English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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 thecomposite optimization problem with the truncated nuclear norm (Gao and Sun,2010, and Zhang et al., 2012), which are known as tight relaxations ofnonconvex optimization problems that have either cardinality or rankconstraints, respectively. First, we extend the trimmed lasso for broaderapplications and conduct a unified analysis of the property of the generalizedtrimmed lasso. Next, the equivalence between local optimality andd-stationarity of the generalized trimmed lasso is shown under a suitableassumption. More generally, the equivalence is shown for problems that minimizea 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 thegeneralized trimmed lasso and the problem with the truncated nuclear normpenalty under mild assumptions. Our exact penalty results are not only new, butalso intuitive, so that we can see what properties of the problems play a rolefor establishing the exact penalization. Lastly, it is shown that existingalgorithms that yield a d-stationary point always converge to a solution thatsatisfies the cardinality or rank constraint.

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