Optimization with Multivariate Conditional Value-at-Risk Constraints

Abstract

For many decision-making problems under uncertainty, it is crucial to develop risk-averse models and specify the decision makers' risk preferences based on multiple stochastic performance measures (or criteria). Incorporating such multivariate preference rules into optimization models is a fairly recent research area. Existing studies focus on extending univariate stochastic dominance rules to the multivariate case. However, enforcing multivariate stochastic dominance constraints can often be overly conservative in practice. As an alternative, we focus on the widely applied risk measure conditional value-at-risk (CVaR), introduce a multivariate CVaR relation, and develop a novel optimization model with multivariate CVaR constraints based on polyhedral scalarization. To solve such problems for finite probability spaces, we develop a cut generation algorithm, where each cut is obtained by solving a mixed-integer problem. We show that a multivariate CVaR constraint reduces to finitely many univariate CVaR constraints, which proves the finite convergence of our algorithm. We also show that our results can be naturally extended to a wider class of coherent risk measures. The proposed approach provides a flexible and computationally tractable way of modeling preferences in stochastic multicriteria decision making. We conduct a computational study for a budget allocation problem to illustrate the effect of enforcing multivariate CVaR constraints and demonstrate the computational performance of the proposed solution methods.

Robust multicriteria risk-averse stochastic programming models

“…Observe that in the multivariate CVaR-constrained problems studied in Noyan and Rudolf (2013) and Küçükyavuz and Noyan (2015), given a random benchmark vector Y, the cut generation problems are given by min c∈C CVaR(c G(z * )) − CVaR(c Y) (we will revisit this cut generation problem in Section 3). Due to the similar structure, we can use the formulations and enhancements given in Noyan and Rudolf (2013) and Küçükyavuz and Noyan (2015) to formulate the cut generation problem (CutGen − Robust) as a mixedinteger program.…”

“…In this context, risk measures are often referred to as acceptability functionals. Our presentation follows along the lines of Pflug and Römisch (2007) and Noyan and Rudolf (2013). Recall that for a univariate random variable X with (not necessarily distinct) realizations x 1 , .…”

“…For example, due to this infeasibility issue, Hu et al (2011) solve such an optimization problem with relaxed SSD constraints. As an alternative, Noyan and Rudolf (2013) propose to use a multivariate preference relation based on CVaR; their approach is motivated by the fact that the univariate SSD relation is equivalent to a continuum of CVaR inequalities (Dentcheva and Ruszczyński, 2006). The authors consider polyhedral scalarization sets and show that their CVaR-based methodology can be extended to optimization problems featuring benchmarking constraints based on a wider class of coherent risk measures.…”

“…In our study, we follow the line of research of Noyan and Rudolf (2013), which provides sufficient flexibility to obtain feasible problem formulations and capture a wide range of risk preferences, including risk-neutral and worst-case approaches.…”

“…For polyhedral scalarization sets, Homem-de-Mello and Mehrotra (2009) and Noyan and Rudolf (2013) show that enforcing the corresponding univariate risk constraint for a finite (exponential) subset of weight vectors is sufficient to model the multivariate SSD and CVaR relations, respectively. These finite representation results allow them to develop finitely convergent delayed cut generation algorithms, where each cut is obtained by solving a mixed-integer programming (MIP) problem.…”