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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

Vladimir Gaitsgory 1, and  Tanya Tarnopolskaya 2,

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.
Keywords: linear programming., threshold value of the penalty parameter, $L_1$-penalization, Conditional value-at-risk (CVaR).
Mathematics Subject Classification: 91G10, 90C05, 90C3.
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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show all references

References:
[1]

in "Risk Measures for the 21st Century" (eds. G. Szego), London: Wiley, (2004), 339-363.  Google Scholar

[2]

Journal of Banking and Finance, 30 (2006), 583-605. doi: 10.1016/j.jbankfin.2005.04.012.  Google Scholar

[3]

in "IEEE Proceedings of the 2003 International Conference on Computational Intelligence for Financial Engineering (CIFE 2003)".  Google Scholar

[4]

Journal of Futures Markets, 30 (2010), 780-794. Google Scholar

[5]

Princeton: Princeton University Press, 1963.  Google Scholar

[6]

European Journal of Operational Research, 191 (2008), 888-911. doi: 10.1016/j.ejor.2007.02.052.  Google Scholar

[7]

Journal of Futures Markets, 26 (2006), 369-390. doi: 10.1002/fut.20195.  Google Scholar

[8]

European Journal of Operational Research, 203 (2010), 185-194. doi: 10.1016/j.ejor.2009.07.010.  Google Scholar

[9]

ALGO Research Quarterly, 1 (1998), 5-20. Google Scholar

[10]

Journal of Risk, 2 (2000), 21-41. Google Scholar

[11]

Journal of Banking and Finance, 26 (2002), 1443-1471. doi: 10.1016/S0378-4266(02)00271-6.  Google Scholar

[12]

Journal of Industrial and Management Optimization, 8 (2012), 343-362. doi: 10.3934/jimo.2012.8.343.  Google Scholar

[13]

in "18th World IMACS Congress and MODSIM09 International Congress on Modelling and Simulation" (eds. R. S. Anderssen et al), (2009), 1559-1565.  Google Scholar

[14]

Journal of Banking and Finance, 26 (2002), 1535-1561. doi: 10.1016/S0378-4266(02)00289-3.  Google Scholar

[15]

Stochastic Optimization: Algorithms and Applications, 54 (2001), 411-435.  Google Scholar

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