  We develop a constructive approach for $\ell_0$-penalized estimation in the
sparse accelerated failure time (AFT) model with high-dimensional covariates.
Our proposed method is based on Stute's weighted least squares criterion
combined with $\ell_0$-penalization. This method is a computational algorithm
that generates a sequence of solutions iteratively, based on active sets
derived from primal and dual information and root finding according to the KKT
conditions. We refer to the proposed method as AFT-SDAR (for support detection
and root finding). An important aspect of our theoretical results is that we
directly concern the sequence of solutions generated based on the AFT-SDAR
algorithm. We prove that the estimation errors of the solution sequence decay
exponentially to the optimal error bound with high probability, as long as the
covariate matrix satisfies a mild regularity condition which is necessary and
sufficient for model identification even in the setting of high-dimensional
linear regression. We also proposed an adaptive version of AFT-SDAR, or
AFT-ASDAR, which determines the support size of the estimated coefficient in a
data-driven fashion. We conduct simulation studies to demonstrate the superior
performance of the proposed method over the lasso and MCP in terms of accuracy
and speed. We also apply the proposed method to a real data set to illustrate
its application.
