spg Projection Functions

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Description

Projection function implementing contraints for spg parameters.

Usage

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     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

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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))