Exact Penalization at D-Stationary Points of Cardinality- or Rank-Constrained Problem

TL;DR

This paper investigates the properties of d-stationary points in nonconvex optimization problems with cardinality or rank constraints, extending existing models and establishing conditions for exact penalization and local optimality.

Contribution

It extends the generalized trimmed lasso framework, proves equivalence between local optimality and d-stationarity, and introduces new results on exact penalty at d-stationary points for these problems.

Findings

Extended the trimmed lasso for broader applications.

Proved equivalence between local optimality and d-stationarity.

Established new results on exact penalization at d-stationary points.

Abstract

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.