admm.enet: Elastic Net Regularization
In ADMM: Algorithms using Alternating Direction Method of Multipliers

Description Usage Arguments Value Iteration History Author(s) References See Also Examples

View source: R/admm.enet.R

Elastic Net regularization is a combination of \ell_2 stability and \ell_1 sparsity constraint simulatenously solving the following,

\textrm{min}_x ~ \frac{1}{2}\|Ax-b\|_2^2 + λ_1 \|x\|_1 + λ_2 \|x\|_2^2

with nonnegative constraints λ_1 and λ_2. Note that if both lambda values are 0, it reduces to least-squares solution.

admm.enet(
  A,
  b,
  lambda1 = 1,
  lambda2 = 1,
  rho = 1,
  abstol = 1e-04,
  reltol = 0.01,
  maxiter = 1000
)

`A`	an (m\times n) regressor matrix
`b`	a length-m response vector
`lambda1`	a regularization parameter for \ell_1 term
`lambda2`	a regularization parameter for \ell_2 term
`rho`	an augmented Lagrangian parameter
`abstol`	absolute tolerance stopping criterion
`reltol`	relative tolerance stopping criterion
`maxiter`	maximum number of iterations

a named list containing

x: a length-n solution vector
history: dataframe recording iteration numerics. See the section for more details.

When you run the algorithm, output returns not only the solution, but also the iteration history recording following fields over iterates,

objval: object (cost) function value
r_norm: norm of primal residual
s_norm: norm of dual residual
eps_pri: feasibility tolerance for primal feasibility condition
eps_dual: feasibility tolerance for dual feasibility condition

In accordance with the paper, iteration stops when both r_norm and s_norm values become smaller than eps_pri and eps_dual, respectively.

Xiaozhi Zhu

\insertRef

zou_regularization_2005aADMM

admm.lasso

## generate underdetermined design matrix
m = 50
n = 100
p = 0.1   # percentange of non-zero elements

x0 = matrix(Matrix::rsparsematrix(n,1,p))
A  = matrix(rnorm(m*n),nrow=m)
for (i in 1:ncol(A)){
  A[,i] = A[,i]/sqrt(sum(A[,i]*A[,i]))
}
b = A%*%x0 + sqrt(0.001)*matrix(rnorm(m))

## run example with both regularization values = 1
output = admm.enet(A, b, lambda1=1, lambda2=1)
niter  = length(output$history$s_norm)
history = output$history

## report convergence plot
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3))
plot(1:niter, history$objval, "b", main="cost function")
plot(1:niter, history$r_norm, "b", main="primal residual")
plot(1:niter, history$s_norm, "b", main="dual residual")
par(opar)