Varying confidence levels for CVaR risk measures and minimax limits

Abstract

Conditional value at risk (CVaR) has been widely studied as a
risk measure. In this paper we add to this work by focusing on the choice of
confidence level and its impact on optimization problems with CVaR appearing
in the objective and also the constraints. We start by considering a problem
in which CVaR is minimized and investigate the way in which it approximates
the minimax robust optimization problem as the confidence level is driven
to one. We make use of a consistent tail condition which ensures that the
CVaR of a random function will converge uniformly to its supremum as the
confidence level increases, and establish an error bound for the CVaR optimal
solution under second order growth conditions. The results are extended to
a minimization problem with a constraint on the CVaR value which in the
limit as the confidence level approaches one coincides with a problem having
semi-infinite constraints. We study the sample average approximation scheme
for the CVaR constraints and establish an exponential rate of convergence for
the sample averaged optimal solution. We propose a procedure to explore the
possibility of varying the confidence level to a lower value which can give an
advantage when there is a need to nd good solutions to CVaR-constrained
problems out of sample. Our numerical results demonstrate that using the
optimal solution to an adjusted problem with lower confidence level can lead
to better overall performance.

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Identifiers

ORCID for Huifu Xu: orcid.org/0000-0001-8307-2920

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