  In this paper, we propose a general class of algorithms for optimizing an
extensive variety of nonsmoothly penalized objective functions that satisfy
certain regularity conditions. The proposed framework utilizes the
majorization-minimization (MM) algorithm as its core optimization engine. The
resulting algorithms rely on iterated soft-thresholding, implemented
componentwise, allowing for fast, stable updating that avoids the need for any
high-dimensional matrix inversion. We establish a local convergence theory for
this class of algorithms under weaker assumptions than previously considered in
the statistical literature. We also demonstrate the exceptional effectiveness
of new acceleration methods, originally proposed for the EM algorithm, in this
class of problems. Simulation results and a microarray data example are
provided to demonstrate the algorithm's capabilities and versatility.
