Sequential Quadratic Programming (SQP)
Sequential (least-squares) quadratic programming (SQP) algorithm for nonlinearly constrained, gradient-based optimization, supporting both equality and inequality constraints.
slsqp(x0, fn, gr = NULL, lower = NULL, upper = NULL, hin = NULL, hinjac = NULL, heq = NULL, heqjac = NULL, nl.info = FALSE, control = list(), ...)
x0 |
starting point for searching the optimum. |
fn |
objective function that is to be minimized. |
gr |
gradient of function |
lower, upper |
lower and upper bound constraints. |
hin |
function defining the inequality constraints, that is
|
hinjac |
Jacobian of function |
heq |
function defining the equality constraints, that is |
heqjac |
Jacobian of function |
nl.info |
logical; shall the original NLopt info been shown. |
control |
list of options, see |
... |
additional arguments passed to the function. |
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.
List with components:
par |
the optimal solution found so far. |
value |
the function value corresponding to |
iter |
number of (outer) iterations, see |
convergence |
integer code indicating successful completion (> 1) or a possible error number (< 0). |
message |
character string produced by NLopt and giving additional information. |
See more infos at http://ab-initio.mit.edu/wiki/index.php/NLopt_Algorithms.
Hans W. Borchers
Dieter Kraft, “A software package for sequential quadratic programming”, Technical Report DFVLR-FB 88-28, Institut fuer Dynamik der Flugsysteme, Oberpfaffenhofen, July 1988.
alabama::auglag, Rsolnp::solnp,
Rdonlp2::donlp2
## Solve the Hock-Schittkowski problem no. 100
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) {
h <- numeric(4)
h[1] <- 127 - 2*x[1]^2 - 3*x[2]^4 - x[3] - 4*x[4]^2 - 5*x[5]
h[2] <- 282 - 7*x[1] - 3*x[2] - 10*x[3]^2 - x[4] + x[5]
h[3] <- 196 - 23*x[1] - x[2]^2 - 6*x[6]^2 + 8*x[7]
h[4] <- -4*x[1]^2 - x[2]^2 + 3*x[1]*x[2] -2*x[3]^2 - 5*x[6] +11*x[7]
return(h)
}
S <- slsqp(x0.hs100, fn = fn.hs100, # no gradients and jacobians provided
hin = hin.hs100,
control = list(xtol_rel = 1e-8, check_derivatives = TRUE))
S
## Optimal value of objective function: 690.622270249131 *** WRONG ***
# Even the numerical derivatives seem to be too tight.
# Let's try with a less accurate jacobian.
hinjac.hs100 <- function(x) nl.jacobian(x, hin.hs100, heps = 1e-2)
S <- slsqp(x0.hs100, fn = fn.hs100,
hin = hin.hs100, hinjac = hinjac.hs100,
control = list(xtol_rel = 1e-8))
S
## Optimal value of objective function: 680.630057392593 *** CORRECT ***Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.