coneB function

Cone Projection -- Constraint Cone

This routine implements the hinge algorithm for cone projection to minimize $||y - \theta||^2$ over the cone $C$ of the form $\{\theta: \theta = v + \sum b_i\delta_i, i = 1,\ldots,m, b_1,\ldots, b_m \ge 0\}$, $v$ is in $V$.

coneB(y, delta, vmat = NULL, w = NULL, face = NULL, msg = TRUE)

Arguments

• y: A vector of length $n$.
• delta: A matrix whose columns are the constraint cone edges. The columns of delta must be irreducible. Its row number must equal the length of $y$. No column of delta is contained in the column space of vmat.
• vmat: A matrix whose columns are the basis of the linear space contained in the constraint cone. Its row number must equal the length of $y$. The columns of vmat must be linearly independent. The default is vmat = NULL
• 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 $(y - \theta)'w(y - \theta)$ 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 $1,\ldots,m$. 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

Details

The routine coneB dynamically loads a C++ subroutine "coneBCpp".

Returns

• df: The dimension of the face of the constraint cone on which the projection lands.

• yhat: The projection of $y$ on the constraint cone.

• steps: The number of iterations before the algorithm converges.

• coefs: The coefficients of the basis of the linear space and the constraint cone edges contained in the constraint cone.

• 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 $1,\ldots,m$.

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.

Author(s)

Mary C. Meyer and Xiyue Liao

See Also

coneA, shapereg

Examples

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

# create the edges of the constraint cone 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
    }

# note that in coneB, the transpose of the edges of the constraint cone is provided
    delta <- crossprod(amat, solve(tcrossprod(amat)))
    
# make the basis of V
    vmat <- matrix(rep(1, n), ncol = 1)

# call coneB
    ans3 <- coneB(y, delta, vmat)
    ans4 <- coneB(y, delta, vmat, w = (1:n)/n)

# make a plot to compare the unweighted fit and weighted fit
    par(mar = c(4, 4, 1, 1))
    plot(y, cex = .7, ylab = "y")
    lines(fitted(ans3), col = 2, lty = 2)
    lines(fitted(ans4), col = 4, lty = 2)
    legend("topleft", bty = "n", c("unweighted fit", "weighted fit"), col = c(2, 4), lty = c(2, 2))
    title("ConeB Example Plot")