Abstract This paper presents a new safety specification method that is robust against errors in the probability distribution of disturbances. Our proposed distributionally robust safe policy maximizes the probability of a system remaining in a desired set for all times, subject to the worst possible disturbance distribution in an ambiguity set. We propose a dynamic game formulation of constructing such policies and identify conditions under which a non-randomized Markov policy is optimal. Based on this existence result, we develop a practical design approach to safety-oriented stochastic controllers with limited information about disturbance distributions. However, an associated Bellman equation involves infinite-dimensional minimax optimization problems since the disturbance distribution may have a continuous density. To alleviate computational issues, we propose a duality-based reformulation method that converts the infinite-dimensional minimax problem into a semi-infinite program that can be solved using existing convergent algorithms. We prove that there is no duality gap, and that this approach thus preserves optimality. The results of numerical tests confirm that the proposed method is robust against distributional errors in disturbances, while a standard stochastic safety verification tool is not.
Distributionally robust optimization is a paradigm for decision making under uncertainty where the uncertain problem data are governed by a probability distribution that is itself subject to uncertainty. The distribution is then assumed to belong to an ambiguity set comprising all distributions that are compatible with the decision maker's prior information. In this paper, we propose a unifying framework for modeling and solving distributionally robust optimization problems. We introduce standardized ambiguity sets that contain all distributions with prescribed conic representable confidence sets and with mean values residing on an affine manifold. These ambiguity sets are highly expressive and encompass many ambiguity sets from the recent literature as special cases. They also allow us to characterize distributional families in terms of several classical and or robust statistical indicators that have not yet been studied in the context of robust optimization. We determine conditions under which distributionally robust optimization problems based on our standardized ambiguity sets are computationally tractable. We also provide tractable conservative approximations for problems that violate these conditions. Stochastic programming can effectively describe many decision-making problems in uncertain environments. Unfortunately, such programs are often computationally demanding to solve. In addition, their solution can be misleading when there is ambiguity in the choice of a distribution for the random parameters. In this paper, we propose a model that describes uncertainty in both the distribution form (discrete, Gaussian, exponential, etc.) and moments (mean and covariance matrix). We demonstrate that for a wide range of cost functions the associated distributionally robust (or min-max) stochastic program can be solved efficiently. Furthermore, by deriving a new confidence region for the mean and the covariance matrix of a random vector, we provide probabilistic arguments for using our model in problems that rely heavily on historical data. These arguments are confirmed in a practical example of portfolio selection, where our framework leads to better-performing policies on the “true” distribution underlying the daily returns of financial assets.  Classical formulations of the portfolio optimization problem, such as mean-variance or Value-at-Risk (VaR) approaches, can result in a portfolio extremely sensitive to errors in the data, such as mean and covariance matrix of the returns. In this paper we propose a way to alleviate this problem in a tractable manner. We assume that the distribution of returns is partially known, in the sense that onlybounds on the mean and covariance matrix are available. We define the worst-case Value-at-Risk as the largest VaR attainable, given the partial information on the returns' distribution. We consider the problem of computing and optimizing the worst-case VaR, and we show that these problems can be cast as semidefinite programs. We extend our approach to various other partial information on the distribution, including uncertainty in factor models, support constraints, and relative entropy information. In this paper, we discuss linear programs in which the data that specify the constraints are subject to random uncertainty. A usual approach in this setting is to enforce the constraints up to a given level of probability. We show that, for a wide class of probability distributions (namely, radial distributions) on the data, the probability constraints can be converted explicitly into convex second-order cone constraints; hence, the probability-constrained linear program can be solved exactly with great efficiency. Next, we analyze the situation where the probability distribution of the data is not completely specified, but is only known to belong to a given class of distributions. In this case, we provide explicit convex conditions that guarantee the satisfaction of the probability constraints for any possible distribution belonging to the given class. We consider Markov decision processes where the values of the parameters are uncertain. This uncertainty is described by a sequence of nested sets (that is, each set contains the previous one), each of which corresponds to a probabilistic guarantee for a different confidence level. Consequently, a set of admissible probability distributions of the unknown parameters is specified. This formulation models the case where the decision maker is aware of and wants to exploit some (yet imprecise) a priori information of the distribution of parameters, and it arises naturally in practice where methods for estimating the confidence region of parameters abound. We propose a decision criterion based on distributional robustness: the optimal strategy maximizes the expected total reward under the most adversarial admissible probability distributions. We show that finding the optimal distributionally robust strategy can be reduced to the standard robust MDP where parameters are known to belong to a single uncertainty set; hence, it can be computed in polynomial time under mild technical conditions.  We investigate the control of constrained stochastic linear systems when faced with limited information regarding the disturbance process, i.e., when only the first two moments of the disturbance distribution are known. We consider two types of distributionally robust constraints. In the first case, we require that the constraints hold with a given probability for all disturbance distributions sharing the known moments. These constraints are commonly referred to as distributionally robust chance constraints. In the second case, we impose conditional value-at-risk (CVaR) constraints to bound the expected constraint violation for all disturbance distributions consistent with the given moment information. Such constraints are referred to as distributionally robust CVaR constraints with second-order moment specifications. We propose a method for designing linear controllers for systems with such constraints that is both computationally tractable and practically meaningful for both finite and infinite horizon problems. We prove in the infinite horizon case that our design procedure produces the globally optimal linear output feedback controller for distributionally robust CVaR and chance constrained problems. The proposed methods are illustrated for a wind blade control design case study for which distributionally robust constraints constitute sensible design objectives. This technical note studies Markov decision processes under parameter uncertainty. We adapt the distributionally robust optimization framework, assume that the uncertain parameters are random variables following an unknown distribution, and seek the strategy which maximizes the expected performance under the most adversarial distribution. In particular, we generalize a previous study [1] which concentrates on distribution sets with very special structure to a considerably more generic class of distribution sets, and show that the optimal strategy can be obtained efficiently under mild technical conditions. This significantly extends the applicability of distributionally robust MDPs by incorporating probabilistic information of uncertainty in a more flexible way.