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In the R package glmnet, does it calculate the Naive form of ElasticNet or is the output rescaled with the term (1 + lambda)?

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No - the glmnet package utilizes the cyclical coordinate descent method to find an elasticnet solution for GLMs as described in Friedman, et al. 2010.

An earlier package written by Zou & Hastie, elasticnet, calculates elasticnet solutions (including the naive elasticnet) for least squares models using the LARS-EN algorithm detailed in their 2005 paper.

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