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January  2013, 9(1): 191-204. doi: 10.3934/jimo.2013.9.191

## Threshold value of the penalty parameter in the minimization of $L_1$-penalized conditional value-at-risk

 1 Centre for Industrial and Applied Mathematics, University of South Australia, Mawson Lakes, SA 5095, Australia 2 CSIRO Mathematics, Informatics and Statistics, North Ryde, NSW, Australia

Received  November 2011 Revised  June 2012 Published  December 2012

A problem of minimization of $L_1$-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.
Citation: Vladimir Gaitsgory, Tanya Tarnopolskaya. Threshold value of the penalty parameter in the minimization of $L_1$-penalized conditional value-at-risk. Journal of Industrial & Management Optimization, 2013, 9 (1) : 191-204. doi: 10.3934/jimo.2013.9.191
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