  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.
