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We propose a subgradient-based method for finding the maximum feasiblesubsystem 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 optimizationproblems to approximate it. The objective of each approximation problem is thesum of the composition of a nonnegative nondecreasing continuouslydifferentiable concave function with the squared distance function to a closedset. Although this objective function is nonsmooth in general, a subgradientcan be obtained in terms of the projections onto the closed sets. Based on thisobservation, we adapt a subgradient projection method to solve theseapproximation problems. Unlike classical subgradient methods, the convergence(clustering to stationary points) of our subgradient method is guaranteed witha {\em nondiminishing stepsize} under mild assumptions. This allows us tofurther study the sequential convergence of the subgradient method undersuitable Kurdyka-{\L}ojasiewicz assumptions. Finally, we illustrate ouralgorithm numerically for solving the MFS$_C$ problems on a collection ofhalfspaces and a collection of unions of halfspaces, respectively, with respectto the set of $s$-sparse vectors.

A subgradient-based approach for finding the maximum feasible subsystem with respect to a set