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For survival data with a large number of explanatory variables, lasso penalized Cox regression is a popular regularization strategy. However, a penalized Cox model may not always provide the best fit to data and can be difficult to estimate in high dimension because of its intrinsic nonlinearity. The semiparametric additive hazards model is a flexible alternative which is a natural survival analogue of the standard linear regression model. Building on this analogy, we develop a cyclic coordinate descent algorithm for fitting the lasso and elastic net penalized additive hazards model. The algorithm requires no nonlinear optimization steps and offers excellent performance and stability. An implementation is available in the R package ahaz. We demonstrate this implementation in a small timing study and in an application to real data.