In this paper we propose a methodology for constructing decision rules for integer and continuous decision variables in multiperiod robust linear optimization problems. This type of problems finds application in, for example, inventory management, lot sizing, and manpower management. We show that by iteratively splitting the uncertainty set into subsets one can differentiate the later-period decisions based on the revealed uncertain parameters. At the same time, the problem's computational complexity stays at the same level as for the static robust problem. This holds also in the non-fixed recourse situation. In the fixed recourse situation our approach can be combined with linear decision rules for the continuous decision variables. We provide theoretical results how to split the uncertainty set by identifying sets of uncertain parameter scenarios to be divided for an improvement in the worst-case objective value. Based on this theory, we propose several splitting heuristics. Numerical examples entailing a capital budgeting and a lot sizing problem illustrate the advantages of the proposed approach.

In optimization problems appearing in fields such as economics, finance, or engineering, it is often important that a risk measure of a decision-dependent random variable stays below a prescribed level. At the same time, the underlying probability distribution determining the risk measure's value is typically known only up to a certain degree and the constraint should hold for a reasonably wide class of probability distributions. In addition to that, the constraint should be computationally tractable. In this paper we review and generalize results on the derivation of tractable counterparts of such constraints for discrete probability distributions. Using established techniques in robust optimization, we show that the derivation of a tractable robust counterpart can be split into two parts: one corresponding to the risk measure and the other to the uncertainty set. This holds for a wide range of risk measures and uncertainty sets for probability distributions defined using statistical goodness-of-fit tests or probability metrics. In this way, we provide a unified framework of reformulating this class of constraints, extending the number of solvable risk measure-uncertainty set combinations considerably, including also risk measures that are nonlinear in the probabilities. To provide a clear overview for the user, we give the computational tractability status for each of the uncertainty set-risk measure pairs of which some have been solved in the literature. Examples, including portfolio optimization and antenna array design, illustrate the proposed approach in a theoretical and numerical setting. block includes the following risk measures: negative mean return, Optimized Certainty Equivalent (with Conditional Value-at-Risk as a special case), Certainty Equivalent, Shortfall Risk, lower partial moments, mean absolute deviation from the median, standard deviation/variance less mean, Sharpe ratio, and the Entropic Value-at-Risk. The second building block encompasses uncertainty sets built using φ-divergences (with the Pearson (χ 2 ) and likelihood ratio (G) tests as special cases), Kolmogorov-Smirnov test, Wasserstein (Kantorovich) distance, Anderson-Darling, Cramer-von Mises, Watson, and Kuiper tests.Secondly, we address a common feature rendering many optimization constraints SOCP SOCP QP Certainty Equivalent Shortfall risk • CP [10],[11] • CP [10][11][38] • CP [10][11][34][54] CP CP • CP [10] CP CP CP Optimized Certainty Equivalent CP CP CP CP CP CP CP CP CP Value-at-Risk • [35] • [35] • [34][35][37] Conditional Value-at-Risk CP • SOCP [14][55] • CP [14][34] • LP [14][55] • LP [45][55][56] • CP [14] • SOCP [14][55] • SOCP [14][55] • LP [14][55] Entropic Value-at-Risk CP CP CP CP CP CP CP CP CP Mean absolute deviation from the mean • [17] Distortion risk measures • [45][56] Coherent risk measures Spectral risk measures

Abstract. In this paper we consider ambiguous stochastic constraints under partial information consisting of means and dispersion measures of the underlying random parameters. Whereas the past literature used the variance as the dispersion measure, here we use the mean absolute deviation from the mean (MAD). This makes it possible to use the 1972 result of Ben-Tal and Hochman (BH) in which tight upper and lower bounds on the expectation of a convex function of a random variable are given. First, we use these results to treat ambiguous expected feasibility constraints to obtain exact reformulations for both functions that are convex and concave in the components of the random variable. This approach requires, however, the independence of the random variables and, moreover, may lead to an exponential number of terms in the resulting robust counterparts. We then show how upper bounds can be constructed that alleviate the independence restriction, and require only a linear number of terms, by exploiting models in which random variables are linearly aggregated. Moreover, using the BH bounds we derive three new safe tractable approximations of chance constraints of increasing computational complexity and quality. In a numerical study, we demonstrate the efficiency of our methods in solving stochastic optimization problems under mean-MAD ambiguity.

In this paper we propose a methodology for constructing decision rules for integer and continuous decision variables in multiperiod robust linear optimization problems. This type of problems finds application in, for example, inventory management, lot sizing, and manpower management. We show that by iteratively splitting the uncertainty set into subsets one can differentiate the later-period decisions based on the revealed uncertain parameters. At the same time, the problem's computational complexity stays at the same level as for the static robust problem. This holds also in the non-fixed recourse situation. In the fixed recourse situation our approach can be combined with linear decision rules for the continuous decision variables. We provide theoretical results how to split the uncertainty set by identifying sets of uncertain parameter scenarios to be divided for an improvement in the worst-case objective value. Based on this theory, we propose several splitting heuristics. Numerical examples entailing a capital budgeting and a lot sizing problem illustrate the advantages of the proposed approach.