Projection function implementing contraints for spg parameters.

1 | ```
projectLinear(par, A, b, meq)
``` |

`par` |
A real vector argument (as for |

`A` |
A matrix. See details. |

`b` |
A vector. See details. |

`meq` |
See 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.

A vector of the constrained parameter values.

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

`spg`

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# 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))
``` |

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