This routine implements the dual method of Goldfarb and Idnani (1982,
1983) for solving quadratic programming problems of the form
min(-d^T b + 1/2 b^T D b) with the
constraints A^T b >= b_0.
matrix appearing in the quadratic function to be minimized.
dvec
vector appearing in the quadratic function to be minimized.
Amat
matrix defining the constraints under which we want to minimize the
quadratic function.
bvec
vector holding the values of b_0 (defaults to zero).
meq
the first meq constraints are treated as equality
constraints, all further as inequality constraints (defaults to 0).
factorized
logical flag: if TRUE, then we are passing
R^(-1) (where D = R^T R) instead of the matrix
D in the argument Dmat.
Value
a list with the following components:
solution
vector containing the solution of the quadratic programming problem.
value
scalar, the value of the quadratic function at the solution
unconstrained.solution
vector containing the unconstrained minimizer of the quadratic
function.
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.
Lagrangian
vector with the Lagragian at the solution.
iact
vector with the indices of the active constraints at the solution.
References
D. Goldfarb and A. Idnani (1982).
Dual and Primal-Dual Methods for Solving Strictly Convex
Quadratic Programs.
In J. P. Hennart (ed.), Numerical Analysis, Springer-Verlag,
Berlin, pages 226–239.
D. Goldfarb and A. Idnani (1983).
A numerically stable dual method for solving strictly convex quadratic
programs.
Mathematical Programming, 27, 1–33.
See Also
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)