We study multistage distributionally robust mixed-integer programs under endogenous uncertainty, where the probability distribution of stage-wise uncertainty depends on the decisions made in previous stages. We first consider two ambiguity sets defined by decision-dependent bounds on the first and second moments of uncertain parameters and by mean and covariance matrix that exactly match decision-dependent empirical ones, respectively. For both sets, we show that the subproblem in each stage can be recast as a mixed-integer linear program (MILP). Moreover, we extend the general moment-based ambiguity set in (Delage and Ye, 2010) to the multistage decision-dependent setting, and derive mixed-integer semidefinite programming (MISDP) reformulations of stage-wise subproblems. We develop methods for attaining lower and upper bounds of the optimal objective value of the multistage MISDPs, and approximate them using a series of MILPs. We deploy the Stochastic Dual Dynamic integer Programming (SDDiP) method for solving the problem under the three ambiguity sets with risk-neutral or risk-averse objective functions, and conduct numerical studies on multistage facility-location instances having diverse sizes under different parameter and uncertainty settings. Our results show that the SDDiP quickly finds optimal solutions for moderate-sized instances under the first two ambiguity sets, and also finds good approximate bounds for the multistage MISDPs derived under the third ambiguity set. We also demonstrate the efficacy of incorporating decision-dependent distributional ambiguity in multistage decision-making processes.
Distributionally robust optimization involves various probability measures in its problem formulation. They can be bundled to constitute a risk functional. For this equivalence, risk functionals constitute a fundamental building block in distributionally robust stochastic programming. Multistage programming requires conditional versions of risk functionals to re-assess future risk after partial realizations and after preceding decisions. This paper discusses a construction of the conditional counterpart of a risk functional by passing its genuine characteristics to its conditional counterparts…
We study multistage distributionally robust optimization (DRO) to hedge against ambiguity in quantifying the underlying uncertainty of a problem. Recognizing that not all the realizations and scenario paths might have an "effect" on the optimal value, we investigate the question of how to define and identify critical scenarios for nested multistage DRO problems. Our analysis extends the work of Rahimian, Bayraksan, and Homem-de-Mello [Math. Program. 173(1--2): 393--430, 2019], which was in the context of a static/two-stage setting, to the multistage setting. To this end, we define the notions …
In this paper, we focus on a data-driven risk-averse multistage stochastic programming (RMSP) model considering distributional robustness. We optimize the RMSP over the worst-case distribution within an ambiguity set of probability distributions constructed directly from historical data samples. The proposed RMSP is intractable due to the multistage nested minimax structure in its objective function, so we reformulate it into a deterministic equivalent that contains a series of convex combination of expectation and conditional value at risk (CVaR), which can be solved by a customized stochasti…
This paper investigates a Multistage Distributionally Robust Optimization (MDRO) approach to water allocation under climate uncertainty. The MDRO is formed by creating sets of conditional distributions (called conditional ambiguity sets) on a finite scenario tree. The distributions in the conditional ambiguity sets remain close to a nominal conditional distribution according a $ϕ$-divergence (e.g., Kullback-Liebler divergence, Hellinger distance, Burg entropy, etc.). The paper discusses a decomposition algorithm to solve the resulting MDRO and applies the modeling and solution techniques to al…
This paper studies distributionally robust optimization (DRO) when the ambiguity set is given by moments for the distributions. The objective and constraints are given by polynomials in decision variables. We reformulate the DRO with equivalent moment conic constraints. Under some general assumptions, we prove the DRO is equivalent to a linear optimization problem with moment and psd polynomial cones. A Moment-SOS relaxation method is proposed to solve it. Its asymptotic and finite convergence are shown under certain assumptions. Numerical examples are presented to show how to solve DRO proble…
We study decision dependent distributionally robust optimization models, where the ambiguity sets of probability distributions can depend on the decision variables. These models arise in situations with endogenous uncertainty. The developed framework includes two-stage decision dependent distributionally robust stochastic programming as a special case. Decision dependent generalizations of five types of ambiguity sets are considered. These sets are based on bounds on moments, Wasserstein metric, $ϕ$-divergence and Kolmogorov-Smirnov test. For the finite support case, we use linear, conic or La…
In this paper, we study a bivariate distributionally robust optimization problem with mean-covariance ambiguity set and half-space support. Under a conventional type of objective function widely adopted in inventory management, option pricing, and portfolio selection, we obtain closed-form tight bounds of the inner problem in six different cases. Through a primal-dual approach, we identify the optimal distributions in each case. As an application in inventory control, we first derive the optimal order quantity and the corresponding worst-case distribution, extending the existing results in the…
Recently, there has been a growing interest in developing inventory control policies which are robust to model misspecification. One approach is to posit that nature selects a worst-case distribution for any stochastic primitives from some pre-specified family. Several communities have observed that a subtle phenomena known as time inconsistency can arise in this framework. In particular, it becomes possible that a policy which is optimal at time zero may not be optimal for the associated optimization problem in which the decision-maker recomputes her policy at each point in time, which has im…
In distributionally robust optimization the probability distribution of the uncertain problem parameters is itself uncertain, and a fictitious adversary, e.g., nature, chooses the worst distribution from within a known ambiguity set. A common shortcoming of most existing distributionally robust optimization models is that their ambiguity sets contain pathological discrete distribution that give nature too much freedom to inflict damage. We thus introduce a new class of ambiguity sets that contain only distributions with sum-of-squares polynomial density functions of known degrees. We show that…
In this paper, we present a sequential sampling-based algorithm for the two-stage distributionally robust linear programming (2-DRLP) models. The 2-DRLP models are defined over a general class of ambiguity sets with discrete or continuous probability distributions. The algorithm is a distributionally robust version of the well-known stochastic decomposition algorithm of Higle and Sen (Math. of OR 16(3), 650-669, 1991) for a two-stage stochastic linear program. We refer to the algorithm as the distributionally robust stochastic decomposition (DRSD) method. The key features of the algorithm incl…
We consider a distributionally robust Partially Observable Markov Decision Process (DR-POMDP), where the distribution of the transition-observation probabilities is unknown at the beginning of each decision period, but their realizations can be inferred using side information at the end of each period after an action being taken. We build an ambiguity set of the joint distribution using bounded moments via conic constraints and seek an optimal policy to maximize the worst-case (minimum) reward for any distribution in the set. We show that the value function of DR-POMDP is piecewise linear conv…
This paper is about a class of distributionally robust model predictive controllers (MPC) for nonlinear stochastic processes that evaluate risk and control performance measures by propagating ambiguity sets in the space of state probability measures. A framework for formulating such ambiguity tube MPC controllers is presented, which is based on modern measure-theoretic methods from the field of optimal transport theory. Moreover, a supermartingale based analysis technique is proposed, leading to stochastic stability results for a large class of distributionally robust controllers for linear an…
This work presents a new Distributionally Robust Optimization approach, using $p$-Wasserstein metrics, to analyze a stochastic program in a general context. The ambiguity set in this approach depends on the decision variable and is represented as a ball where both the center and the radius depend on the decision variable. We show that, under Lipschitz's assumptions for the objective function, our approach can be reformulated as a finite-dimensional optimization problem, which is sometimes convex. In addition, we numerically compare our proposed approach with the standard formulation of distrib…
Given $X \subset R^n$, $\varepsilon \in (0,1)$, a parametrized family of probability distributions $(μ\_{a})\_{a\in A}$ on $Ω\subset R^p$, we consider the feasible set $X^*\_\varepsilon\subset X$ associated with the {\em distributionally robust} chance-constraint \[X^*\_\varepsilon\,=\,\{x \in X :\:{\rm Prob}\_μ[f(x,ω)\,>\,0]> 1-\varepsilon,\,\forallμ\in M\_a\},\]where $M\_a$ is the set of all possibles mixtures of distributions $μ\_a$, $a\in A$.For instance and typically, the family$M\_a$ is the set of all mixtures ofGaussian distributions on $R$ with mean and standard deviation $a=(a,σ)$ in …
Chance constrained optimal power flow (OPF) has been recognized as a promising framework to manage the risk from variable renewable energy (VRE). In presence of VRE uncertainties, this paper discusses a distributionally robust chance constrained approximate AC-OPF. The power flow model employed in the proposed OPF formulation combines an exact AC power flow model at the nominal operation point and an approximate linear power flow model to reflect the system response under uncertainties. The ambiguity set employed in the distributionally robust formulation is the Wasserstein ball centered at th…
We consider exact deterministic mixed-integer programming (MIP) reformulations of distributionally robust chance-constrained programs (DR-CCP) with random right-hand sides over Wasserstein ambiguity sets. The existing MIP formulations are known to have weak continuous relaxation bounds, and, consequently, for hard instances with small radius, or with large problem sizes, the branch-and-bound based solution processes suffer from large optimality gaps even after hours of computation time. This significantly hinders the practical application of the DR-CCP paradigm. Motivated by these challenges, …
Aggregation of heating, ventilation, and air conditioning (HVAC) loads can provide reserves to absorb volatile renewable energy, especially solar photo-voltaic (PV) generation. In this paper, we decide HVAC control schedules under uncertain PV generation, using a distributionally robust chance-constrained (DRCC) building load control model under two typical ambiguity sets: the moment-based and Wasserstein ambiguity sets. We derive mixed-integer linear programming (MILP) reformulations for DRCC problems under both sets. Especially, for the Wasserstein ambiguity set, we utilize the right-hand si…
In this paper, we consider a multistage expected utility maximization problem where the decision maker's utility function at each stage depends on historical data and the information on the true utility function is incomplete. To mitigate adverse impact arising from ambiguity of the true utility, we propose a maximin robust model where the optimal policy is based on the worst-case sequence of utility functions from an ambiguity set constructed with partially available information about the decision maker's preferences. We then show that the multistage maximin problem is time consistent when th…
Distributionally robust chance-constrained programs (DR-CCP) over Wasserstein ambiguity sets exhibit attractive out-of-sample performance and admit big-$M$-based mixed-integer programming (MIP) reformulations with conic constraints. However, the resulting formulations often suffer from scalability issues as sample size increases. To address this shortcoming, we derive stronger formulations that scale well with respect to the sample size. Our focus is on ambiguity sets under the so-called left-hand side (LHS) uncertainty, where the uncertain parameters affect the coefficients of the decision va…
We study elective surgery planning in flexible operating rooms (ORs) where emergency patients are accommodated in the existing elective surgery schedule. Specifically, elective surgeries can be scheduled weeks or months in advance. In contrast, an emergency surgery arrives randomly and must be performed on the day of arrival. Probability distributions of the actual durations of elective and emergency surgeries are unknown, and only a possibly small set of historical realizations may be available. To address distributional uncertainty, we first construct an ambiguity set that encompasses all po…
Adaptive robust optimization problems are usually solved approximately by restricting the adaptive decisions to simple parametric decision rules. However, the corresponding approximation error can be substantial. In this paper we show that two-stage robust and distributionally robust linear programs can often be reformulated exactly as conic programs that scale polynomially with the problem dimensions. Specifically, when the ambiguity set constitutes a 2-Wasserstein ball centered at a discrete distribution, then the distributionally robust linear program is equivalent to a copositive program (…
We consider data-driven approaches that integrate a machine learning prediction model within distributionally robust optimization (DRO) given limited joint observations of uncertain parameters and covariates. Our framework is flexible in the sense that it can accommodate a variety of regression setups and DRO ambiguity sets. We investigate asymptotic and finite sample properties of solutions obtained using Wasserstein, sample robust optimization, and phi-divergence-based ambiguity sets within our DRO formulations, and explore cross-validation approaches for sizing these ambiguity sets. Through…
Chance constrained optimal power flow (CC-OPF) formulations have been proposed to minimize operational costs while controlling the risk arising from uncertainties like renewable generation and load consumption. To solve CC-OPF, we often need access to the (true) joint probability distribution of all uncertainties, which is rarely known in practice. A solution based on a biased estimate of the distribution can result in poor reliability. To overcome this challenge, recent work has explored distributionally robust chance constraints, in which the chance constraints are satisfied over a family of…
To manage renewable generation and load consumption uncertainty, chance-constrained optimal power flow (OPF) formulations and various solution methodologies have been proposed. However, conventional solution approaches often rely on accurate estimates of uncertainty distributions, which may not exist. When the distributions are not known but can be limited to a set of plausible distributions, termed an ambiguity set, distributionally robust (DR) optimization can be used to ensure that chance constraints hold for all distributions in that set. However, DR OPF yields conservative solutions if th…
We propose a distributionally robust return-risk model for Markov decision processes (MDPs) under risk and reward ambiguity. The proposed model optimizes the weighted average of mean and percentile performances, and it covers the distributionally robust MDPs and the distributionally robust chance-constrained MDPs (both under reward ambiguity) as special cases. By considering that the unknown reward distribution lies in a Wasserstein ambiguity set, we derive the tractable reformulation for our model. In particular, we show that that the return-risk model can also account for risk from uncertain…
We study stochastic optimization problems with chance and risk constraints, where in the latter, risk is quantified in terms of the conditional value-at-risk (CVaR). We consider the distributionally robust versions of these problems, where the constraints are required to hold for a family of distributions constructed from the observed realizations of the uncertainty via the Wasserstein distance. Our main results establish that if the samples are drawn independently from an underlying distribution and the problems satisfy suitable technical assumptions, then the optimal value and optimizers of …
This work is devoted to the development of a distributionally robust active fault diagnosis approach for a class of nonlinear systems, which takes into account any ambiguity in distribution information of the uncertain model parameters. More specifically, a new approach is presented using the total variation distance metric as an information constraint, and as a measure for the separation of multiple models based on the similarity of their output probability density functions. A practical aspect of the proposed approach is that different levels of ambiguity may be assigned to the models pertai…
Topology design is a critical task for the reliability, economic operation, and resilience of distribution systems. This paper proposes a distributionally robust optimization (DRO) model for designing the topology of a new distribution system facing random contingencies (e.g., imposed by natural disasters). The proposed DRO model optimally configures the network topology and integrates distributed generation to effectively meet the loads. Moreover, we take into account the uncertainty of contingency. Using the moment information of distribution line failures, we construct an ambiguity set of t…
This paper presents a distributionally robust stochastic model predictive control (SMPC) approach for linear discrete-time systems subject to unbounded and correlated additive disturbances. We consider hard input constraints and state chance constraints, which are approximated as distributionally robust (DR) Conditional Value-at-Risk (CVaR) constraints over a Wasserstein ambiguity set. The computational complexity is reduced by resorting to a tube-based MPC scheme with indirect feedback, such that the error scenarios can be sampled offline. Recursive feasibility is guaranteed by softening the …
