Adjustable Robust Optimization Research Articles

Designing tractable piecewise affine policies for multi-stage adjustable robust optimization

AbstractWe study piecewise affine policies for multi-stage adjustable robust optimization (ARO) problems with non-negative right-hand side uncertainty. First, we construct new dominating uncertainty sets and show how a multi-stage ARO problem can be solved efficiently with a linear program when uncertainty is replaced by these new sets. We then demonstrate how solutions for this alternative problem can be transformed into solutions for the original problem. By carefully choosing the dominating sets, we prove strong approximation bounds for our policies and extend many previously best-known bounds for the two-staged problem variant to its multi-stage counterpart. Moreover, the new bounds are—to the best of our knowledge—the first bounds shown for the general multi-stage ARO problem considered. We extensively compare our policies to other policies from the literature and prove relative performance guarantees. In two numerical experiments, we identify beneficial and disadvantageous properties for different policies and present effective adjustments to tackle the most critical disadvantages of our policies. Overall, the experiments show that our piecewise affine policies can be computed by orders of magnitude faster than affine policies, while often yielding comparable or even better results.

Exact SDP Reformulations for Adjustable Robust Quadratic Optimization with Affine Decision Rules

Exact SDP Reformulations for Adjustable Robust Quadratic Optimization with Affine Decision Rules

Adjustable Robust Low-Carbon Unit Commitment With Nonanticipativity by Linear Programming

Thermal power generation is the main source of carbon emissions. Adopting low-carbon dispatching and developing renewable energy sources (RESs) are effective ways to reduce carbon emissions. This article constructs a mixed-integer programming (MIP) model of low-carbon full-scenario unit commitment with nonanticipativity. To overcome the curse of dimensionality problem caused by the use of a massive number of scenarios, we employ the adjustable robust optimization approach (AROA) to reformulate the full-scenario unit commitment as a deterministic robust model. In addition, we establish the precise adjustable robust convex hull of generation constraints in a higher dimensional space and generate a relaxed linear programming (LP) formulation of the original MIP model. Then, we design a heuristic method for obtaining a near-optimal feasible solution by converting a large-scale MIP into an LP model that can be solved in polynomial time. The numerical experiments presented in this article demonstrate the effectiveness and efficiency of the proposed method.

Adjustable robust optimization with nonlinear recourses

Adjustable robust optimization with nonlinear recourses

Adjustable Robust Optimization for Water Distribution System Operation Under Uncertainty

AbstractThe optimal operation of water distribution systems (WDS) is a paramount objective for water utilities due to the substantial energy consumption associated with pumping. A major challenge in optimizing WDS operation is addressing uncertainties such as those related to consumer demands. Real‐time operation under uncertainty necessitates a dynamic approach that can utilize the newly observed information and adjust the operational policy accordingly. This study presents an adjustable robust optimization (ARO) approach to tackle this challenge. Unlike static optimization methods, ARO generates a decision rule policy that is dynamically adjusted as new data becomes available and the operational horizon evolves, thereby ensuring adaptability to changing conditions. Furthermore, the study includes a quantitative analysis of typical demand uncertainty that supports the formulation of the ARO model. The proposed method is evaluated through two case studies and compared with traditional folding horizon approaches. The results indicate that the ARO method is competitive with traditional methods in terms of objective value and surpasses them in terms of robustness. An additional advantage of the method is its offline operation capability which enables it to produce decision rules independent of real‐time programs. This feature facilitates various practical applications such as what‐if analyses, maintenance work planning, and preparation for other special events.

Decision‐dependent distributionally robust integrated generation, transmission, and storage expansion planning: An enhanced Benders decomposition approach

AbstractIntegrated generation, transmission, and storage expansion planning (IGT&SP) is the cornerstone to realize low‐carbon transition considering security constraints in the long run. A novel IGT&SP planning scheme is proposed to balance the planning cost, i.e. renewable energy sources (RESs) and energy storage systems, based on the distributionally ambiguity sets. A novel decision‐dependent ambiguity set is proposed to capture the relation between the uncertainties of RES output and long‐term planning. A two‐stage risk‐averse distributionally robust optimization is formulated, where the RESs, energy storage systems, and transmission line expansion are optimized in the first stage and a unit commitment problem is proposed in the second‐stage optimization to assess the performance of the expanded system. This problem is reformulated into a two‐stage optimization problem with complete mixed‐integer recourse, where the state variable is binary. A novel enhanced Benders decomposition algorithm is proposed to solve the IGT&SEP efficiently, where the cutting planes are generated by a primal‐dual relaxation of the recourse problem. Simulations are conducted on the modified IEEE‐30 test system and modified IEEE‐118 test system. Compared with adjustable robust optimization and L1‐norm Wasserstein distance‐based distributionally robust optimization, numerical results verify the effectiveness of the proposed IGT&SP, together with the solution algorithm.

Adjustable Robust Optimization with Discrete Uncertainty

In this paper, we study adjustable robust optimization (ARO) problems with discrete uncertainty. Under a very general modeling framework, we show that such two-stage robust problems can be exactly reformulated as ARO problems with objective uncertainty only. This reformulation is valid with and without the fixed recourse assumption and is not limited to continuous wait-and-see decision variables unlike most of the existing literature. Additionally, we extend an enumerative algorithm akin to a branch-and-cut scheme for which we study the asymptotic convergence. We discuss how to apply the reformulation on two variants of well-known optimization problems, a facility location problem in which uncertainty may affect the capacity values and a multiple knapsack problem with uncertain weights, and we report extensive computational results demonstrating the effectiveness of the approach. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms – Discrete. Funding: This work was supported by the Air Force Office of Scientific Research [Grants FA8655-20-1-7012, FA8655-20-1-7019].

An oracle-based algorithm for robust planning of production routing problems in closed-loop supply chains of beverage glass bottles

The demand for glass bottles is exhibiting an upward trend over time. The manufacturing of glass bottles is costlier in terms of time and resources and is associated with a higher level of heat generation and environmental pollution compared to recycling processes. In response to the aforementioned challenges, companies that use glass bottles need to implement strategies to manage their reverse supply chains in conjunction with their traditional supply chains, as the economic and environmental benefits of returned products are unquestionable. Closed-loop supply chains (CLSCs) integrate forward and reverse flows of products and information. This integration helps companies to have a broader view of the whole chain. Despite these advantages, managing CLSCs can be challenging as they are exposed to many uncertainties regarding supply and demand processes, travel times, and quantity/quality of returned products.In this study, we consider the production planning, inventory management, and vehicle routing decisions of a CLSC of beverage glass bottles. We propose an MILP model and rely on a multi-stage adjustable robust optimization (ARO) formulation to deal with the randomness in both the demand for filled bottles and the requests for pickups of empty bottles. We develop an exact oracle-based algorithm to solve the ARO problem and propose a heuristic search algorithm to reduce the solution time. Our numerical experiments not only show the incompetency of the customary method, namely the affine decision rule approach, but also illustrate how our algorithms can solve the small-size problems and significantly improve the quality of the obtained solution for large problems. Furthermore, our numerical results show that robust plans tend to be sparse, meaning the routes are chosen so that empty bottles are transported to production sites in such a way that fewer new bottles need to be ordered. Thus, robust planning makes the CLSCs more environmentally friendly.

Robust optimal energy management with dynamic price response: A non-cooperative multi-community aggregative game perspective

Energy management system gradually becomes an effective means in smart grid for paving the way to low carbon economy. However, information privacy, a large population of users, and the uncertainties of renewable energy sources and consumers’ behaviors have led to significant challenges for energy management system operation in terms of security and robustness. To conquer such difficulties, a bilevel energy management scheme for the day-ahead optimal scheduling of multi-community system combined with distributed energy sources is first proposed. Meanwhile, uncertainties induced by renewable energy sources generation and load consumption are handled through adjustable robust optimization. Secondly, a non-cooperative multi-community aggregative game is formulated to describe the interaction of numerous residential users which are coupled through the dynamic electricity price. Then, to seek the ɛ-Nash Equilibrium of the proposed game, an improved decentralized iterative algorithm based on Mean-Field control and consensus is presented which is benchmarked with centralized algorithm and a decentralized optimization method based on quadratic programming. Since each player in the proposed algorithm does not need to exchange information directly with other players, the information privacy is fully preserved. Also, the convergence of the proposed algorithm is provided. Case studies of a five-community system are conducted and the comparison results show that our proposed approach has better performance in terms of computational time and electricity cost.

LP-based approximations for disjoint bilinear and two-stage adjustable robust optimization

LP-based approximations for disjoint bilinear and two-stage adjustable robust optimization

Adjustable robust optimization with objective uncertainty

In this work, we study optimization problems where some cost parameters are not known at decision time and the decision flow is modeled as a two-stage process within a robust optimization setting. We address general problems in which all constraints (including those linking the first and the second stages) are defined by convex functions and involve mixed-integer variables, thus extending the existing literature to a much wider class of problems. We show how these problems can be reformulated using Fenchel duality, allowing to derive an enumerative exact algorithm, for which we prove asymptotic convergence in the general case, and finite convergence for cases where the first-stage variables are all integer.An implementation of the resulting algorithm, embedding a column generation scheme, is then computationally evaluated on a variant of the Capacitated Facility Location Problem with uncertain transportation costs, using instances that are derived from the existing literature. To the best of our knowledge, this is the first approach providing results on the practical solution of this class of problems.

A Low-Carbon and Economic Dispatch Strategy for a Multi-Microgrid Based on a Meteorological Classification to Handle the Uncertainty of Wind Power

In a modern power system, reducing carbon emissions has become a significant goal in mitigating the impact of global warming. Therefore, renewable energy sources, particularly wind-power generation, have been extensively implemented in the system. Despite the advantages of wind power, its uncertainty and randomness lead to critical security, stability, and economic issues in the power system. Recently, multi-microgrid systems (MMGSs) have been considered as a suitable wind-power deployment candidate. Although wind power can be efficiently utilized by MMGSs, uncertainty and randomness still have a significant impact on the dispatching and operation of the system. Therefore, to address the wind power uncertainty issue and achieve an optimal dispatching strategy for MMGSs, this paper presents an adjustable robust optimization (ARO) model based on meteorological clustering. Firstly, the maximum relevance minimum redundancy (MRMR) method and the CURE clustering algorithm are employed for meteorological classification in order to better identify wind patterns. Secondly, a conditional generative adversarial network (CGAN) is adopted to enrich the wind-power datasets with different meteorological patterns, resulting in the construction of ambiguity sets. Thirdly, the uncertainty sets that are finally employed by the ARO framework to establish a two-stage cooperative dispatching model for MMGS can be derived from the ambiguity sets. Additionally, stepped carbon trading is introduced to control the carbon emissions of MMGSs. Finally, the alternative direction method of multipliers (ADMM) and the column and constraint generation (C&CG) algorithm are adopted to achieve a decentralized solution for the dispatching model of MMGSs. Case studies indicate that the presented model has a great performance in improving the wind-power description accuracy, increasing cost efficiency, and reducing system carbon emissions. However, the case studies also report that the approach consumes a relative long running time. Therefore, in future research, the solution algorithm will be further improved for the purpose of raising the efficiency of the solution.

Optimal Robust Policy for Feature-Based Newsvendor

We study policy optimization for the feature-based newsvendor, which seeks an end-to-end policy that renders an explicit mapping from features to ordering decisions. Most existing works restrict the policies to some parametric class that may suffer from suboptimality (such as affine class) or lack of interpretability (such as neural networks). Differently, we aim to optimize over all functions of features. In this case, the classic empirical risk minimization yields a policy that is not well-defined on unseen feature values. To avoid such degeneracy, we consider a Wasserstein distributionally robust framework. This leads to an adjustable robust optimization, whose optimal solutions are notoriously difficult to obtain except for a few notable cases. Perhaps surprisingly, we identify a new class of policies that are proven to be exactly optimal and can be computed efficiently. The optimal robust policy is obtained by extending an optimal robust in-sample policy to unobserved feature values in a particular way and can be interpreted as a Lipschitz regularized critical fractile of the empirical conditional demand distribution. We compare our method with several benchmarks using synthetic and real data and demonstrate its superior empirical performance. This paper was accepted by J. George Shanthikumar, data science. Supplemental Material: The online appendix and data are available at https://doi.org/10.1287/mnsc.2023.4810 .

Robust optimization of seasonal, day-ahead and real time operation of aggregated energy systems

This work proposes an approach for the robust operational optimization of Aggregated Energy Systems (AES) on three key time scales: seasonal, day-ahead and real-time hourly operation. The evaluation of all the three time-scales is fundamental for AESs featuring seasonal storage systems and/or units (such as combined heat and power plants) with yearly-basis constraints on relevant performance indexes or emissions. The approach consists in a rolling horizon algorithm based on an Affine Adjustable Robust Optimization model for optimizing both day ahead schedule (commitment and economic dispatch) and the decision rules to adjust the real-time operation. The robust optimization model takes as input (i) the day-ahead forecasts of renewable production and energy demands with their corresponding uncertainty, (ii) past and future expected performance of the units with yearly constraints, and (iii) target end-of-the-day charge levels for the seasonal storage system. These long-term targets are estimated by optimizing the operation over representative years defined on the basis of the past measured data. The proposed methodology is tested on three real-world case studies, featuring up to four short-term uncertain parameters (energy demands and non-dispatchable generation), yearly constraints and seasonal storage. Results shows that the proposed methodology meets the yearly constraints and safely manages the seasonal storage without shortages, while always meeting the energy demands (no shedding). In addition, the cost of short- and long-term uncertainty were evaluated by comparing the results of the robust rolling horizon with other two deterministic approaches, proving a limited increase.

Improving robustness of water supply system using a multi-objective robust optimization framework

Due to the uncertainty of rainfall and water demand, the water supply of various participants has been challenged in such a way that this challenge has accelerated the failure of water supply system. Thus, this study proposes a multi-stage Adjustable Robust Optimization integrated to the multi-objective programming framework to drive water supply system to the failure safety zone and thereby improve robustness of system under different scenarios. Indeed, Adjustable Robust Optimization framework is applied to investigate the two uncertain factors of rainfall and water demand. A real arid area of Sistan basin in southeastern Iran is considered to analyze the proposed multi-objective programming model. Next, various comparative feasibilities under different levels of uncertainty are carried out to examine the robustness status in more detail. In the following, due to the deterioration of climatic patterns in the coming years, some managerial insights are highlighted. According to the final outputs, the domestic sector has reached more optimal value compared to that of the agricultural and industrial participants in all objectives due to less water intake, and as a result, it has a significant impact on the robustness of water supply system.

Budget allocation of food procurement for natural disaster response

This paper studies a variant of the lot sizing problem that arises in the context of disaster management. In this problem, a fixed budget has to be allocated efficiently over multiple time periods to procure large quantities of a staple food that will be stored and later delivered to people affected by disaster strikes whose numbers are unknown in advance. Starting from the deterministic model where perfect information is assumed, different formulations to address the uncertainties are constructed: classical robust optimisation, risk-minimisation stochastic programming, and adjustable robust optimisation. Experiments conducted using data from West Java, Indonesia allow us to discuss the advantages and drawbacks of each method. Our methods constitute a toolbox to support decision makers with making procurement decisions and answering managerial questions such as which annual budget is fair and safe, or when storage peaks are likely to occur.

Humanitarian Relief Distribution Problem: An Adjustable Robust Optimization Approach

Management of humanitarian logistics operations is one of the most critical planning problems to be addressed immediately after a disaster. The response phase covers the first 12 hours after the disaster and is prone to uncertainties because of debris and gridlock traffic influencing the dispatching operations of relief logistics teams in the areas affected. Moreover, the teams have limited time and resources, and they must provide equitable distribution of supplies to affected people. This paper proposes an adjustable robust optimization approach for the associated humanitarian logistics problem. The approach creates routes for relief logistics teams and decides the service times of the visited sites to distribute relief supplies by taking the uncertainty in travel times into account. The associated model allows relief logistics teams to adjust their service decisions according to the revealed information during the process. Hence, our solutions are robust for the worst-case realization of travel times, but still more flexible and less conservative than those of static robust optimization. We propose novel reformulation techniques to model these adjustable decisions. The resulting models are computationally challenging optimization problems to be solved by exact methods, and, hence, we propose heuristic algorithms. The state-of-the-art heuristic, which is based on clustering and a dedicated decision-rule algorithm, yields near-optimal results for medium-sized instances and is scalable even for large-sized instances. We have also shown the effectiveness of our approach in a case study using a data set obtained from an earthquake that hit the Van province of Turkey in 2011. History: This paper has been accepted for the Transportation Science Special Issue on Emerging Topics in Transportation Science and Logistics. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2023.1204 .

Affinely adjustable robust optimization for radiation therapy under evolving data uncertainty via semi-definite programming

Static robust optimization has played an important role in radiotherapy, where the decisions aim to safeguard against all possible realizations of uncertainty. However, it may lead to overly conservative decisions or too expensive treatment plans, such as delivering significantly more dose than necessary. Motivated by the success of adjustable robust optimization in reducing highly conservative decision-making of static robust optimization in applications, in this paper, we present an affinely adjustable robust optimization (AARO) model for hypoxia-based radiation treatment planning in the face of evolving data uncertainty. We establish an exact semi-definite program reformulation of the model under a so-called affine decision rule and evaluate our model and approach on a liver cancer case as a proof-of-concept. Our AARO model incorporates uncertainties both in dose influence matrix and re-oxygenation data as well as inexactness of the revealed (re-oxygenation) data. Our numerical experiments demonstrate that the adjustable model successfully handles uncertainty in both re-oxygenation and the dose matrix. They also show that, by utilizing information halfway through the treatment plan, the adjustable solutions of the AARO model outperform a static method while maintaining a similar total dose.

Multi-timescale Affinely Adjustable Robust Reactive Power Dispatch of Distribution Networks Integrated with High Penetration of PV

Photovoltaic (PV) power generation has highly penetrated in distribution networks, providing clean and sustainable energy. However, its uncertain and intermittent power outputs significantly impair network operation, leading to unexpected power loss and voltage fluctuation. To address the uncertainties, this paper proposes a multi-timescale affinely adjustable robust reactive power dispatch (MTAAR-RPD) method to reduce the network power losses as well as alleviate voltage deviations and fluctuations. The MTAAR-RPD aims to coordinate on-load tap changers (OLTCs), capacitor banks (CBs), and PV inverters through a three-stage structure which covers multiple timescales of “hour-minute-second”. The first stage schedules CBs and OLTCs hourly while the second stage dispatches the base reactive power outputs of PV inverter every 15 min. The third stage affinely adjusts the inverter reactive power output based on an optimized Q-P droop controller in real time. The three stages are coordinately optimized by an affinely adjustable robust optimization method. A solution algorithm based on a cutting plane algorithm is developed to solve the optimization problem effectively. The proposed method is verified through theoretical analysis and numerical simulations.

Stochastic Robust Real-Time Power Dispatch With Wind Uncertainty Using Difference- of-Convexity Optimization

This paper proposes a novel stochastic robust real-time power dispatch model considering wind uncertainty. In this dispatch model that incorporates automatic power generation control, system constraints are constructed with affinely adjustable robust optimization to enhance the system security, while the objective concerning the expected value of generation cost is optimized to improve the cost efficiency. The optimized objective is consistent with stochastic optimization, while the formulation for the objective is different. A Nataf conversion-based three-point estimate (NC-TPE) is adopted, which uses an approximating function to formulate the expected generation cost with deterministic estimate points. Compared to traditional stochastic sampling methods, the deterministic assessment method NC-TPE achieves an accurate formulation with more stable and efficient computation performance. The dispatch model is derived as a problem with nonconvex objective and bilinear constraints using the dual theory. To address the highly nonconvex problem, the objective and bilinear constraints are first converted to difference-of-convexity functions, and then a tailored difference-of-convexity optimization is employed to convexify the nonconvex functions, such that the original problem can be solved with an iterative quadratically constrained program. Numerical simulations demonstrate the economic superiority and robustness of the proposed model, as well as the effectiveness of the convexified solution.