quadprog (version 1.4-1)

solve.QP: Solve a Quadratic Programming Problem

Description

This routine implements the dual method of Goldfarb and Idnani (1982, 1983) for solving quadratic programming problems.

Usage

solve.QP        (Dmat, dvec, Amat,       bvec, meq=0, factorized=FALSE)
solve.QP.compact(Dmat, dvec, Amat, Aind, bvec, meq=0, factorized=FALSE)

Arguments

Dmat

matrix appearing in the quadratic function to be minimized.

dvec

vector appearing in the quadratic function to be minimized.

Amat

For solve.QP(): matrix defining the constraints under which we want to minimize the quadratic function.

For solve.QP.compact(): matrix containing the non-zero elements of the matrix A that defines the constraints. I

Aind

matrix of integers. The first element of each column gives the number of non-zero elements in the corresponding column of the matrix A. The following entries in each column contain the indexes of the rows in which these non-zero elements are

bvec

vector holding the values of b0 (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.

item

• unconstrained.solution
• iterations

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 27, 1-33.

See Also

Matrix Inversion with solve.

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)
bvec       <- c(-8,2,0)

Amat0       <- matrix(c(-4,-3,0, 2,1,0, 0,-2,1),3,3)
solve.QP(Dmat,dvec, Amat0, bvec=bvec)

# Now with solve.QP.compact :
#
Aind <- rbind(c(2,2,2),
              c(1,1,2),
              c(2,2,3))
Amat <- rbind(c(-4,2,-2),
              c(-3,1, 1))
solve.QP.compact(Dmat,dvec,Amat,Aind,bvec=bvec)