1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
|
\name{solve.QP.compact}
\alias{solve.QP.compact}
\title{
Solve a Quadratic Programming Problem
}
\usage{
solve.QP.compact(Dmat, dvec, Amat, Aind, 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 containing the non-zero elements of the matrix \eqn{A} that defines
the constraints. If \eqn{m_i} denotes the number of non-zero elements in the
\eqn{i}-th column of \eqn{A} then the first \eqn{m_i} entries of the
\eqn{i}-th column of \code{Amat} hold these non-zero elements.
(If \eqn{maxmi} denotes the maximum of all \eqn{m_i}, then each column of
\code{Amat} may have arbitrary elements from row \eqn{m_i+1} to row
\eqn{maxmi} in the \eqn{i}-th column.)
}
\item{Aind}{
matrix of integers. The first element of each column gives the number of
non-zero elements in the corresponding column of the matrix \eqn{A}. The
following entries in each column contain the indexes of the rows in
which these non-zero elements are.
}
\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.
}}
\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}.
}
\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}}
}
\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.compact as follows:
#
Dmat <- matrix(0,3,3)
diag(Dmat) <- 1
dvec <- c(0,5,0)
Aind <- rbind(c(2,2,2),c(1,1,2),c(2,2,3))
Amat <- rbind(c(-4,2,-2),c(-3,1,1))
bvec <- c(-8,2,0)
solve.QP.compact(Dmat,dvec,Amat,Aind,bvec=bvec)
}
\keyword{optimize}
% Converted by Sd2Rd version 0.2-a5.
|
