Worst-case Distributions Research Articles

Robustly Optimal Mechanisms for Selling Multiple Goods

Abstract We study robustly optimal mechanisms for selling multiple items. The seller maximizes revenue against a worst-case distribution of a buyer’s valuations within a set of distributions, called an “ambiguity” set. We identify the exact forms of robustly optimal selling mechanisms and the worst-case distributions when the ambiguity set satisfies various moment conditions on the values of subsets of goods. The analysis reveals general properties of the ambiguity set that justifies categorical bundling, which includes separate sales and pure bundling as special cases.

Kernel Robust Hypothesis Testing

The problem of robust hypothesis testing is studied, where under the null and the alternative hypotheses, the data-generating distributions are assumed to be in some uncertainty sets, and the goal is to design a test that performs well under the worst-case distributions over the uncertainty sets. In this paper, uncertainty sets are constructed in a data-driven manner using kernel method, i.e., they are centered around empirical distributions of training samples from the null and alternative hypotheses, respectively; and are constrained via the distance between kernel mean embeddings of distributions in the reproducing kernel Hilbert space, i.e., maximum mean discrepancy (MMD). The Bayesian setting and the Neyman-Pearson setting are investigated. For the Bayesian setting where the goal is to minimize the worst-case error probability, an optimal test is firstly obtained when the alphabet is finite. When the alphabet is infinite, a tractable approximation is proposed to quantify the worst-case average error probability, and a kernel smoothing method is further applied to design test that generalizes to unseen samples. A direct robust kernel test is also proposed and proved to be exponentially consistent. For the Neyman-Pearson setting, where the goal is to minimize the worst-case probability of miss detection subject to a constraint on the worst-case probability of false alarm, an efficient robust kernel test is proposed and is shown to be asymptotically optimal. Numerical results are provided to demonstrate the performance of the proposed robust tests.

Comprehensive Performance Study of Hashing Functions

Most literature on hashing functions speaks in terms of hashing functions being either ‘good’ or ‘bad’. In this paper, we demonstrate how a hashing function that gives good results for one key set, performs badly for another. We also demonstrate that, for a single key set, we can find hashing functions that hash the keys with varying performances ranging from perfect to worst distributions. We present a study on the effect of changing the prime number ‘$p$’ on the performance of a hashing function from $H_1$ Class of Universal Hashing Functions. This paper then explores a way to characterize hashing functions by studying their performance over all subsets of a chosen Universe. We compare the performance of some popular hashing functions based on the average search performance and the number of perfect and worst-case distributions over different key sets chosen from a Universe. The experimental results show that the division-remainder method provides the best distribution for most key sets of the Universe when compared to other hashing functions including functions from $H_1$ Class of Universal Hashing Functions.

Optimal low-carbon operation of regional integrated energy systems: A data-driven hybrid stochastic-distributionally robust optimization approach

Uncertainties arising from the increasing renewable energy source (RES) penetration induced by the imposition of carbon emission penalties introduce substantial challenges to the optimal operations of regional integrated energy system(RIES). The present work addresses this issue by developing a data-driven hybrid stochastic-distributionally robust optimization approach for RIESs, in which carbon emission costs and uncertainties in RES output are considered. The approach involves a tri-level optimization problem that is transformed into three single-level problems by the column-and-constraint generation algorithm, and the stochastic optimization problem is solved in the master problem with the worst-case probability distributions obtained by the subproblems. In addition, the stochastic programming of the master problem is built on clusters of RES output scenarios with similar characteristics, and the distributionally robust programming of the subproblems is built on the scenarios in each cluster. Numerical results demonstrate that the proposed hybrid optimization approach results in lower operation cost than conventional distributionally robust approaches and exhibits higher robustness than conventional stochastic approaches. In addition, we investigate the impacts of carbon emission costs and the level of RES penetration on the obtained low-carbon operating strategy.

Distributionally Robust Stochastic Optimization with Wasserstein Distance

Distributionally robust stochastic optimization (DRSO) is an approach to optimization under uncertainty in which, instead of assuming that there is a known true underlying probability distribution, one hedges against a chosen set of distributions. In this paper, we first point out that the set of distributions should be chosen to be appropriate for the application at hand and some of the choices that have been popular until recently are, for many applications, not good choices. We next consider sets of distributions that are within a chosen Wasserstein distance from a nominal distribution. Such a choice of sets has two advantages: (1) The resulting distributions hedged against are more reasonable than those resulting from other popular choices of sets. (2) The problem of determining the worst-case expectation over the resulting set of distributions has desirable tractability properties. We derive a strong duality reformulation of the corresponding DRSO problem and construct approximate worst-case distributions (or an exact worst-case distribution if it exists) explicitly via the first-order optimality conditions of the dual problem. Our contributions are fourfold. (i) We identify necessary and sufficient conditions for the existence of a worst-case distribution, which are naturally related to the growth rate of the objective function. (ii) We show that the worst-case distributions resulting from an appropriate Wasserstein distance have a concise structure and a clear interpretation. (iii) Using this structure, we show that data-driven DRSO problems can be approximated to any accuracy by robust optimization problems, and thereby many DRSO problems become tractable by using tools from robust optimization. (iv) Our strong duality result holds in a very general setting. As examples, we show that it can be applied to infinite dimensional process control and intensity estimation for point processes.

Robust Stochastic Facility Location: Sensitivity Analysis and Exact Solution

This work focuses on a broad class of facility location problems in the context of adaptive robust stochastic optimization under the state-dependent demand uncertainty. The demand is assumed to be significantly affected by related state information, such as the seasonal or socio-economic information. In particular, a state-wise ambiguity set is adopted for modeling the distributional uncertainty associated with the demand in different states. The conditional distributional characteristics in each state are described by a support, as well as by mean and dispersion measures, which are assumed to be conic representable. A robust sensitivity analysis is performed, in which, on the one hand, we analyze the impact of the change in ambiguity-set parameters (e.g., state probabilities, mean value abounds, and dispersion bounds in different states) onto the optimal worst-case expected total cost using the ambiguity dual variables. On the other hand, we analyze the impact of the change in location design onto the worst-case expected second-stage cost and show that the sensitivity bounds are fully described as the worst-case expected shadow-capacity cost. As for the solution approach, we propose a nested Benders decomposition algorithm for solving the model exactly, which leverages the subgradients of the worst-case expected second-stage cost at the location decisions formed insightfully by the associated worst-case distributions. The nested Benders decomposition approach ensures a finite-step convergence, which can also be regarded as an extension of the classic L-shaped algorithm for two-stage stochastic programming to our state-wise, robust stochastic facility location problem with conic representable ambiguity. Finally, the results of a series of numerical experiments are presented that justify the value of the state-wise distributional information incorporated in our robust stochastic facility location model, the robustness of the model, and the performance of the exact solution approach.

Data-driven distributionally robust unbalanced operation of distribution networks with high penetration of photovoltaic generation and electric vehicles

The uncertainties associated with the large-scale integration of electric vehicles (EV) and renewable energy resources introduce new challenges to operating the unbalanced three-phase distribution networks. In this paper, a data-driven distributionally robust optimization framework is proposed for the operation of the distribution network considering the uncertainties associated with the interconnected EV fleets and solar photovoltaic (PV) generation. The proposed framework leverages the column-and-constraint generation (C&CG) approach to minimize the operation cost considering the worst-case probability distributions of PV generation, the available energy in EV fleets, the arrival and departure times of EV fleets as well as their minimum and maximum energy capacities. The proposed approach is applied to the modified three-phase unbalanced IEEE 34-bus and IEEE 123-bus networks. To evaluate the performance of the proposed distributionally robust optimization framework, the results are compared to those procured by solving scenario-based stochastic programming and robust optimization problems. Furthermore, the impact of vehicle-to-grid capability on the operation of the distribution network is investigated and the in-sample and out-of-sample performances of the proposed framework are evaluated.

Multi-level coordinated energy management for energy hub in hybrid markets with distributionally robust scheduling

Maintaining energy balance and economical operation is significant for multi-energy systems such as the energy hub (EH). However, it is usually challenged by the frequently changing and unpredictable uncertain parameters at different timescales. Under this scope, this paper investigates the coordinated energy management problem for day-ahead and intra-day conditions considering uncertainties of source-load and market prices concurrently. Note that the precise knowledge of distributions about uncertainties may be unaccessible before the decision-making in day-ahead phase. A two-stage chance-constrained model based on distributionally robust approach with ambiguous moment information is proposed to immunize scheduling strategies against the worst-case probability distributions. The first stage is dedicated to obtaining more energy arbitrage and operation flexibility by optimizing bidding strategies in day-ahead power, natural gas and carbon trading markets. The second stage focuses on the optimization of the worst-case expected operation cost. It provides a robust energy equipment and load scheduling strategy for the reference of subsequent intra-day arrangements. With respect to different variations of electrical and thermal components, an intra-day two-timescale coordination is implemented step by step. The energy scheduling is re-dispatched circularly to minimize the operation and penalty costs. The possible energy imbalance is also compensated by this way. As the energy management program is nonlinear, chance-constrained and multi-stage, some linearization and dual transformation techniques are designed to enhance tractability of the program. Experimental results show that the developed multi-level framework results in a carbon emission decrease of 37%, and reduces energy cost averagely 3% compared with corresponding contrasting cases. The obtained strategy validates a good tradeoff between robustness and optimality.

MAD Dispersion Measure Makes Extremal Queue Analysis Simple

A notorious problem in queueing theory is to compute the worst possible performance of the GI/G/1 queue under mean-dispersion constraints for the interarrival- and service-time distributions. We address this extremal queue problem by measuring dispersion in terms of mean absolute deviation (MAD) instead of the more conventional variance, making available methods for distribution-free analysis. Combined with random walk theory, we obtain explicit expressions for the extremal interarrival- and service-time distributions and, hence, the best possible upper bounds for all moments of the waiting time. We also obtain tight lower bounds that, together with the upper bounds, provide robust performance intervals. We show that all bounds are computationally tractable and remain sharp also when the mean and MAD are not known precisely but are estimated based on available data instead. Summary of Contribution: Queueing theory is a classic OR topic with a central role for the GI/G/1 queue. Although this queueing system is conceptually simple, it is notoriously hard to determine the worst-case expected waiting time when only knowing the first two moments of the interarrival- and service-time distributions. In this setting, the exact form of the extremal distribution can only be determined numerically as the solution to a nonconvex nonlinear optimization problem. Our paper demonstrates that using mean absolute deviation (MAD) instead of variance alleviates the computational intractability of the extremal GI/G/1 queue problem, enabling us to state the worst-case distributions explicitly.

Optimal coordination of thyristor controlled series compensation and transmission expansion planning: Distributionally robust optimization approach

This paper proposes a distributionally robust optimization method for assessing economic benefits of transmission expansions with thyristor controlled series compensation devices. To accommodate the uncertainty of wind generations and electrical vehicle loads, a distributionally robust optimization method is proposed to use historical data instead of predefined distribution probabilities. In the proposed framework, firstly, a confidence set for unknown distributions are constructed based on the historical data. Then, based on worst-case distributions within a constructed confidence set, the optimal results of transmission expansion planning and allocation of thyristor controlled series compensation device are obtained. In order to take advantage of the flexible alternative current transmission devices (FACTS) in both economic and technical considerations, the optimal allocation of thyristor controlled series compensation is integrated into the transmission expansion planning. This requires AC-based transmission expansion planning model. AC power flow suffers from high computational complexity. To tackle this issue, the linearized AC power flow model is utilized. Furthermore, to solve the optimization problem, the combined technique including Bender's decomposition and cut-and-column are utilized. The numerical results on the Garver 6-bus and IEEE 118-bus systems demonstrate the effectiveness of the proposed method in comparison with scenario-based stochastic and robust optimization methods. Besides, the impacts of the thyristor controlled series compensation on total cost reduction has been investigated.

Distributionally Robust Optimization under Distorted Expectations

robust optimization (DRO) has arose as an important paradigm to address the issue of distributional ambiguity in decision optimization. In its standard form, DRO seeks an optimal solution against the worst-possible expected value evaluated based on a set of candidate distributions. In the case where a decision maker is not risk neutral, the most common scheme applied in DRO to capture one’s risk attitude is employing an expected utility functional. In this paper, we propose to address a decision maker’s risk attitude in DRO by following an alternative scheme known as “dual expected utility”. In this scheme, a distortion function is applied to convert physical probabilities to subjective probabilities so that the resulting expectation, also known as distorted expectation, captures the decision maker’s risk attitude. Unlike an expected utility functional which is linear in probability, in the dual scheme a distorted expectation is generally non-linear in probability. We distinguish DRO based on distorted expectations by terming it Distributionally Robust Risk Optimization” (DRRO), and show that DRRO can be equally, if not more, tractable to solve than DRO based on utility functionals. Our tractability results hold for any distortion function, and hence our scheme provides more flexibility to capture more realistic forms of risk attitudes. These include, as an important example, the inverse S-shaped distortion functionals that play a prominent role in Cumulative Prospect Theory (CPT), and several other non-convex risk measures developed more recently. Central to our development is the characterization of worst-case distributions based on the notion of convex envelope, which enables us to discover “hidden convexity in DRRO. We demonstrate through a numerical example that a production manager who overly weights “very good and “very bad outcomes may act as if (s)he is risk-averse when taking into account distributional ambiguity. Worst-case distributions are presented that can provide further explanation of such risk-averse behaviour.

Learning-Based Risk-Averse Model Predictive Control for Adaptive Cruise Control with Stochastic Driver Models

We propose a learning-based, distributionally robust model predictive control approach towards the design of adaptive cruise control (ACC) systems. We model the preceding vehicle as an autonomous stochastic system, using a hybrid model with continuous dynamics and discrete, Markovian inputs. We estimate the (unknown) transition probabilities of this model empirically using observed mode transitions and simultaneously determine sets of probability vectors (ambiguity sets) around these estimates, that contain the true transition probabilities with high confidence. We then solve a risk-averse optimal control problem that assumes the worst-case distributions in these sets. We furthermore derive a robust terminal constraint set and use it to establish recursive feasibility of the resulting MPC scheme. We validate the theoretical results and demonstrate desirable properties of the scheme through closed-loop simulations.

Distributionally Robust Model of Energy and Reserve Dispatch Based on Kullback–Leibler Divergence

This paper proposes a distance-based distributionally robust energy and reserve (DB-DRER) dispatch model via Kullback–Leibler (KL) divergence, considering the volatile of renewable energy generation. Firstly, a two-stage optimization model is formulated to minimize the expected total cost of energy and reserve (ER) dispatch. Then, KL divergence is adopted to establish the ambiguity set. Distinguished from conventional robust optimization methodology, the volatile output of renewable power generation is assumed to follow the unknown probability distribution that is restricted in the ambiguity set. DB-DRER aims at minimizing the expected total cost in the worst-case probability distributions of renewables. Combining with the designed empirical distribution function, the proposed DB-DRER model can be reformulated into a mixed integer nonlinear programming (MINLP) problem. Furthermore, using the generalized Benders decomposition, a decomposition method is proposed and sample average approximation (SAA) method is applied to solve this problem. Finally, simulation result of the proposed method is compared with those of stochastic optimization and conventional robust optimization methods on the 6-bus system and IEEE 118-bus system, which demonstrates the effectiveness and advantages of the method proposed.

Shear deformation effects on stability of unbraced steel frames in variable loading

The generalized lateral stiffness equation of a semi-rigidly connected frame with accounting for shear deformation is derived in this paper. The lateral stiffness equation can be used to determine the critical loads of frames, as instability occurs when the lateral stiffness diminishes to zero. The proposed method is numerically exact, and is obtained via solving the Timoshenko system of governing differential equations for a semi-rigidly connected column. It is demonstrated in the derivation that the effect of shear deformation can conveniently be accounted for via use of the shear flexibility coefficient, η. The proposed equations are validated via finite element analysis and demonstrated via numerical examples. The concept of variable loading originally proposed by Xu is also generalized to include the effect of shear deformation, whereby the worst and best case distributions of axial loads in frames can be determined via solving a minimization problem. Although the effect of shear deformation can generally be neglected in practice regardless of whether proportional or variable loading are considered, this study demonstrates that the effect of shear deformation on the lateral stiffness and critical loads can be significant and detrimental if the frames contain supporting columns with low slenderness ratios and/or lean-on columns. From the results of the numerical examples, the critical load of a lean-on frame containing a low-slenderness supporting column (L/r < 40) can be reduced by up to 41% even when the supporting column is not loaded.

Towards the Clinical Implementation of Pencil Beam Scanned Proton Therapy (PBS-PT) for Thoracic Indications: A Comprehensive 4D Robustness Evaluation Based on Patient and Machine Specific Information

Due to anticipated clinical benefits, thoracic tumors are potential future indications for PBS-PT. However, the concern of plan robustness as a result of motion hamper its clinical implementation. Here we present a comprehensive evaluation of different planning strategies based on patient and machine specific data in order to define optimal PBS-PT protocols for lung and esophageal tumors. This study includes 3 non-small cell lung and 3 esophageal cancer patients with a planning as well as 5 weekly repeated 4DCTs available. For all patients, the CTV volume and motion were monitored throughout the treatment course. Layered rescanned (5 times) 3D and 4D robust optimized PBS-PT plans were created, and approved clinically. These plans were then delivered in dry runs at our proton facility to obtain machine log files, and subsequently evaluated through our 4D robustness evaluation method (4DREM). The 4DREM assesses the plan robustness for the combination of (1) setup and range errors, (2) machine errors, (3) anatomy changes, (4) breathing motion, and (5) interplay effects. The disturbing effects (2), (3), (4), (5) are considered by calculating subplan doses on particular 4DCT phases and performing 4D dose accumulation in all 4DCTs. The subplans were split from the nominal plan by retrieving log files information. For each evaluated plan, 14 4D accumulated scenario dose distributions were obtained, representing 14 possible full fractionated treatment courses. Only small differences in V95(CTV) between nominal and voxel-wise worst-case dose distributions were observed for all 3D / 4D robust optimized plans (consistently ≥ 99.89 %). Similarly, averaged Dmean(lungs-GTV) over all scenarios considered within the 4DREM changed only slightly (Table 1). For 6 patients with lung / esophageal cancer, our 3D / 4D PBS-PT planning protocol showed to be clinically suitable. After confirming these results in an extended patient cohort, we will proceed to investigate less conservative, more conformal planning strategies. The experience gained through comprehensive robustness evaluations based on patient and machine specific data is essential for the definition of optimal PBS-PT protocols for patients with thoracic tumors.Abstract 3762; Table 1Patient characteristics (mean ± SD volume and motion variations) and plan statisticsLungPt. 1Pt. 2Pt. 3Volume [cm3]108.2 ± 16.254.6 ± 5.755.6 ± 2.9Motion [mm]3.9 ± 0.84.6 ± 1.43.8 ± 1.23D4D3D4D3D4DDmean(lungs-GTV) [GyRBE] Nominal9.4010.478.497.359.078.47 Scenarios9.8310.688.617.598.927.96EsophagusPt. 4Pt. 5Pt. 6Volume [cm3]367.4 ± 14.0133.6 ± 10.3209.9 ± 2.6Motion [mm]6.5 ± 0.96.5 ± 0.73.1 ± 0.63D4D3D4D3D4DDmean(lungs-GTV) [GyRBE] Nominal4.304.281.381.411.911.98 Scenarios4.354.531.521.641.972.00 Open table in a new tab

Strategic planning of dedicated autonomous vehicle lanes and autonomous vehicle/toll lanes in transportation networks

Employing vehicle communication and automated control technologies, autonomous vehicles (AVs) can safely drive closer together than human-driven vehicles (HVs), thereby potentially improving traffic efficiency. Separation between AV and HV traffic through the deployment of dedicated AV lanes is foreseen as an effective method of amplifying the benefits of AVs and promoting their adoption. However, it is important to consider mixed AV and HV traffic in a transportation network. On the one hand, it may be impractical to deploy dedicated AV lanes throughout the network, while on the other hand, dedicated AV lanes may even reduce the total traffic efficiency of a road segment when the AV flow rate is low. In this study, we considered a new form of managed lanes for AVs, designated as autonomous vehicle/toll (AVT) lanes, which grant free access to AVs while allowing HVs to access the lanes by paying a toll. We investigated the optimal deployment of dedicated AV lanes and AVT lanes in transportation networks with mixed AV and HV flows. The user equilibrium (UE) problem in a transportation network with mixed flows of AVs and HVs is first explored. We formulated the UE problem as a link-based variational inequality (VI) and identified that, with different impacts of AVs on road capacity, the UE problem can have unique or non-unique flow patterns. Considering that the UE problem may have non-unique flow distributions, we proposed a robust optimal deployment model, which is a generalized semi-infinite min-max program, to deploy the dedicated AV lanes and AVT lanes so that the system performance under the worst-case flow distributions is optimized. We proposed effective solution algorithms to solve these models and presented numerical studies to demonstrate the models and the solution algorithms. The results show that the system performance can be significantly improved through the deployment of dedicated AV lanes and AVT lanes.

The distortion principle for insurance pricing: properties, identification and robustness

Distortion (Denneberg in ASTIN Bull 20(2):181–190, 1990) is a well known premium calculation principle for insurance contracts. In this paper, we study sensitivity properties of distortion functionals w.r.t. the assumptions for risk aversion as well as robustness w.r.t. ambiguity of the loss distribution. Ambiguity is measured by the Wasserstein distance. We study variances of distances for probability models and identify some worst case distributions. In addition to the direct problem we also investigate the inverse problem, that is how to identify the distortion density on the basis of observations of insurance premia.

Technical Note—Closed-Form Solutions for Worst-Case Law Invariant Risk Measures with Application to Robust Portfolio Optimization

Worst-case risk measures provide a means of calculating the largest value of risk when only partial information of the underlying distribution is available. For popular risk measures such as value-at-risk (VaR) and conditional value-at-risk (CVaR) it is now known that their worst-case counterparts can be evaluated in closed form when only the first two moments are known. We show in this paper that closed-form solutions exist for a general class of law invariant coherent risk measures, which consist of spectral risk measures (and thus CVaR also) as special cases. Moreover, we provide worst-case distributions characterized in terms of risk spectrums, which can take any form of distribution bounded from below. As applications of the closed-form results, new formulas are derived for calculating the worst-case values of higher order risk measures and higher order semideviation, and new robust portfolio optimization models are provided. The online appendix is available at https://doi.org/10.1287/opre.2018.1736 .

An approximation framework for two-stage ambiguous stochastic integer programs under mean-MAD information

We consider two-stage recourse models in which only limited information is available on the probability distributions of the random parameters in the model. If all decision variables are continuous, then we are able to derive the worst-case and best-case probability distributions under the assumption that only the means and mean absolute deviations of the random parameters are known. Contrary to most existing results in the literature, these probability distributions are the same for every first-stage decision. The ambiguity set that we use in this paper also turns out to be particularly suitable for ambiguous recourse models involving integer decisions variables. For such problems, we develop a general approximation framework and derive error bounds for using these approximatons. We apply this approximation framework to mixed-ambiguous mixed-integer recourse models in which some of the probability distributions of the random parameters are known and others are ambiguous. To illustrate these results we carry out numerical experiments on a surgery block allocation problem.

Dynamic charging infrastructure deployment for plug-in hybrid electric trucks

Inspired by the rapid development of charging-while-driving (CWD) technology, plans are ongoing in government agencies worldwide for the development of electrified road freight transportation systems through the deployment of dynamic charging lanes. This en route method for the charging of plug-in hybrid electric trucks is expected to supplement the more conventional charging technique, thus enabling significant reduction in fossil fuel consumption and pollutant emission from road freight transportation. In this study, we investigated the optimal deployment of dynamic charging lanes for plug-in hybrid electric trucks. First, we developed a multi-class multi-criteria user equilibrium model of the route choice behaviors of truck and passenger car drivers and the resultant equilibrium flow distributions. Considering that the developed user equilibrium model may have non-unique flow distributions, a robust deployment of dynamic charging lanes that optimizes the system performance under the worst-case flow distributions was targeted. The problem was formulated as a generalized semi-infinite min-max program, and a heuristic algorithm for solving it was proposed. This paper includes numerical examples that were used to demonstrate the application of the developed models and solution algorithms.