Distributionally Robust Optimization: A Review

The concepts of risk-aversion, chance-constrained optimization, and robust optimization have developed significantly over the last decade. Statistical learning community has also witnessed a rapid theoretical and applied growth by relying on these concepts. A modeling framework, called distributionally robust optimization (DRO), has recently received significant attention in both the operations research and statistical learning communities. This paper surveys main concepts and contributions to DRO, and its relationships with robust optimization, risk-aversion, chance-constrained optimization, and function regularization

The Impact of a Target on Newsvendor Decisions

Goal achieving is a commonly observed phenomenon in practice, and it plays an important role in decision making. In this paper, we investigate the impact of a target on newsvendor decisions. We take into account the risk and model the effect of a target by maximizing the satisficing measure of a newsvendor’s profit with respect to that target. We study two satisficing measures: (i) conditional value at risk (CVaR) satisficing measure that evaluates the highest confidence level of CVaR achieving the target; (ii) entropic satisficing measure that assesses the smallest risk tolerance level under which the certainty equivalent for exponential utility function achieves the target. For both satisficing measures, we find that the optimal ordering quantity increases with the target level. We determine an optimal order quantity for a target-based newsvendor and characterize its properties with respect to, for example, product’s profit margin

Evaluating alternative estimators for optimal order quantities in the newsvendor model with skewed demand

This paper considers the classical Newsvendor model, also known as the Newsboy problem, with the demand to be fully observed and to follow in successive inventory cycles one of the Exponential, Rayleigh, and Log-Normal distributions. For each distribution, appropriate estimators for the optimal order quantity are considered, and their sampling distributions are derived. Then, through Monte-Carlo simulations, we evaluate the performance of corresponding exact and asymptotic confidence intervals for the true optimal order quantity. The case where normality for demand is erroneously assumed is also investigated. Asymptotic confidence intervals produce higher precision, but to attain equality between their actual and nominal confidence level, samples of at least a certain size should be available. This size depends upon the coefficients of variation, skewness and kurtosis. The paper concludes that having available data on the skewed demand for enough inventory cycles enables (i) to trace non-normality, and (ii) to use the right asymptotic confidence intervals in order the estimates for the optimal order quantity to be valid and precise.Inventory Control; Newsboy Problem; Skewed Demand; Exact and Asymptotic Confidence Intervals; Monte-Carlo Simulations

The newsvendor problem with convex risk

The newsvendor problem is a classical topic in Management Science and Operations Research. It deals with purchases and price strategies when a least one deadline is involved. In this paper we will assume that the decision is driven by an optimization problem involving both expected pro ts and risks. As a main novelty, risks will be given by a convex risk measure, including the usual utility functions. This approach will allow us to nd necessary and su¢ cient optimality conditions under very general frameworks, since we will not need any speci c assumption about the demand distribution.Research partially supported by Ministerio de Economía (grant ECO2012-39031-C02-01, Spain)

Risk trading and endogenous probabilities in investment equilibria

A risky design equilibrium problem is an equilibrium system that involves N designers who invest in risky assets, such as production plants, evaluate these using convex or coherent risk measures, and also trade financial securities in order to manage their risk. Our main finding is that in a complete risk market - when all uncertainties can be replicated by financial products - a risky design equilibrium problem collapses to what we call a risky design game, i.e., a stochastic Nash game in which the original design agents act as risk neutral and there emerges an additional system risk agent. The system risk agent simultaneously prices risk and determines the probability density used by the other agents for their risk neutral evaluations. This situation is stochastic-endogenous: the probability density used by agents to value uncertain investments is endogenous to the risky design equilibrium problem. This result is most striking when design agents use coherent risk measures in which case the intersection of their risk sets turns out to be a risk set for the system risk agent, thereby extending existing results for risk markets. We also investigate existence of equilibria in both the complete and incomplete cases