论文标题
在D-Stationary的基数或等级约束问题的确切处罚
Exact Penalization at D-Stationary Points of Cardinality- or Rank-Constrained Problem
论文作者
论文摘要
This paper studies the properties of d-stationary points of the trimmed lasso (Luo et al., 2013, Huang et al., 2015, and Gotoh et al., 2018) and the composite optimization problem with the truncated nuclear norm (Gao and Sun, 2010, and Zhang et al., 2012), which are known as tight relaxations of nonconvex optimization problems that have either cardinality or rank constraints, respectively.首先，我们将修剪的拉索扩展以进行更广泛的应用，并对广义修剪的拉索的性质进行统一分析。接下来，在合适的假设下显示了广义修剪的套索的局部最优性与D平稳性之间的等效性。更一般而言，对于最小化由有限的许多凸函数定义的函数的问题显示了等效性。然后，我们提出了普遍修剪的套索的D平稳点的确切惩罚的新结果，以及在轻度假设下截断的核量规范罚款的问题。我们的确切惩罚结果不仅是新的，而且是直观的，因此我们可以看到问题的特性在确定确切的惩罚中起着作用。最后，还讨论了与算法有关的问题。
This paper studies the properties of d-stationary points of the trimmed lasso (Luo et al., 2013, Huang et al., 2015, and Gotoh et al., 2018) and the composite optimization problem with the truncated nuclear norm (Gao and Sun, 2010, and Zhang et al., 2012), which are known as tight relaxations of nonconvex optimization problems that have either cardinality or rank constraints, respectively. First, we extend the trimmed lasso for broader applications and for conducting a unified analysis of the property of the generalized trimmed lasso. Next, the equivalence between local optimality and d-stationarity of the generalized trimmed lasso is shown under a suitable assumption. More generally, the equivalence is shown for problems that minimize a function defined by the pointwise minimum of finitely many convex functions. Then, we present new results of the exact penalty at d-stationary points of the generalized trimmed lasso and the problem with the truncated nuclear norm penalty under mild assumptions. Our exact penalty results are not only new, but also intuitive, so that we can see what properties of the problems play a role for establishing the exact penalization. Lastly, matters relating to algorithms are also discussed.
