# Cone Projection -- Polar Cone This routine implements the hinge algorithm for cone projection to minimize $||y - \theta||^2$ over the cone $C$ of the form $\{\theta: A\theta \ge 0\}$. ```r coneA(y, amat, w = NULL, face = NULL, msg = TRUE) ``` ## 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 $(y - \theta)'w(y - \theta)$ 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 coneA dynamically loads a C++ subroutine "coneACpp". The rows of $- A$ are the edges of the polar cone $\Omega^o$. This routine first projects $y$ onto $\Omega^o$ 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 $\Omega^o$ to get the estimation of $\theta$. See references cited in this section for more details about the relationship between polar cone and constraint cone. ## Returns - **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 $1,\ldots,m$. ## 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 `coneB`, `constreg`, `qprog` ## Examples ```r # 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") ```

coneA function