coneA: Cone Projection -- Polar Cone

Description

This routine implements the hinge algorithm for cone projection to minimize $| | y - θ | |^{2}$ over the cone $C$ of the form ${θ : A θ \geq 0}$ .

Usage

coneA(y, amat, w = NULL, face = NULL, msg = TRUE)

Value

df: The dimension of the face of the constraint cone on which the projection lands.
thetahat: The projection of $y$ on the constraint cone.
steps: The number of iterations before the algorithm converges.
xmat: The rows of the matrix are the edges of the face of the polar cone on which the residual of the projection onto the constraint cone lands.
face: A vector of the positions of edges, which define the face on which the final projection lands on. For example, when there are $m$ cone edges, then face is a subset of $1, \dots, m$ .

Arguments

y: A vector of length $n$ .
amat: A constraint matrix. The rows of amat must be irreducible. The column number of amat must equal the length of $y$ .
w: An optional nonnegative vector of weights of length $n$ . If w is not given, all weights are taken to equal 1. Otherwise, the minimization of $(y - θ)^{'} w (y - θ)$ over $C$ is returned. The default is w = NULL.
face: A vector of the positions of edges, which define the initial face for the cone projection. For example, when there are $m$ cone edges, then face is a subset of $1, \dots, m$ . The default is face = NULL.
msg: A logical flag. If msg is TRUE, then a warning message will be printed when there is a non-convergence problem; otherwise no warning message will be printed. The default is msg = TRUE

Author

Mary C. Meyer and Xiyue Liao

Details

The routine coneA dynamically loads a C++ subroutine "coneACpp". The rows of $- A$ are the edges of the polar cone $Ω^{o}$ . This routine first projects $y$ onto $Ω^{o}$ to get the residual of the projection onto the constraint cone $C$ , and then uses the fact that $y$ is equal to the sum of the projection of $y$ onto $C$ and the projection of $y$ onto $Ω^{o}$ to get the estimation of $θ$ . See references cited in this section for more details about the relationship between polar cone and constraint cone.

References

Meyer, M. C. (1999) An extension of the mixed primal-dual bases algorithm to the case of more constraints than dimensions. Journal of Statistical Planning and Inference 81, 13--31.

Meyer, M. C. (2013b) A simple new algorithm for quadratic programming with applications in statistics. Communications in Statistics 42(5), 1126--1139.

Liao, X. and M. C. Meyer (2014) coneproj: An R package for the primal or dual cone projections with routines for constrained regression. Journal of Statistical Software 61(12), 1--22.

Examples

Run this code

# generate y
    set.seed(123)
    n <- 50
    x <- seq(-2, 2, length = 50)
    y <- - x^2 + rnorm(n)

# create the constraint matrix to make the first half of y monotonically increasing
# and the second half of y monotonically decreasing
    amat <- matrix(0, n - 1, n)
    for(i in 1:(n/2 - 1)){
       amat[i, i] <- -1; amat[i, i + 1] <- 1
    }
    for(i in (n/2):(n - 1)){
       amat[i, i] <- 1; amat[i, i + 1] <- -1
    }

# call coneA
    ans1 <- coneA(y, amat)
    ans2 <- coneA(y, amat, w = (1:n)/n)

# make a plot to compare the unweighted fit and the weighted fit
    par(mar = c(4, 4, 1, 1))
    plot(y, cex = .7, ylab = "y")
    lines(fitted(ans1), col = 2, lty = 2)
    lines(fitted(ans2), col = 4, lty = 2)
    legend("topleft", bty = "n", c("unweighted fit", "weighted fit"), col = c(2, 4), lty = c(2, 2))
    title("ConeA Example Plot")

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