isres function

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.

Returns

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 $n+1$.

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)

## 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

sum(S$par)

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(s)

Hans W. Borchers