  In this paper, we propose a novel variable selection approach in the
framework of high-dimensional linear models where the columns of the design
matrix are highly correlated. It consists in rewriting the initial
high-dimensional linear model to remove the correlation between the columns of
the design matrix and in applying a generalized Elastic Net criterion since it
can be seen as an extension of the generalized Lasso. The properties of our
approach called gEN (generalized Elastic Net) are investigated both from a
theoretical and a numerical point of view. More precisely, we provide a new
condition called GIC (Generalized Irrepresentable Condition) which generalizes
the EIC (Elastic Net Irrepresentable Condition) of Jia and Yu (2010) under
which we prove that our estimator can recover the positions of the null and non
null entries of the coefficients when the sample size tends to infinity. We
also assess the performance of our methodology using synthetic data and compare
it with alternative approaches. Our numerical experiments show that our
approach improves the variable selection performance in many cases.
