Package 'LowRankQP'

Solve Low Rank Quadratic Programming Problems

Description

This routine implements a primal-dual interior point method solving quadratic programming problems of the form

min	$d^T alpha + 1/2 alpha^T H alpha$
such that	$A alpha = b$
	$0 <= alpha <= u$

with dual

min	$1/2 alpha^T H alpha + beta^T b + xi^T u$
such that	$H alpha + c + A^T beta - zeta + xi = 0$
	$xi, zeta >= 0$

where $H=V$ if $V$ is square and $H=VV^T$ otherwise.

Usage

LowRankQP(Vmat,dvec,Amat,bvec,uvec,method="PFCF",verbose=FALSE,niter=200,epsterm=1.0E-8)

Arguments

Vmat	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 $b$ (defaults to zero).
uvec	vector holding the values of $u$.
method	Method used for inverting H+D where D is full rank diagonal. If $V$ is square: 'LU': Use LU factorization. (More stable) 'CHOL': Use Cholesky factorization. (Faster) If $V$ is not square: 'SMW': Use Sherman-Morrison-Woodbury (Faster) 'PFCF': Use Product Form Cholesky Factorization (More stable)
verbose	Display iterations and termination statistics.
niter	Number of iteration to perform.
epsterm	Termination tolerance. See equation (12) of Ormerod et al (2008).

Value

a list with the following components:

alpha	vector containing the solution of the quadratic programming problem.
beta	vector containing the solution of the dual of quadratic programming problem.
xi	vector containing the solution of the dual quadratic programming problem.
zeta	vector containing the solution of the dual quadratic programming problem.

References

Ormerod, J.T., Wand, M.P. and Koch, I. (2008). Penalised spline support vector classifiers: computational issues, Computational Statistics, 23, 623-641.

Boyd, S. and Vandenberghe, L. (2004). Convex Optimization. Cambridge University Press.

Ferris, M. C. and Munson, T. S. (2003). Interior point methods for massive support vector machines. SIAM Journal on Optimization, 13, 783-804.

Fine, S. and Scheinberg, K. (2001). Efficient SVM training using low-rank kernel representations. Journal of Machine Learning Research, 2, 243-264.

B. Sch\"olkopf and A. J. Smola. (2002). Learning with Kernels. The MIT Press, Cambridge, Massachusetts.

Examples

library(LowRankQP)

# Assume we want to minimize: (0 -5 0 0 0 0) %*% alpha + 1/2 alpha[1:3]^T alpha[1:3]
# under the constraints:      A^T alpha = b
# with b = (-8,  2,  0 )^T
# and      (-4   2   0 ) 
#      A = (-3   1  -2 )
#          ( 0   0   1 )
#          (-1   0   0 )
#          ( 0  -1   0 )
#          ( 0   0  -1 )
#  alpha >= 0
#
# (Same example as used in quadprog)
#
# we can use LowRankQP as follows:

Vmat          <- matrix(0,6,6)
diag(Vmat)    <- c(1, 1,1,0,0,0)
dvec          <- c(0,-5,0,0,0,0)
Amat          <- matrix(c(-4,-3,0,-1,0,0,2,1,0,0,-1,0,0,-2,1,0,0,-1),6,3)
bvec          <- c(-8,2,0)
uvec          <- c(100,100,100,100,100,100)
LowRankQP(Vmat,dvec,t(Amat),bvec,uvec,method="CHOL")

# Now solve the same problem except use low-rank V

Vmat          <- matrix(c(1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0),6,3)
dvec          <- c(0,-5,0,0,0,0)
Amat          <- matrix(c(-4,-3,0,-1,0,0,2,1,0,0,-1,0,0,-2,1,0,0,-1),6,3)
bvec          <- c(-8,2,0)
uvec          <- c(100,100,100,100,100,100)
LowRankQP(Vmat,dvec,t(Amat),bvec,uvec,method="SMW")

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