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Problem definition: Vehicle-to-grid increases the low utilization rate of privately owned electric vehicles by making their batteries available to electricity grids. We formulate a robust optimization problem that maximizes a vehicle owner's expected profit from selling primary frequency regulation to the grid and guarantees that market commitments are met at all times for all frequency deviation trajectories in a functional uncertainty set that encodes applicable legislation. Faithfully modeling the energy conversion losses during battery charging and discharging renders this optimization problem nonconvex. Methodology/results: By exploiting a total unimodularity property of the uncertainty set and an exact linear decision rule reformulation, we prove that this nonconvex robust optimization problem with functional uncertainties is equivalent to a tractable linear program. Through extensive numerical experiments using real-world data, we quantify the economic value of vehicle-to-grid and elucidate the financial incentives of vehicle owners, aggregators, equipment manufacturers, and regulators. Managerial implications: We find that the prevailing penalties for nondelivery of promised regulation power are too low to incentivize vehicle owners to honor the delivery guarantees given to grid operators. Funding: This work was supported by the Institut Vedecom. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.0154. [ABSTRACT FROM AUTHOR]

In the era of modern business analytics, data-driven optimization has emerged as a popular modeling paradigm to transform data into decisions. By constructing an ambiguity set of the potential data-generating distributions and subsequently hedging against all member distributions within this ambiguity set, data-driven optimization effectively combats the ambiguity with which real-life data sets are plagued. Chen et al. (2022) study data-driven, chance-constrained programs in which a decision has to be feasible with high probability under every distribution within a Wasserstein ball centered at the empirical distribution. The authors show that the problem admits an exact deterministic reformulation as a mixed-integer conic program and demonstrate (in numerical experiments) that the reformulation compares favorably to several state-of-the-art data-driven optimization schemes. 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 ∞ -norm, the cone is the nonnegative orthant, and the chance-constrained program can be reformulated as a mixed-integer linear program. Our reformulation compares favorably to several state-of-the-art data-driven optimization schemes in our numerical experiments. Funding: The authors gratefully acknowledge financial support from the Hong Kong Research Grants Council [Early Career Scheme CityU 21502820], the Swiss National Science Foundation [Grant BSCGI0_157733] as well as the Engineering and Physical Sciences Research Council [Grant EP/N020030/1]. [ABSTRACT FROM AUTHOR]

We study a robust monopoly pricing problem with a minimax regret objective, where a seller endeavors to sell multiple goods to a single buyer, only knowing that the buyer's values for the goods range over a rectangular uncertainty set. We interpret this pricing problem as a zero-sum game between the seller, who chooses a selling mechanism, and a fictitious adversary or 'nature', who chooses the buyer's values from within the uncertainty set. Using duality techniques rooted in robust optimization, we prove that this game admits a Nash equilibrium in mixed strategies that can be computed in closed form. The Nash strategy of the seller is a randomized posted price mechanism under which the goods are sold separately, while the Nash strategy of nature is a distribution on the uncertainty set under which the buyer's values are comonotonic. We further show that the restriction of the pricing problem to deterministic mechanisms is solved by a deteministic posted price mechanism under which the goods are sold separately. [ABSTRACT FROM AUTHOR]

Wasserstein balls, which contain all probability measures within a pre-specified Wasserstein distance to a reference measure, have recently enjoyed wide popularity in the distributionally robust optimization and machine learning communities to formulate and solve data-driven optimization problems with rigorous statistical guarantees. In this technical note we prove that the Wasserstein ball is weakly compact under mild conditions, and we offer necessary and sufficient conditions for the existence of optimal solutions. We also characterize the sparsity of solutions if the Wasserstein ball is centred at a discrete reference measure. In comparison with the existing literature, which has proved similar results under different conditions, our proofs are self-contained and shorter, yet mathematically rigorous, and our necessary and sufficient conditions for the existence of optimal solutions are easily verifiable in practice. [ABSTRACT FROM AUTHOR]

We study stochastic programs where the decision maker cannot observe the distribution of the exogenous uncertainties but has access to a finite set of independent samples from this distribution. In this setting, the goal is to find a procedure that transforms the data to an estimate of the expected cost function under the unknown data-generating distribution, that is, a predictor, and an optimizer of the estimated cost function that serves as a near-optimal candidate decision, that is, a prescriptor. As functions of the data, predictors and prescriptors constitute statistical estimators. We propose a meta-optimization problem to find the least conservative predictors and prescriptors subject to constraints on their out-of-sample disappointment. The out-of-sample disappointment quantifies the probability that the actual expected cost of the candidate decision under the unknown true distribution exceeds its predicted cost. Leveraging tools from large deviations theory, we prove that this meta-optimization problem admits a unique solution: The best predictor-prescriptor-pair is obtained by solving a distributionally robust optimization problem over all distributions within a given relative entropy distance from the empirical distribution of the data. This paper was accepted by Chung Piaw Teo, optimization. [ABSTRACT FROM AUTHOR]

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 distributions 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 (SOS) polynomial density functions of known degrees. We show that these ambiguity sets are highly expressive as they conveniently accommodate distributional information about higher-order moments, conditional probabilities, conditional moments or marginal distributions. Exploiting the theoretical properties of a measure-based hierarchy for polynomial optimization due to Lasserre (SIAM J Optim 21(3):864–885, 2011), we prove that certain worst-case expectation constraints are polynomial-time solvable under these new ambiguity sets. We also show how SOS densities can be used to approximately solve the general problem of moments. We showcase the applicability of the proposed approach in the context of a stylized portfolio optimization problem and a risk aggregation problem of an insurance company. [ABSTRACT FROM AUTHOR]

We study a mechanism design problem in which an indivisible good is auctioned to multiple bidders for each of whom it has a private value that is unknown to the seller and the other bidders. The agents perceive the ensemble of all bidder values as a random vector governed by an ambiguous probability distribution, which belongs to a commonly known ambiguity set. The seller aims to design a revenue-maximizing mechanism that is not only immunized against the ambiguity of the bidder values, but also against the uncertainty about the bidders' attitude toward ambiguity. We argue that the seller achieves this goal by maximizing the worst-case expected revenue across all value distributions in the ambiguity set and by positing that the bidders have Knightian preferences. For ambiguity sets containing all distributions supported on a hypercube, we show that the Vickrey auction is the unique mechanism that is optimal, efficient, and Pareto robustly optimal. If the bidders' values are additionally known to be independent, then the revenue of the (unknown) optimal mechanism does not exceed that of a second-price auction with only one additional bidder. For ambiguity sets under which the bidders' values are dependent and characterized through moment bounds, on the other hand, we provide a new class of randomized mechanisms, the highest-bidder lotteries, whose revenues cannot be matched by any second-price auction with a constant number of additional bidders. Moreover, we show that the optimal highest-bidder lottery is a 2-approximation of the (unknown) optimal mechanism, whereas the best second-price auction fails to provide any constant-factor approximation guarantee. This paper was accepted by Yinyu Ye, optimization. [ABSTRACT FROM AUTHOR]

Stochastic programming and distributionally robust optimization seek deterministic decisions that optimize a risk measure, possibly in view of the most adverse distribution in an ambiguity set. We investigate under which circumstances such deterministic decisions are strictly outperformed by random decisions, which depend on a randomization device producing uniformly distributed samples that are independent of all uncertain factors affecting the decision problem. We find that, in the absence of distributional ambiguity, deterministic decisions are optimal if both the risk measure and the feasible region are convex or alternatively, if the risk measure is mixture quasiconcave. We show that some risk measures, such as mean (semi-)deviation and mean (semi-)moment measures, fail to be mixture quasiconcave and can, therefore, give rise to problems in which the decision maker benefits from randomization. Under distributional ambiguity, however, we show that, for any ambiguity-averse risk measure satisfying a mild continuity property, we can construct a decision problem in which a randomized decision strictly outperforms all deterministic decisions. [ABSTRACT FROM AUTHOR]

We consider stochastic programs where the distribution of the uncertain parameters is only observable through a finite training dataset. Using the Wasserstein metric, we construct a ball in the space of (multivariate and non-discrete) probability distributions centered at the uniform distribution on the training samples, and we seek decisions that perform best in view of the worst-case distribution within this Wasserstein ball. The state-of-the-art methods for solving the resulting distributionally robust optimization problems rely on global optimization techniques, which quickly become computationally excruciating. In this paper we demonstrate that, under mild assumptions, the distributionally robust optimization problems over Wasserstein balls can in fact be reformulated as finite convex programs—in many interesting cases even as tractable linear programs. Leveraging recent measure concentration results, we also show that their solutions enjoy powerful finite-sample performance guarantees. Our theoretical results are exemplified in mean-risk portfolio optimization as well as uncertainty quantification. [ABSTRACT FROM AUTHOR]

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, the distributionally robust linear program is equivalent to a copositive program (if the problem has complete recourse) or can be approximated arbitrarily closely by a sequence of copositive programs (if the problem has sufficiently expensive recourse). These results directly extend to the classical robust setting and motivate strong tractable approximations of two-stage problems based on semidefinite approximations of the copositive cone. We also demonstrate that the two-stage distributionally robust optimization problem is equivalent to a tractable linear program when the ambiguity set constitutes a 1-Wasserstein ball centered at a discrete distribution and there are no support constraints. [ABSTRACT FROM AUTHOR]

We study joint chance constraints where the distribution of the uncertain parameters is only known to belong to an ambiguity set characterized by the mean and support of the uncertainties and by an upper bound on their dispersion. This setting gives rise to pessimistic (optimistic) ambiguous chance constraints, which require the corresponding classical chance constraints to be satisfied for every (for at least one) distribution in the ambiguity set. We demonstrate that the pessimistic joint chance constraints are conic representable if (i) the constraint coefficients of the decisions are deterministic, (ii) the support set of the uncertain parameters is a cone, and (iii) the dispersion function is of first order, that is, it is positively homogeneous. We also show that pessimistic joint chance constrained programs become intractable as soon as any of the conditions (i), (ii) or (iii) is relaxed in the mildest possible way. We further prove that the optimistic joint chance constraints are conic representable if (i) holds, and that they become intractable if (i) is violated. We show in numerical experiments that our results allow us to solve large-scale project management and image reconstruction models to global optimality. [ABSTRACT FROM AUTHOR]

Although stochastic programming problems were always believed to be computationally challenging, this perception has only recently received a theoretical justification by the seminal work of Dyer and Stougie (Math Program A 106(3):423-432, 2006). Amongst others, that paper argues that linear two-stage stochastic programs with fixed recourse are #P-hard even if the random problem data is governed by independent uniform distributions. We show that Dyer and Stougie's proof is not correct, and we offer a correction which establishes the stronger result that even the approximate solution of such problems is #P-hard for a sufficiently high accuracy. We also provide new results which indicate that linear two-stage stochastic programs with random recourse seem even more challenging to solve. [ABSTRACT FROM AUTHOR]

Over the last two decades, robust optimization has emerged as a computationally attractive approach to formulate and solve single-stage decision problems affected by uncertainty. More recently, robust optimization has been successfully applied to multistage problems with continuous recourse. This paper takes a step toward extending the robust optimization methodology to problems with integer recourse, which have largely resisted solution so far. To this end, we approximate two-stage robust binary programs by their corresponding K-adaptability problems, in which the decision maker precommits to K second-stage policies, here -and-now, and implements the best of these policies once the uncertain parameters are observed. We study the approximation quality and the computational complexity of the K-adaptability problem, and we propose two mixed-integer linear programming reformulations that can be solved with off-the-shelf software. We demonstrate the effectiveness of our reformulations for stylized instances of supply chain design, route planning, and capital budgeting problems. [ABSTRACT FROM AUTHOR]

In a stochastic interdiction game a proliferator aims to minimize the expected duration of a nuclear weapons development project, and an interdictor endeavors to maximize the project duration by delaying some of the project tasks. We formulate static and dynamic versions of the interdictor's decision problem where the interdiction plan is either precommitted or adapts to new information revealed over time, respectively. The static model gives rise to a stochastic program, whereas the dynamic model is formalized as a multiple optimal stopping problem in continuous time and with decision-dependent information. Under a memoryless probabilistic model for the task durations, we prove that the static model reduces to a mixed-integer linear program, whereas the dynamic model reduces to a finite Markov decision process in discrete time that can be solved via efficient value iteration. We then generalize the dynamic model to account for uncertainty in the outcomes of the interdiction actions. We also discuss a crashing game where the proliferator can use limited resources to expedite tasks so as to counterbalance the interdictor's efforts. The resulting problem can be formulated as a robust Markov decision process. [ABSTRACT FROM AUTHOR]

Markov decision processes (MDPs) are powerful tools for decision making in uncertain dynamic environments. However, the solutions of MDPs are of limited practical use because of their sensitivity to distributional model parameters, which are typically unknown and have to be estimated by the decision maker. To counter the detrimental effects of estimation errors, we consider robust MDPs that offer probabilistic guarantees in view of the unknown parameters. To this end, we assume that an observation history of the MDP is available. Based on this history, we derive a confidence region that contains the unknown parameters with a prespecified probability 1 - β. Afterward, we determine a policy that attains the highest worst-case performance over this confidence region. By construction, this policy achieves or exceeds its worst-case performance with a confidence of at least 1 - β. Our method involves the solution of tractable conic programs of moderate size. [ABSTRACT FROM AUTHOR]

The purpose of software partitioning is to assign code segments of a given computer program to a range of execution locations such as general-purpose processors or specialist hardware components. These execution locations differ in speed, communication characteristics, and size. In particular, hardware components offering high speed tend to accommodate only few code segments. The goal of software partitioning is to find an assignment of code segments to execution locations that minimizes the overall program run time and respects the size constraints. In this paper we demonstrate that an additional speedup is obtained if we allow code segments to be instantiated on more than one location. We further show that the program run time not only depends on the execution frequency of the code segments but also on their execution order if there are multiply instantiated code segments. Unlike frequency information, however, sequence information is not available at the time when the software partition is selected. This motivates us to formulate the software-partitioning problem as a robust optimization problem with decision-dependent uncertainty. We transform this problem to a mixed- integer linear program of moderate size and report on promising numerical results. [ABSTRACT FROM AUTHOR]

Abstract: We propose a tractable approximation scheme for convex (not necessarily linear) multi-stage robust optimization problems. We approximate the adaptive decisions by finite linear combinations of prescribed basis functions and demonstrate how one can optimize over these decision rules at low computational cost through constraint randomization. We obtain a-priori probabilistic guarantees on the feasibility properties of the optimal decision rule by applying existing constraint sampling techniques to the semi-infinite problem arising from the decision rule approximation. We demonstrate that for a suitable choice of basis functions, the approximation converges as the size of the basis and the number of sampled constraints tend to infinity. The approach yields an algorithm parameterized in the basis size, the probability of constraint violation and the confidence that this probability will not be exceeded. These three parameters serve to tune the trade-off between optimality and feasibility of the decision rules and the computational cost of the algorithm. We assess the convergence and scalability properties of our approach in the context of two inventory management problems. [Copyright &y& Elsevier]