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\name{solve.QP}
\alias{solve.QP}
\title{Solve a Quadratic Programming Problem}
\description{
This routine implements the dual method of Goldfarb and Idnani (1982,
1983) for solving quadratic programming problems of the form
\eqn{\min(-d^T b + 1/2 b^T D b)} with the constraints \eqn{A^T b >=
b_0}.
}
\usage{
solve.QP(Dmat, dvec, Amat, bvec, meq=0, factorized=FALSE)
}
\arguments{
\item{Dmat}{
matrix appearing in the quadratic function to be minimized.
}
\item{dvec}{
vector appearing in the quadratic function to be minimized.
}
\item{Amat}{
matrix defining the constraints under which we want to minimize the
quadratic function.
}
\item{bvec}{
vector holding the values of \eqn{b_0} (defaults to zero).
}
\item{meq}{
the first \code{meq} constraints are treated as equality
constraints, all further as inequality constraints (defaults to 0).
}
\item{factorized}{
logical flag: if \code{TRUE}, then we are passing
\eqn{R^{-1}}{R^(-1)} (where \eqn{D = R^T R}) instead of the matrix
\eqn{D} in the argument \code{Dmat}.
}
}
\value{
a list with the following components:
\item{solution}{
vector containing the solution of the quadratic programming problem.
}
\item{value}{
scalar, the value of the quadratic function at the solution
}
\item{unconstrained.solution}{
vector containing the unconstrained minimizer of the quadratic function.
}
\item{iterations}{
vector of length 2, the first component contains the number of
iterations the algorithm needed, the second indicates how often
constraints became inactive after becoming active first.
vector with the indices of the active constraints at the solution.
}}
\references{
Goldfarb, D. and Idnani, A. (1982).
Dual and Primal-Dual Methods for Solving Strictly Convex
Quadratic Programs. In
Numerical Analysis
J.P. Hennart, ed. Springer-Verlag, Berlin. pp. 226-239.
Goldfarb, D. and Idnani, A. (1983).
A numerically stable dual method for solving strictly convex quadratic
programs.
Mathematical Programming
\bold{27}, 1-33.
}
\seealso{
\code{\link{solve.QP.compact}}
}
\examples{
#
# Assume we want to minimize: -(0 5 0) \%*\% b + 1/2 b^T b
# under the constraints: A^T b >= b0
# with b0 = (-8,2,0)^T
# and (-4 2 0)
# A = (-3 1 -2)
# ( 0 0 1)
# we can use solve.QP as follows:
#
Dmat <- matrix(0,3,3)
diag(Dmat) <- 1
dvec <- c(0,5,0)
Amat <- matrix(c(-4,-3,0,2,1,0,0,-2,1),3,3)
bvec <- c(-8,2,0)
solve.QP(Dmat,dvec,Amat,bvec=bvec)
}
\keyword{optimize}
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