How is glmnet dealing with high number of unpenalized parameters?

Question

I'm trying to fit a poisson model with glmnet where I know that a lot of the parameters have to be included in the model and then unpenalized. I'm setting their penalty.factor to 0. Unfortunately, glmnet has issue with convergence as the number of unpenalized parameters is high. Here is an example from the glmnet manual where the algorithm doesn't converge if the number of unpenalized parameters is high.

    library(glmnet)
    N=500; p=300
    nzc=5
    x=matrix(rnorm(N*p),N,p)
    beta=rnorm(nzc)
    f = x[,seq(nzc)]%*%beta
    mu=exp(f)
    y=rpois(N,mu)


    pena=rep(0,ncol(x))
    pena[1:10]=1 # penalized only the first ten parameters

    fit=glmnet(x,y,family="poisson",penalty.factor=pena,alpha=0.9)
    # fit is not converging 

    pena=rep(0,ncol(x))
    pena[1:250]=1 # penalized the first 250 parameters

    fit=glmnet(x,y,family="poisson",penalty.factor=pena,alpha=0.9)
    # fit is converging 

I know that the model doesn't make sense for this example but at least I can reproduce the same error as in my data. It looks like lambda_max is set to infinity. Does it mean that the algorithm can't find a minimum lambda_max where all the penalized parameters are zero? How can I fit a model where a lot of the parameters are unpenalized?

Thanks a lot