black", "blue"))