constrOptim Linearly Constrained Optimization

Description

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

Usage

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)

Arguments

theta	numeric (vector) starting value (of length p): must be in the feasible region.
f	function to minimise (see below).
grad	gradient of f (a function as well), or NULL (see below).
ui	constraint matrix (k x p), see below.
ci	constraint vector of length k (see below).
mu	(Small) tuning parameter.
control, method, hessian	passed to optim.
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 f and grad: needs to be passed through optim so should not match its argument names.

Details

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.

Value

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.

References

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

See Also

optim, especially method = "L-BFGS-B" which does box-constrained optimisation.

Examples

## 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))