标题： 具有基数惩罚的非光滑凸-凹鞍点问题 显示英文标题

标题： Nonsmooth convex-concave saddle point problems with cardinality penalties

摘要： 本文中，我们关注一类带有基数惩罚的凸约束非光滑凸-凹鞍点问题。 尽管此类非光滑非凸-非凹和不连续的极小极大问题可能没有鞍点，但我们证明它们具有局部鞍点和全局极小极大点，一些局部鞍点具有下界性质。 我们基于下界性质定义了一类强局部鞍点，以保证变量选择的稳定性。 此外，我们提出一个框架，基于卷积构造不连续极小极大问题的连续松弛形式，使得它们与原问题具有相同的鞍点。 我们还建立了连续松弛问题与原问题在局部鞍点、全局极小极大点、局部极小极大点和驻点之间的关系。 最后，我们通过分布鲁棒稀疏凸回归、稀疏鲁棒债券组合构建和稀疏凸-凹逻辑回归鞍点问题来说明我们的结果。 显示英文摘要

摘要： In this paper, we focus on a class of convexly constrained nonsmooth convex-concave saddle point problems with cardinality penalties. Although such nonsmooth nonconvex-nonconcave and discontinuous min-max problems may not have a saddle point, we show that they have a local saddle point and a global minimax point, and some local saddle points have the lower bound properties. We define a class of strong local saddle points based on the lower bound properties for stability of variable selection. Moreover, we give a framework to construct continuous relaxations of the discontinuous min-max problems based on the convolution, such that they have the same saddle points with the original problem. We also establish the relations between the continuous relaxation problems and the original problems regarding local saddle points, global minimax points, local minimax points and stationary points. Finally, we illustrate our results with distributionally robust sparse convex regression, sparse robust bond portfolio construction and sparse convex-concave logistic regression saddle point problems.