SIAM Journal on Optimization, Volume 30, Issue 2, Page 1274-1299, January 2020.
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 nondiminishing stepsize under mild assumptions. This allows us to further study the sequential convergence of the subgradient method under suitable Kurdyka--Łojasiewicz 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.

中文翻译：

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一种基于梯度的方法来找到关于集合的最大可行子系统

SIAM优化杂志，第30卷，第2期，第1274-1299页，2020年1月。
我们提出一种基于次梯度的方法，用于针对给定的封闭集$ C $（MFS $ _C $）在封闭集的集合中找到最大可行子系统。在这种方法中，我们将MFS $ _C $问题重新表述为$ \ ell_0 $优化问题，并构造一系列连续优化问题以对其进行近似。每个逼近问题的目标是非负非递减连续可微凹函数与距离函数平方平方成一个闭集的总和。尽管该目标函数通常是不平滑的，但可以根据闭合集上的投影获得次梯度。基于此观察，我们采用次梯度投影方法来解决这些近似问题。与经典的次梯度方法不同，在温和的假设下，我们的次梯度方法的收敛性（聚集到静止点）可以通过不减小的步长来保证。这使我们可以在适当的Kurdyka-Łojasiewicz假设下进一步研究次梯度方法的顺序收敛性。最后，我们针对$ s $-稀疏向量的集合，分别用数字方式说明用于求解半空间集合和半空间并集集合上的MFS $ _C $问题的算法。