  We present a powerful and easy-to-implement iterative algorithm for solving
large-scale optimization problems that involve $L_1$/total-variation (TV)
regularization. The method is based on combining the Alternating Directions
Method of Multipliers (ADMM) with a Conjugate Directions technique in a way
that allows reusing conjugate search directions constructed by the algorithm
across multiple iterations of the ADMM. The new method achieves fast
convergence by trading off multiple applications of the modeling operator for
the increased memory requirement of storing previous conjugate directions. We
illustrate the new method with a series of imaging and inversion applications.
