A SUBGRADIENT-BASED APPROACH FOR FINDING THE MAXIMUM FEASIBLE SUBSYSTEM WITH RESPECT TO A SET

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 (MFSC). In this method, we reformulate the MFSC problem as an l(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 non-decreasing continuously differentiable concave function with the squared distance function to a closed set. Although this objective function is non-smooth 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 nondiminishing stepsize under mild assumptions. This allows us to further study the sequential convergence of the subgradient method under suitable Kurdyka-Lojasiewicz assumptions. Finally, we illustrate our algorithm numerically for solving the MFSC problems on a collection of halfspaces and a collection of unions of halfspaces, respectively, with respect to the set of s-sparse vectors.