solve.QP {quadprog}R Documentation

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. If mi denotes the number of non-zero elements in the i-th column of A then the first mi entries of the i-th column of Amat hold these non-zero elements. (If maxmi denoted the maximum of all mi, then each column of Amat may have arbitrary elements from row mi+1 to row maxmi in the i-th column)
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.

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)