  A new spectral conjugate subgradient method is presented to solve nonsmooth
unconstrained optimization problems. The method combines the spectral conjugate
gradient method for smooth problems with the spectral subgradient method for
nonsmooth problems. We study the effect of two different choices of line
search, as well as three formulas for determining the conjugate directions. In
addition to numerical experiments with standard nonsmooth test problems, we
also apply the method to several image reconstruction problems in computed
tomography, using total variation regularization. Performance profiles are used
to compare the performance of the algorithm using different line search
strategies and conjugate directions to that of the original spectral
subgradient method. Our results show that the spectral conjugate subgradient
algorithm outperforms the original spectral subgradient method, and that the
use of the Polak-Ribiere formula for conjugate directions provides the best and
most robust performance.
