  Gradient sampling (GS) has proved to be an effective methodology for the
minimization of objective functions that may be nonconvex and/or nonsmooth. The
most computationally expensive component of a contemporary GS method is the
need to solve a convex quadratic subproblem in each iteration. In this paper, a
strategy is proposed that allows the use of inexact solutions of these
subproblems, which, as proved in the paper, can be incorporated without the
loss of theoretical convergence guarantees. Numerical experiments show that by
exploiting inexact subproblem solutions, one can consistently reduce the
computational effort required by a GS method. Additionally, a strategy is
proposed for aggregating gradient information after a subproblem is solved
(potentially inexactly), as has been exploited in bundle methods for nonsmooth
optimization. It is proved that the aggregation scheme can be introduced
without the loss of theoretical convergence guarantees. Numerical experiments
show that incorporating this gradient aggregation approach substantially
reduces the computational effort required by a GS method.
