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

TL;DR

This paper introduces a subgradient-based algorithm for identifying the largest feasible subsystem within a collection of closed sets, reformulating the problem as an $ ext{l}_0$ optimization and ensuring convergence under mild conditions.

Contribution

It presents a novel subgradient projection method for the maximum feasible subsystem problem, with convergence guarantees and numerical validation on specific set collections.

Findings

Convergence to stationary points is guaranteed with nondiminishing stepsize.

The method effectively solves MFS$_C$ problems on halfspaces and unions of halfspaces.

Numerical experiments demonstrate the algorithm's practical applicability.

Abstract

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.