CVaR norm and applications in optimization

Abstract

This paper introduces the family of CVaR norms in ${\mathbb {R}}^{n}$, based on the CVaR concept. The CVaR norm is defined in two variations: scaled and non-scaled. The well-known $L_{1}$ and $L_{\infty }$ norms are limiting cases of the new family of norms. The D-norm, used in robust optimization, is equivalent to the non-scaled CVaR norm. We present two relatively simple definitions of the CVaR norm: (i) as the average or the sum of some percentage of largest absolute values of components of vector; (ii) as an optimal solution of a CVaR minimization problem suggested by Rockafellar and Uryasev. CVaR norms are piece-wise linear functions on ${\mathbb {R}}^{n}$ and can be used in various applications where the Euclidean norm is typically used. To illustrate, in the computational experiments we consider the problem of projecting a point onto a polyhedral set. The CVaR norm allows formulating this problem as a convex or linear program for any level of conservativeness.