  The ability to make optimal decisions under uncertainty remains important
across a variety of disciplines from portfolio management to power engineering.
This generally implies applying some safety margins on uncertain parameters
that may only be observable through a finite set of historical samples.
Nevertheless, the optimized decisions must be resilient to all probable
outcomes, while ideally providing some measure of severity of any potential
violations in the less probable outcomes.It is known that the conditional
value-at-risk (CVaR) can be used to quantify risk in an optimization task,
though may also impose overly conservative margins. Therefore, this paper
develops a means of co-controlling the value-at-risk (VaR) level associated
with the CVaR to guarantee resilience in probable cases while providing a
measure of the average violation in less probable cases. To further combat
uncertainty, the CVaR and VaR co-control is extended in a distributionally
robust manner using the Wasserstein metric to establish an ambiguity set
constructed from finite samples, which is guaranteed to contain the true
distribution with a certain confidence.
