Threshold value of the penalty parameter in the minimization of L1-penalized conditional value-at-risk

A problem of minimization of L₁-penalized conditional value-at-risk (CVaR) is considered. It is shown that there exists a non-negative threshold value of the penalty parameter such that the optimal value of the penalized problem is unbounded if the penalty parameter is less than the threshold value, and it is bounded if the penalty parameter is greater or equal than this value. It is established that the threshold value can be found via the solution of a linear programming problem, and, therefore, readily computable. Theoretical results are illustrated by numerical examples.