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 find 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.

Distributionally robust optimization with matrix moment constraints: Lagrange duality and cutting plane methods

“…Proof The thrust of the proof is to use CVaR and its sample average approximation to approximate sup ξ ∈ g(x, ξ) of problem (3.3) which is in line with the convergence analysis carried out in [1]. However, there are a couple of important differences: (a) the convergence here is for the randomization scheme (3.4) rather than the sample average approximation of the CVaR approximation of sup ξ ∈ g(x, ξ), (b) g is not necessarily a convex function.…”

“…• We propose a discretization scheme based on Monte Carlo sampling for approximating the semiinfinite constraints of the Lagrange dual of the inner maximization problem. The approach is in line with the randomization scheme considered by Campi and Calafiore [10] and Anderson, Xu and Zhang [1] for mathematical programs with robust convex constraints. Under some moderate conditions, we demonstrate convergence of the optimal values, the optimal solutions and the stationary points obtained from the approximate problems as sample size increases (Theorems 3.1 and 3.2).…”

“…where ρ(·) denotes the density function of the random variable ξ , (c) + = max(0, c) for c ∈ IR. In the literature, CVaR β (g(x, ξ)) is known as conditional value at risk at a specified confidence level β and v N β (x) is its sample average approximation, see [1,35]. It is well known that the maximum w.r.t.…”

“…First, condition (a) explicitly ensures g(x, y) the essential supremum of g(x, ξ) being bounded for each fixed x ∈ X . Condition (b) is considered by Anderson, Xu and Zhang [1]. Inequality (3.5) is guaranteed when g(x, ·) is Hölder continuous over .…”

“…Condition (c) is fulfilled when the density function of ξ is bounded away from zero around ξ 0 . A combination of (b) and (c) provide a sufficient condition for the so-call consistent tail behaviour for g(x, ξ), see [1] for a detailed discussion.…”