R: spg Projection Functions

project {BB} | R Documentation

## spg Projection Functions

### Description

Projection function implementing contraints for spg parameters.

### Usage

         projectLinear(par, A, b, meq)

### Arguments

`par` | A real vector argument (as for `fn`), indicating the parameter values to which the constraint should be applied.
`A` | A matrix. See details.
`b` | A vector. See details.
`meq` | See details.

### Details

The function `projectLinear` can be used by `spg` to define the constraints of the problem. It projects a point in R^n onto a region that defines the constraints. It takes a real vector `par` as argument and returns a real vector of the same length.

The function `projectLinear` incorporates linear equalities and inequalities in nonlinear optimization using a projection method, where an infeasible point is projected onto the feasible region using a quadratic programming solver. The inequalities are defined such that: `A %*% x - b > 0` . The first ‘meq’ rows of A and the first ‘meq’ elements of b correspond to equality constraints.

### Value

A vector of the constrained parameter values.

### Note

We are grateful to Berwin Turlach for creating a special version of solve.QP function that is specifically tailored to solving the projection problem.

### See Also

`spg`

### Examples

    # Example
    fn <- function(x) (x[1] - 3/2)^2 + (x[2] - 1/8)^4

    gr <- function(x) c(2 * (x[1] - 3/2) , 4 * (x[2] - 1/8)^3)

    # This is the set of inequalities
    # x[1] - x[2] >= -1
    # x[1] + x[2] >= -1
    # x[1] - x[2] <= 1
    # x[1] + x[2] <= 1

    # The inequalities are written in R such that:  Amat %*% x  >= b
    Amat <- matrix(c(1, -1, 1, 1, -1, 1, -1, -1), 4, 2, byrow=TRUE)
    b <- c(-1, -1, -1, -1)
    meq <- 0  # all 4 conditions are inequalities

    p0 <- rnorm(2)
    spg(par=p0, fn=fn, gr=gr, project="projectLinear",
          projectArgs=list(A=Amat, b=b, meq=meq))

    meq <- 1  # first condition is now an equality
    spg(par=p0, fn=fn, gr=gr, project="projectLinear",
          projectArgs=list(A=Amat, b=b, meq=meq))

    # box-constraints can be incorporated as follows:
    # x[1] >= 0
    # x[2] >= 0
    # x[1] <= 0.5
    # x[2] <= 0.5

    Amat <- matrix(c(1, 0, 0, 1, -1, 0, 0, -1), 4, 2, byrow=TRUE)
    b <- c(0, 0, -0.5, -0.5)

    meq <- 0
    spg(par=p0, fn=fn, gr=gr, project="projectLinear",
       projectArgs=list(A=Amat, b=b, meq=meq))

    # Note that the above is the same as the following:
    spg(par=p0, fn=fn, gr=gr, lower=0, upper=0.5)

    # An example showing how to impose other constraints in spg()

    fr <- function(x) { ## Rosenbrock Banana function
      x1 <- x[1]
      x2 <- x[2]
      100 * (x2 - x1 * x1)^2 + (1 - x1)^2
      }

    # Impose a constraint that sum(x) = 1

    proj <- function(x){ x / sum(x) }

    spg(par=runif(2), fn=fr, project="proj")

    # Illustration of the importance of `projecting' the constraints, rather
    #   than simply finding a feasible point:

    fr <- function(x) { ## Rosenbrock Banana function
    x1 <- x[1]
    x2 <- x[2]
    100 * (x2 - x1 * x1)^2 + (1 - x1)^2
    }
    # Impose a constraint that sum(x) = 1

    proj <- function(x){
    # Although this function does give a feasible point it is
    #  not a "projection" in the sense of the nearest feasible point to `x'
    x / sum(x)
    }

    p0 <- c(0.93, 0.94)

    # Note, the starting value is infeasible so the next
    #   result is "Maximum function evals exceeded"

    spg(par=p0, fn=fr, project="proj")

    # Correct approach to doing the projection using the `projectLinear' function

    spg(par=p0, fn=fr, project="projectLinear", projectArgs=list(A=matrix(1, 1, 2), b=1, meq=1))

    # Impose additional box constraint on first parameter

    p0 <- c(0.4, 0.94)    # need feasible starting point

    spg(par=p0, fn=fr,  lower=c(-0.5, -Inf), upper=c(0.5, Inf),
      project="projectLinear", projectArgs=list(A=matrix(1, 1, 2), b=1, meq=1))

[Package BB version 2014.10-1 Index]