solve.QP.compact: Solve a Quadratic Programming Problem

In quadprog: Functions to Solve Quadratic Programming Problems

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.

See Also

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)

Example output

$solution
[1] 0.4761905 1.0476190 2.0952381

$value
[1] -2.380952

$unconstrained.solution
[1] 0 5 0

$iterations
[1] 3 0

$Lagrangian
[1] 0.0000000 0.2380952 2.0952381

$iact
[1] 3 2

quadprog documentation built on Nov. 20, 2019, 9:06 a.m.