Constructing Uncertainty Sets for Robust Linear Optimization

Abstract

In this paper, we propose a methodology for constructing uncertainty sets within the framework of robust optimization for linear optimization problems with uncertain parameters. Our approach relies on decision maker risk preferences. Specifically, we utilize the theory of coherent risk measures initiated by Artzner et al. (1999) [Artzner, P., F. Delbaen, J. Eber, D. Heath. 1999. Coherent measures of risk. Math. Finance 9 203-228.], and show that such risk measures, in conjunction with the support of the uncertain parameters, are equivalent to explicit uncertainty sets for robust optimization. We explore the structure of these sets in detail. In particular, we study a class of coherent risk measures, called distortion risk measures, which give rise to polyhedral uncertainty sets of a special structure that is tractable in the context of robust optimization. In the case of discrete distributions with rational probabilities, which is useful in practical settings when we are sampling from data, we show that the class of all distortion risk measures (and their corresponding polyhedral sets) are generated by a finite number of conditional value-at-risk (CVaR) measures. A subclass of the distortion risk measures corresponds to polyhedral uncertainty sets symmetric through the sample mean. We show that this subclass is also finitely generated and can be used to find inner approximations to arbitrary, polyhedral uncertainty sets.

Tight Approximations of Dynamic Risk Measures

“…Since, in practice, it may be far easier to elicit or estimate a single static risk measure µ I , characterizing and computing α ‹ µ C ,µ I and α ‹ µ I ,µ C constitutes the first step towards constructing the time-consistent risk measure µ C that is "closest" to a given µ I . We note that a similar concept of inner and outer approximations by means of distortion risk measures appears in [Bertsimas and Brown, 2009]. However, the goal and analysis there are quite different, since the question is to approximate a static risk measure by means of another static distortion risk measure.…”

Online Modified Greedy algorithm for storage control under uncertainty

“…Another possible approach is to estimate the supports using the forecasts of δ t and p t , which in turn are based on historical observation of the processes. Techniques that are used to determine the uncertainty sets for robust optimization can be used here; interested readers are referred to [27] for more details. As in general a smaller Dg − Dg leads to better performance guarantees, it is beneficial to obtain a tight estimate for the supports of the stochastic parameters.…”

From CVaR to Uncertainty Set: Implications in Joint Chance-Constrained Optimization

“…Recent works of Bertsimas and Brown [8] and Natarajan et al [24] have uncovered the relation between financial risk measures and uncertainty sets in robust optimization. The CVaR constraint,…”