Improved Stochastic Ranking Evolution Strategy

The Improved Stochastic Ranking Evolution Strategy (ISRES) is an algorithm for nonlinearly constrained global optimization, or at least semi-global, although it has heuristics to escape local optima.

isres(
  x0,
  fn,
  lower,
  upper,
  hin = NULL,
  heq = NULL,
  maxeval = 10000,
  pop.size = 20 * (length(x0) + 1),
  xtol_rel = 1e-06,
  nl.info = FALSE,
  deprecatedBehavior = TRUE,
  ...
)

Arguments

x0

initial point for searching the optimum.

fn

objective function that is to be minimized.

lower, upper

lower and upper bound constraints.

hin

function defining the inequality constraints, that is hin <= 0 for all components.

heq

function defining the equality constraints, that is heq = 0 for all components.

maxeval

maximum number of function evaluations.

pop.size

population size.

xtol_rel

stopping criterion for relative change reached.

nl.info

logical; shall the original NLopt info be shown.

deprecatedBehavior

logical; if TRUE (default for now), the old behavior of the Jacobian function is used, where the equality is \(\ge 0\) instead of \(\le 0\). This will be reversed in a future release and eventually removed.

...

additional arguments passed to the function.

Value

List with components:

par

the optimal solution found so far.

value

the function value corresponding to par.

iter

number of (outer) iterations, see maxeval.

convergence

integer code indicating successful completion (> 0) or a possible error number (< 0).

message

character string produced by NLopt and giving additional information.

Details

The evolution strategy is based on a combination of a mutation rule—with a log-normal step-size update and exponential smoothing—and differential variation—a Nelder-Mead-like update rule). The fitness ranking is simply via the objective function for problems without nonlinear constraints, but when nonlinear constraints are included the stochastic ranking proposed by Runarsson and Yao is employed.

This method supports arbitrary nonlinear inequality and equality constraints in addition to the bounds constraints.

Note

The initial population size for CRS defaults to \(20x(n+1)\) in \(n\) dimensions, but this can be changed. The initial population must be at least \(n+1\).

References

Thomas Philip Runarsson and Xin Yao, “Search biases in constrained evolutionary optimization,” IEEE Trans. on Systems, Man, and Cybernetics Part C: Applications and Reviews, vol. 35 (no. 2), pp. 233-243 (2005).

Author

Hans W. Borchers

Examples

## Rosenbrock Banana objective function

rbf <- function(x) {(1 - x[1]) ^ 2 + 100 * (x[2] - x[1] ^ 2) ^ 2}

x0 <- c(-1.2, 1)
lb <- c(-3, -3)
ub <- c(3,  3)

## The function as written above has a minimum of 0 at (1, 1)

isres(x0 = x0, fn = rbf, lower = lb, upper = ub)
#> $par
#> [1] 1.000015 1.000029
#> 
#> $value
#> [1] 2.785245e-10
#> 
#> $iter
#> [1] 10000
#> 
#> $convergence
#> [1] 5
#> 
#> $message
#> [1] "NLOPT_MAXEVAL_REACHED: Optimization stopped because maxeval (above) was reached."
#> 

## Now subject to the inequality that x[1] + x[2] <= 1.5

hin <- function(x) {x[1] + x[2] - 1.5}

S <- isres(x0 = x0, fn = rbf, hin = hin, lower = lb, upper = ub,
           maxeval = 2e5L, deprecatedBehavior = FALSE)

S
#> $par
#> [1] 0.8231316 0.6768683
#> 
#> $value
#> [1] 0.03132831
#> 
#> $iter
#> [1] 13126
#> 
#> $convergence
#> [1] 4
#> 
#> $message
#> [1] "NLOPT_XTOL_REACHED: Optimization stopped because xtol_rel or xtol_abs (above) was reached."
#> 

sum(S$par)
#> [1] 1.5