R: Linearly Constrained Optimization

constrOptim {stats} | R Documentation

## 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))

[Package stats version 2.15.1 Index]