constrOptim {stats} | R Documentation |

Minimise a function subject to linear inequality constraints using an adaptive barrier algorithm.

constrOptim(theta, f, grad, ui, ci, mu = 1e-04, control = list(), method = if(is.null(grad)) "Nelder-Mead" else "BFGS", outer.iterations = 100, outer.eps = 1e-05, ..., hessian = FALSE)

`theta` |
numeric (vector) starting value (of length |

`f` |
function to minimise (see below). |

`grad` |
gradient of |

`ui` |
constraint matrix ( |

`ci` |
constraint vector of length |

`mu` |
(Small) tuning parameter. |

`control, method, hessian` |
passed to |

`outer.iterations` |
iterations of the barrier algorithm. |

`outer.eps` |
non-negative number; the relative convergence tolerance of the barrier algorithm. |

`...` |
Other named arguments to be passed to |

The feasible region is defined by `ui %*% theta - ci >= 0`

. The
starting value must be in the interior of the feasible region, but the
minimum may be on the boundary.

A logarithmic barrier is added to enforce the constraints and then
`optim`

is called. The barrier function is chosen so that
the objective function should decrease at each outer iteration. Minima
in the interior of the feasible region are typically found quite
quickly, but a substantial number of outer iterations may be needed
for a minimum on the boundary.

The tuning parameter `mu`

multiplies the barrier term. Its precise
value is often relatively unimportant. As `mu`

increases the
augmented objective function becomes closer to the original objective
function but also less smooth near the boundary of the feasible
region.

Any `optim`

method that permits infinite values for the
objective function may be used (currently all but "L-BFGS-B").

The objective function `f`

takes as first argument the vector
of parameters over which minimisation is to take place. It should
return a scalar result. Optional arguments `...`

will be
passed to `optim`

and then (if not used by `optim`

) to
`f`

. As with `optim`

, the default is to minimise, but
maximisation can be performed by setting `control$fnscale`

to a
negative value.

The gradient function `grad`

must be supplied except with
`method = "Nelder-Mead"`

. It should take arguments matching
those of `f`

and return a vector containing the gradient.

As for `optim`

, but with two extra components:
`barrier.value`

giving the value of the barrier function at the
optimum and `outer.iterations`

gives the
number of outer iterations (calls to `optim`

).
The `counts`

component contains the *sum* of all
`optim()$counts`

.

K. Lange *Numerical Analysis for Statisticians.* Springer
2001, p185ff

`optim`

, especially `method = "L-BFGS-B"`

which
does box-constrained optimisation.

## from optim fr <- function(x) { ## Rosenbrock Banana function x1 <- x[1] x2 <- x[2] 100 * (x2 - x1 * x1)^2 + (1 - x1)^2 } grr <- function(x) { ## Gradient of 'fr' x1 <- x[1] x2 <- x[2] c(-400 * x1 * (x2 - x1 * x1) - 2 * (1 - x1), 200 * (x2 - x1 * x1)) } optim(c(-1.2,1), fr, grr) #Box-constraint, optimum on the boundary constrOptim(c(-1.2,0.9), fr, grr, ui = rbind(c(-1,0), c(0,-1)), ci = c(-1,-1)) # x <= 0.9, y - x > 0.1 constrOptim(c(.5,0), fr, grr, ui = rbind(c(-1,0), c(1,-1)), ci = c(-0.9,0.1)) ## Solves linear and quadratic programming problems ## but needs a feasible starting value # # from example(solve.QP) in 'quadprog' # no derivative fQP <- function(b) {-sum(c(0,5,0)*b)+0.5*sum(b*b)} Amat <- matrix(c(-4,-3,0,2,1,0,0,-2,1), 3, 3) bvec <- c(-8, 2, 0) constrOptim(c(2,-1,-1), fQP, NULL, ui = t(Amat), ci = bvec) # derivative gQP <- function(b) {-c(0, 5, 0) + b} constrOptim(c(2,-1,-1), fQP, gQP, ui = t(Amat), ci = bvec) ## Now with maximisation instead of minimisation hQP <- function(b) {sum(c(0,5,0)*b)-0.5*sum(b*b)} constrOptim(c(2,-1,-1), hQP, NULL, ui = t(Amat), ci = bvec, control = list(fnscale = -1))

[Package *stats* version 4.1.1 Index]