Chance constraints yield non-convex feasible regions in general. In particular, when the uncertain parameters are modeled by a Wasserstein ball, arXiv:1806.07418 and arXiv:1809.00210 showed that the distributionally robust (pessimistic) chance constraint admits a mixed-integer conic representation. This paper identifies sufficient conditions that lead to convex feasible regions of chance constraints with Wasserstein ambiguity. First, when uncertainty arises from the right-hand side of a pessimistic joint chance constraint, we show that the ensuing feasible region is convex if the Wasserstein ball is centered around a log-concave distribution (or, more generally, an $α$-concave distribution with $α\geq -1$). In addition, we propose a block coordinate ascent algorithm and prove its convergence to global optimum, as well as the rate of convergence. Second, when uncertainty arises from the left-hand side of a pessimistic two-sided chance constraint, we show the convexity if the Wasserstein ball is centered around an elliptical and star-unimodal distribution. In addition, we propose a family of second-order conic inner approximations, and we bound their approximation error and prove their asymptotic exactness. Furthermore, we extend the convexity results to optimistic chance constraints.
We provide an exact deterministic reformulation for data-driven chance constrained programs over Wasserstein balls. For individual chance constraints as well as joint chance constraints with right-hand side uncertainty, our reformulation amounts to a mixed-integer conic program. In the special case of a Wasserstein ball with the $1$-norm or the $\infty$-norm, the cone is the nonnegative orthant, and the chance constrained program can be reformulated as a mixed-integer linear program. Our reformulation compares favourably to several state-of-the-art data-driven optimization schemes in our numer…
Distributionally robust chance constrained programs minimize a deterministic cost function subject to the satisfaction of one or more safety conditions with high probability, given that the probability distribution of the uncertain problem parameters affecting the safety condition(s) is only known to belong to some ambiguity set. We study three popular approximation schemes for distributionally robust chance constrained programs over Wasserstein balls, where the ambiguity set contains all probability distributions within a certain Wasserstein distance to a reference distribution. The first app…
In this paper, we develop an exact reformulation and a deterministic approximation for distributionally robust joint chance-constrained programmings (DRCCPs) with a general class of convex uncertain constraints under data-driven Wasserstein ambiguity sets. It is known that robust chance constraints can be conservatively approximated by worst-case conditional value-at-risk (CVaR) constraints. It is shown that the proposed worst-case CVaR approximation model can be reformulated as an optimization problem involving biconvex constraints for joint DRCCP. We then derive a convex relaxation of this a…
We study a generalized distributionally robust chance-constrained set covering problem (DRC) with a Wasserstein ambiguity set, where both decisions and uncertainty are binary-valued. We establish the NP-hardness of DRC and recast it as a two-stage stochastic program, which facilitates decomposition algorithms. Furthermore, we derive two families of valid inequalities. The first family targets the hypograph of a "shifted" submodular function, which is associated with each scenario of the two-stage reformulation. We show that the valid inequalities give a complete description of the convex hull …
This paper studies a distributionally robust chance constrained program (DRCCP) with Wasserstein ambiguity set, where the uncertain constraints should be satisfied with a probability at least a given threshold for all the probability distributions of the uncertain parameters within a chosen Wasserstein distance from an empirical distribution. In this work, we investigate equivalent reformulations and approximations of such problems. We first show that a DRCCP can be reformulated as a conditional value-at-risk constrained optimization problem, and thus admits tight inner and outer approximation…
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…
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…
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 …
In this paper we discuss distributional robustness in the context of stochastic model predictive control (SMPC) for linear time-invariant systems. We derive a simple approximation of the MPC problem under an additive zero-mean i.i.d. noise with quadratic cost. Due to the lack of distributional information, chance constraints are enforced as distributionally robust (DR) chance constraints, which we opt to unify with the concept of probabilistic reachable sets (PRS). For Wasserstein ambiguity sets, we propose a simple convex optimization problem to compute the DR-PRS based on finitely many distu…
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 …
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…
As a natural approach to modeling system safety conditions, chance constraint (CC) seeks to satisfy a set of uncertain inequalities individually or jointly with high probability. Although a joint CC offers stronger reliability certificate, it is oftentimes much more challenging to compute than individual CCs. Motivated by the application of optimal power flow, we study a special joint CC, named two-sided CC. We model the uncertain parameters through a Wasserstein ball centered at a Gaussian distribution and derive a hierarchy of conservative approximations based on second-order conic constrain…
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…
Markov decision process (MDP) is a decision making framework where a decision maker is interested in maximizing the expected discounted value of a stream of rewards received at future stages at various states which are visited according to a controlled Markov chain. Many algorithms including linear programming methods are available in the literature to compute an optimal policy when the rewards and transition probabilities are deterministic. In this paper, we consider an MDP problem where the transition probabilities are known and the reward vector is a random vector whose distribution is part…
The key of the post-disaster humanitarian logistics (PD-HL) is to build a good facility location and capacity planning (FLCP) model for delivering relief supplies to affected areas in time. To fully exploit the historical PD data, this paper adopts the data-driven distributionally robust (DR) approach and proposes a novel multi-period FLCP model under the $\infty$-Wasserstein joint chance constraints (MFLCP-W). Specifically, we sequentially decide locations from a candidate set to build facilities with supply capacities, which are expanded if more economical, and use a finite number of histori…
This paper considers a security constrained dispatch problem involving generation and line contingencies in the presence of the renewable generation. The uncertainty due to renewables is modeled using joint chance-constraint and the mismatch caused by contingencies and renewables are handled using reserves. We consider a distributionally robust approach to solve the chance-constrained program. We assume that samples of the uncertainty are available. Using them, we construct a set of distributions, termed ambiguity set, containing all distributions that are close to the empirical distribution u…
The University of Sydney（悉尼大学）；University of Wisconsin–Madison（威斯康星大学麦迪逊分校）
We study a model for adversarial classification based on distributionally robust chance constraints. We show that under Wasserstein ambiguity, the model aims to minimize the conditional value-at-risk of the distance to misclassification, and we explore links to adversarial classification models proposed earlier and to maximum-margin classifiers. We also provide a reformulation of the distributionally robust model for linear classification, and show it is equivalent to minimizing a regularized ramp loss objective. Numerical experiments show that, despite the nonconvexity of this formulation, st…
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 (…
Increasing integration of renewable generation poses significant challenges to ensure robustness guarantees in real-time energy system decision-making. This work aims to develop a robust optimal transmission switching (OTS) framework that can effectively relieve grid congestion and mitigate renewable curtailment. We formulate a two-stage distributionally robust chance-constrained (DRCC) problem that assures limited constraint violations for any uncertainty distribution within an ambiguity set. Here, the second-stage recourse variables are represented as linear functions of uncertainty, yieldin…
This paper studies distributionally robust chance constrained programs (DRCCPs), where the uncertain constraints must be satisfied with at least a probability of a prespecified threshold for all probability distributions from the Wasserstein ambiguity set. As DRCCPs are often nonconvex and challenging to solve optimally, researchers have been developing various convex inner approximations. Recently, ALSO-X has been proven to outperform the conditional value-at-risk (CVaR) approximation of a regular chance constrained program when the deterministic set is convex. In this work, we relax this ass…
This paper is motivated by addressing open questions in distributionally robust chance-constrained programs (DRCCP) using the popular Wasserstein ambiguity sets. Specifically, the computational techniques for those programs typically place restrictive assumptions on the constraint functions and the size of the Wasserstein ambiguity sets is often set using costly cross-validation (CV) procedures or conservative measure concentration bounds. In contrast, we propose a practical DRCCP algorithm using kernel maximum mean discrepancy (MMD) ambiguity sets, which we term MMD-DRCCP, to treat general no…
City University of Hong Kong（香港城市大学）；Tencent（腾讯）；JD Digits（京东数科）
This paper provides a non-robust interpretation of the distributionally robust optimization (DRO) problem by relating the distributional uncertainties to the chance probabilities. Our analysis allows a decision-maker to interpret the size of the ambiguity set, which is often lack of business meaning, through the chance parameters constraining the objective function. We first show that, for general $ϕ$-divergences, a DRO problem is asymptotically equivalent to a class of mean-deviation problems. These mean-deviation problems are not subject to uncertain distributions, and the ambiguity radius i…
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；TU Delft（代尔夫特理工大学）
We study a distributionally robust mean square error estimation problem over a nonconvex Wasserstein ambiguity set containing only normal distributions. We show that the optimal estimator and the least favorable distribution form a Nash equilibrium. Despite the non-convex nature of the ambiguity set, we prove that the estimation problem is equivalent to a tractable convex program. We further devise a Frank-Wolfe algorithm for this convex program whose direction-searching subproblem can be solved in a quasi-closed form. Using these ingredients, we introduce a distributionally robust Kalman filt…
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 consider chance-constrained binary programs, where each row of the inequalities that involve uncertainty needs to be satisfied probabilistically. Only the information of the mean and covariance matrix is available, and we solve distributionally robust chance-constrained binary programs (DCBP). Using two different ambiguity sets, we equivalently reformulate the DCBPs as 0-1 second-order cone (SOC) programs. We further exploit the submodularity of 0-1 SOC constraints under special and general covariance matrices, and utilize the submodularity as well as lifting to derive extended polymatroid …
Markov decision processes (MDPs) are known to be sensitive to parameter specification. Distributionally robust MDPs alleviate this issue by allowing for \emph{ambiguity sets} which give a set of possible distributions over parameter sets. The goal is to find an optimal policy with respect to the worst-case parameter distribution. We propose a framework for solving Distributionally robust MDPs via first-order methods, and instantiate it for several types of Wasserstein ambiguity sets. By developing efficient proximal updates, our algorithms achieve a convergence rate of $O\left(NA^{2.5}S^{3.5}\…
Chance-constrained programming (CCP) is one of the most difficult classes of optimization problems that has attracted the attention of researchers since the 1950s. In this survey, we focus on cases when only a limited information on the distribution is available, such as a sample from the distribution, or the moments of the distribution. We first review recent developments in mixed-integer linear formulations of chance-constrained programs that arise from finite discrete distributions (or sample average approximation). We highlight successful reformulations and decomposition techniques that en…
School of Automation and Intelligent Sensing, Shanghai Jiao Tong University（自动化与智能感知学院，上海交通大学）；Institute of Medical Robotics, Shanghai Jiao Tong University（医学机器人研究所，上海交通大学）；Key Laboratory of System Control and Information Processing, Ministry of Education of China（教育部系统控制与信息处理重点实验室）；Shanghai Engineering Research Center of Intelligent Control and Management, Shanghai（智能控制与管理上海工程研究中心）
