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

Vladimir Gaitsgory; Tanya Tarnopolskaya

doi:10.3934/jimo.2013.9.191

Article Contents

2013, Volume 9, Issue 1: 191-204. Doi: 10.3934/jimo.2013.9.191

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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
2.
CSIRO Mathematics, Informatics and Statistics, North Ryde, NSW

Received: October 31, 2011

Revised: May 31, 2012

Published: December 2012

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Abstract

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:

Conditional value-at-risk (CVaR),
$L_1$-penalization,
threshold value of the penalty parameter,
linear programming.

Mathematics Subject Classification: 91G10, 90C05, 90C31.

Citation:

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