solve.QP.compact: 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 \(\min(-d^T b + 1/2 b^T D b)\) with the constraints \(A^T b >= b_0\).

Usage

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

matrix containing the non-zero elements of the matrix \(A\) that defines the constraints. If \(m_i\) denotes the number of non-zero elements in the \(i\)-th column of \(A\) then the first \(m_i\) entries of the \(i\)-th column of Amat hold these non-zero elements. (If \(maxmi\) denotes the maximum of all \(m_i\), then each column of Amat may have arbitrary elements from row \(m_i+1\) 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 \(b_0\) (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.

Examples

Run this code

# NOT RUN {
##
## 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)
# }