slsqp: Sequential Quadratic Programming (SQP)

Description

Sequential (least-squares) quadratic programming (SQP) algorithm for nonlinearly constrained, gradient-based optimization, supporting both equality and inequality constraints.

Usage

slsqp(
  x0,
  fn,
  gr = NULL,
  lower = NULL,
  upper = NULL,
  hin = NULL,
  hinjac = NULL,
  heq = NULL,
  heqjac = NULL,
  nl.info = FALSE,
  control = list(),
  deprecatedBehavior = TRUE,
  ...
)

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 (> 1) or a possible error number (< 0).
message: character string produced by NLopt and giving additional information.

Arguments

x0: starting point for searching the optimum.
fn: objective function that is to be minimized.
gr: gradient of function fn; will be calculated numerically if not specified.
lower, upper: lower and upper bound constraints.
hin: function defining the inequality constraints, that is hin <= 0 for all components. This is new behavior in line with the rest of the nloptr arguments. To use the old behavior, please set deprecatedBehavior to TRUE.
hinjac: Jacobian of function hin; will be calculated numerically if not specified.
heq: function defining the equality constraints, that is heq = 0 for all components.
heqjac: Jacobian of function heq; will be calculated numerically if not specified.
nl.info: logical; shall the original NLopt info been shown.
control: list of options, see nl.opts for help.
deprecatedBehavior: logical; if TRUE (default for now), the old behavior of the Jacobian function is used, where the equality is $\geq 0$ instead of $\leq 0$ . This will be reversed in a future release and eventually removed.
...: additional arguments passed to the function.

Author

Hans W. Borchers

Details

The algorithm optimizes successive second-order (quadratic/least-squares) approximations of the objective function (via BFGS updates), with first-order (affine) approximations of the constraints.

References

Dieter Kraft, ``A software package for sequential quadratic programming'', Technical Report DFVLR-FB 88-28, Institut fuer Dynamik der Flugsysteme, Oberpfaffenhofen, July 1988.

Examples

Run this code


##  Solve the Hock-Schittkowski problem no. 100 with analytic gradients
##  See https://apmonitor.com/wiki/uploads/Apps/hs100.apm

x0.hs100 <- c(1, 2, 0, 4, 0, 1, 1)
fn.hs100 <- function(x) {(x[1] - 10) ^ 2 + 5 * (x[2] - 12) ^ 2 + x[3] ^ 4 +
                         3 * (x[4] - 11) ^ 2 + 10 * x[5] ^ 6 + 7 * x[6] ^ 2 +
                         x[7] ^ 4 - 4 * x[6] * x[7] - 10 * x[6] - 8 * x[7]}

hin.hs100 <- function(x) {c(
2 * x[1] ^ 2 + 3 * x[2] ^ 4 + x[3] + 4 * x[4] ^ 2 + 5 * x[5] - 127,
7 * x[1] + 3 * x[2] + 10 * x[3] ^ 2 + x[4] - x[5] - 282,
23 * x[1] + x[2] ^ 2 + 6 * x[6] ^ 2 - 8 * x[7] - 196,
4 * x[1] ^ 2 + x[2] ^ 2 - 3 * x[1] * x[2] + 2 * x[3] ^ 2 + 5 * x[6] -
 11 * x[7])
}

S <- slsqp(x0.hs100, fn = fn.hs100,   # no gradients and jacobians provided
     hin = hin.hs100,
     nl.info = TRUE,
     control = list(xtol_rel = 1e-8, check_derivatives = TRUE),
     deprecatedBehavior = FALSE)

##  The optimum value of the objective function should be 680.6300573
##  A suitable parameter vector is roughly
##  (2.330, 1.9514, -0.4775, 4.3657, -0.6245, 1.0381, 1.5942)

S

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