coneproj (version 1.2)

qprog: Quadratic Programming

Description

Given a positive definite $n$ by $n$ matrix $Q$ and a constant vector $c$ in $R^n$, the object is to find $\theta$ in $R^n$ to minimize $\theta'Q\theta - 2c'\theta$ subject to $A\theta \ge b$, for an irreducible constraint matrix $A$. This routine transforms into a cone projection problem for the constrained solution.

Usage

qprog(q, c, amat, b)

Arguments

q

A $n$ by $n$ positive definite matrix.

c

A vector of length $n$.

amat

A $m$ by $n$ constraint matrix. The rows of amat must be irreducible.

b

A vector of length $m$. Its default value is 0.

Value

• df
  The dimension of the face of the constraint cone on which the projection lands.
• thetahat
  A vector minimizing $\theta'Q\theta - 2c'\theta$.
• steps
  The number of iterations before the algorithm converges.

Details

To get the constrained solution to $\theta'Q\theta - 2c'\theta$ subject to $A\theta \ge b$, this routine makes the Cholesky decomposition of $Q$. Let $U'U = Q$, and define $\phi = U\theta$ and $z = U^{-1}c$, where $U^{-1}$ is the inverse of $U$. Then we minimize $||z - \phi||^2$, subject to $B\phi \ge 0$, where $B = AU^{-1}$. It is now a cone projection problem with the constraint cone $C$ of the form ${\phi: B\phi \ge 0 }$. This routine gives the estimation of $\theta$, which is $U^{-1}$ times the estimation of $\phi$. The routine qprog dynamically loads a C++ subroutine "qprogCpp".

References

Goldfarb, D. and A. Idnani (1983) A numerically stable dual method for solving strictly convex quadratic programs. Mathematical Programming 27, 1--33. Fraser, D. A. S. and H. Massam (1989) A mixed primal-dual bases algorithm for regression under inequality constraints application to concave regression. Scandinavian Journal of Statistics 16, 65--74. Fang,S.-C. and S. Puthenpura (1993) Linear Optimization and Extensions. Englewood Cliffs, New Jersey: Prentice Hall. Silvapulle, M. J. and P. Sen (2005) Constrained Statistical Inference. John Wiley and Sons. Meyer, M. C. (2013b) A simple new algorithm for quadratic programming with applications in statistics. Communications in Statistics 42(5), 1126--1139.

See Also

coneA

Examples

#load the cubic data set
    data(cubic)

#extract x
    x <- cubic$x

#extract y
    y <- cubic$y

#make the design matrix
    xmat <- cbind(1, x, x^2, x^3)

#make the q matrix
    q <- crossprod(xmat)

#make the c vector
    c <- crossprod(xmat, y)

#make the constraint matrix to constrain the regression to be increasing, nonnegative and convex
    amat <- matrix(0, 4, 4)
    amat[1, 1] <- 1; amat[2, 2] <- 1
    amat[3, 3] <- 1; amat[4, 3] <- 1
    amat[4, 4] <- 6
    b <- rep(0, 4)

#call qprog 
    ans <- qprog(q, c, amat, b)

#get the constrained fit of y
    betahat <- ans$thetahat
    fitc <- crossprod(t(xmat), betahat)

#get the unconstrained fit of y
    fitu <- lm(y ~ x + I(x^2) + I(x^3))

# make a plot to compare fitc and fitu
    par(mar = c(4, 4, 1, 1))
    plot(x, y, cex = .7, xlab = "x", ylab = "y")
    lines(x, fitted(fitu))
    lines(x, fitc, col = 2, lty = 4)
    legend("topleft", c("constr.fit", "unconstr.fit"), lty = c(4, 1), col = c(2, 1))
    title("Qprog Example Plot")