# constrOptim: Linearly Constrained Optimization

### Description

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

### Usage

 ```1 2 3 4``` ```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

`optim`, especially `method = "L-BFGS-B"` which does box-constrained optimisation.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37``` ```## 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)) ```