  We propose a subgradient-based method for finding the maximum feasible
subsystem in a collection of closed sets with respect to a given closed set $C$
(MFS$_C$). In this method, we reformulate the MFS$_C$ problem as an $\ell_0$
optimization problem and construct a sequence of continuous optimization
problems to approximate it. The objective of each approximation problem is the
sum of the composition of a nonnegative nondecreasing continuously
differentiable concave function with the squared distance function to a closed
set. Although this objective function is nonsmooth in general, a subgradient
can be obtained in terms of the projections onto the closed sets. Based on this
observation, we adapt a subgradient projection method to solve these
approximation problems. Unlike classical subgradient methods, the convergence
(clustering to stationary points) of our subgradient method is guaranteed with
a {\em nondiminishing stepsize} under mild assumptions. This allows us to
further study the sequential convergence of the subgradient method under
suitable Kurdyka-{\L}ojasiewicz assumptions. Finally, we illustrate our
algorithm numerically for solving the MFS$_C$ problems on a collection of
halfspaces and a collection of unions of halfspaces, respectively, with respect
to the set of $s$-sparse vectors.
