Bilevel Programming Research Articles

Coordinative dispatching of shared and public transportation under passenger flow outburst

In the passenger flow outburst area (e.g., metro emergency, dispersal of sporting events, and concerts), passengers put forward large demand for ride-sourcing services. Insufficient ride-sourcing supply of the area induces increasing matching time, and the ride-sourcing influx from outside picking up passengers leads to heavier road traffic congestion in the flow outburst area. To solve this problem, we propose a novel coordinative dispatching method of ride-sourcing and multi-modal transit, in which subsidies and buses are provided for passengers in flow outburst areas to encourage taking other public transportation to low passenger flow areas first and then taking ride-sourcing to their destinations. In this paper, there are two options for passengers: (a) wait for ride-sourcing to pick them up; (b) take other public transportation first and then take ride-sourcing. To obtain the optimal dispatching and subsidy schemes, we develop a bi-level mixed integer programming model based on network flow theory and design the corresponding iterative algorithm to solve it. Considering the high uncertainty of ride-sourcing demand in the area, we further develop a robust optimization model to obtain more reliable schemes. Case studies based on a real-problem-scale dataset are conducted. The results demonstrate that our approach can be carried out in real time, and encouraging passengers to take multi-modal transit shows great potential in reducing affected users’ delays. The robust optimization model offers a more reliable and competitive solution when demand varies widely. Our method offers a win–win–win way for passengers, ride-sourcing service providers, and public transportation systems.

Decarbonizing Maritime Transport through Green Fuel-Powered Vessel Retrofitting: A Game-Theoretic Approach

Addressing the urgent global challenge of man-made greenhouse gas emissions and climate change necessitates collaborative action between shipping lines and government regulatory agencies. Aligning with the International Maritime Organization’s emissions reduction strategy, this paper presents a novel bi-level programming model that unifies these stakeholders. On the upper level of the proposed bi-level model, a number of shipping lines optimize retrofitting plans for their vessels to maximize economic benefits. On the lower level, the regulatory agency responds to the carbon reduction efforts by setting retrofitting subsidies and emission penalty rates. This framework represents a multi-leader–single-follower game involving shipping lines and the regulatory agency, and its equilibrium is determined through an equilibrium problem with equilibrium constraints (EPEC). The EPEC comprises multiple single-leader–follower problems, each of which can be formulated as a mathematical program with equilibrium constraints (MPEC). The diagonalization algorithm (DM) is employed for its solution. Simulation studies performed based on a ten-year planning period show that the proposed approach can effectively promote vessel retrofitting and the use of green fuels, which leads to an annual emission reduction of over 50%.

Low-carbon oriented planning of shared photovoltaics and energy storage systems in distribution networks via carbon emission flow tracing

With the targets of carbon peaking and carbon neutrality, carbon emission reduction measures have become significant issues in new-type power systems. Distribution networks (DNs), which multiple power sources and flexible loads access to, play an essential role in exploiting the potential of reducing carbon emissions on the demand side. To achieve a global carbon emission reduction considering the carbon quota of each customer, shared photovoltaics (PVs) and energy storage systems (ESSs) are allocated with a centralized calculation and optimization conducted by DNs. To solve two key points in demand-side planning of shared PVs and ESSs in distribution networks, i.e., the accuracy of carbon emission flow (CEF) calculation and carbon quota-oriented optimization planning, this paper proposes a low-carbon oriented planning method for shared PVs and ESSs via CEF tracing. Firstly, a novel CEF calculation method is proposed based on the forward current sweep and backward voltage sweep algorithm, which considers the tracing of reactive power and power losses. Next, carbon emission index and economic index of allocating shared PVs and ESSs are defined. Subsequently, a bi-level optimization model is presented, considering the excess carbon emissions of each load based on carbon accounting and carbon quotas. Finally, the superiority of the proposed method is verified through a modified IEEE 33-node distribution network, and the effects of various investment constraints and carbon quotas are discussed.

On the complexity of inverse bivariate multi-unit assignment valuation problems

Inverse and bilevel optimization problems play a central role in both theory and applications. These two classes are known to be closely related due to the pioneering work of Dempe and Lohse (2006), and thus have often been discussed together ever since. In this paper, we consider inverse problems for multi-unit assignment valuations. Multi-unit assignment valuations form a subclass of strong-substitutes valuations that can be represented by edge-weighted complete bipartite graphs. These valuations play a key role in auction theory as the strong substitutes condition implies the existence of a Walrasian equilibrium. A recent line of research concentrated on the problem of deciding whether a bivariate valuation function is an assignment valuation or not. In this paper, we consider an inverse variant of the problem: we are given a bivariate function g, and our goal is to find a bivariate multi-unit assignment valuation function f that is as close to g as possible. The difference between f and g can be measured either in ℓ 1 - or ℓ ∞ -norm. Using tools from discrete convex analysis, we show that the problem is strongly NP -hard. On the other hand, we derive linear programming formulations that solve relaxed versions of the problem.

Momentum recursive DARTS

DARTS has emerged as a popular method for neural architecture search (NAS) owing to its efficiency and simplicity. It employs gradient-based bi-level optimization to iteratively optimize the upper-level architecture parameters and lower-level super-network weights. The key challenge in DARTS is the accurate estimation of gradients for two-level object functions, leading to significant errors in gradient approximation. To address this issue, we propose a new approach, MR-DARTS, that incorporates a momentum term and a recursive scheme to improve gradient estimation. Specifically, we leverage historical information by using a running average of past observed gradients to enhance the quality of current gradient estimation in both upper-level and lower-level functions. Our theoretical analysis shows that the variance of our estimated gradient decreases with each iteration. By utilizing momentum and a recursive scheme, MR-DARTS effectively controls the error in stochastic gradient updates that result from inaccurate gradient estimation. Furthermore, we utilize the Neumann series approximation and Hessian Vector Product scheme to reduce computational requirements and memory usage. We evaluate our proposed method on several benchmarks and demonstrate its effectiveness through comprehensive experiments.

Designing a collaborative maintenance planning mechanism between airline and maintenance company using a bilevel model – Considering flexible maintenance schedule and service charges

Periodic aircraft Maintenance Repair and Overhaul (MRO) activities are compulsory. To capture the features of MRO operations between airline and maintenance company under MRO outsourcing mode, we consider a collaborative aircraft maintenance planning problem, in which the airline outsources its self-operated maintenance checks to an independent maintenance company. Under such context, MRO planning conflicts between two collaborates exists especially during maintenance peak period, as airline aims to maximize flight operations revenues by receiving timely MRO service. On the other hand, maintenance company strives to accomplish heterogenous demands without causing tardiness and involving excessive MRO resources cost. Therefore, finding an effective mechanism to eliminate the conflicts between maintenance demands (from airline company) and maintenance service supply capability (from maintenance company) is indispensable for collaboration. A bi-level optimization model is formulated to describe the conflicting objectives and maintenance planning features of two independent stakeholders. To facilitates a synchronization of demand and supply and reconcile MRO conflicts, flexible maintenance schedules and service charges decisions from two independent collaborators is involved in the model, enabling a maintenance planning negotiation for finding mutual beneficial maintenance plan. An iterative algorithm deploying bounding updating strategy is developed, which generates a series of convergent-oriented bounding constraints along the iterative negotiation and guides the adjustment of maintenance demand and supply towards equilibrium. Based on the computational results in solving various maintenance planning scenarios, the impacts of different features (e.g. resource availability, resource demands level, and penalty cost of service tardiness) on airline and service company’s planning performance are analysed, revealing that maintenance service company’s own interest can be assured and maintained though it is a follower in the collaboration framework Afterwards, the managerial insights for facilitating the maintenance collaboration are further discussed.

Collaborative Planning and Comprehensive Evaluation Method for Energy Storage System and Flexible Interconnection in Distribution Networks Considering Island Operation

Introduction: When the superior power grid fails and loses power, due to the traditional single power supply radial grid structure of low-voltage substation areas, all loads in the lowvoltage substation area will lose power, causing a poor electricity consumption experience for low-voltage users. To enhance the resilience of low-voltage distribution areas against faults and power outages from higher-tier power grids, a collaborative planning and comprehensive evaluation method for grid-forming energy storage systems (GFM-ESS) and flexible interconnection in distribution networks, considering island operation, is proposed. Method: This method integrates GFM-ESS and flexible interconnections, taking islanding operations into account. Firstly, the structure of GFM-ESS and voltage source converters is analyzed. The control modes of GFM-ESS in parallel or off-grid scenarios are also scrutinized. Secondly, minimizing the annual comprehensive cost and the annual power outage load are the objective functions. A bilevel programming model for GFM-ESS and flexible interconnection of lowvoltage distribution networks, considering island operation, has been established. Moreover, a comprehensive evaluation method based on the analytic hierarchy process for proposed lowvoltage distribution network structural schemes is introduced. An evaluation index system is established to assess the rationality of these proposed network structural planning schemes. Finally, the superiority of the proposed collaborative planning method is verified through a comparative analysis of three planning methods. Result: The proposed collaborative planning method can effectively ensure the continuous power supply of low-voltage distribution networks under the fault scenario of the upper power network. Conclusion: Moreover, the rationality of the results obtained from the proposed collaborative planning method is verified through a comprehensive evaluation method of grid structure schemes based on the analytic hierarchy process. The proposed comprehensive evaluation method can determine which two areas are optimal for grid planning. result: The superiority of the proposed collaborative planning method was verified by comparative analysis of three planning methods. And the rationality of the proposed collaborative planning method results was verified by the comprehensive evaluation method of grid structure schemes based on analytic hierarchy process.

Integrated Optimization of Production Scheduling and Haulage Route Planning in Open-Pit Mines

In mining, deposits are divided into blocks, forming the basis for open-pit mine planning, covering production and haulage route planning. Current studies often stage optimization and lack the consideration of road capacity, leading to suboptimal solutions. A novel approach integrates production scheduling and haulage route planning through a bilevel optimization model. The upper-level model integrates ore mining constraints to establish a mixed-integer production scheduling model, minimizing haulage costs. Spatiotemporal correlation constraints for block mining are determined using a two-stage algorithm. The lower-level model incorporates road capacity, forming a haulage route optimization model based on multicommodity network flow. A solution algorithm with a distance penalty strategy facilitates feedback between the upper and lower levels, achieving optimal solutions. Tested on a real open-pit coal mine with over 5 million blocks, this approach reduces haulage costs by 10.06% compared to stage optimization. Additionally, this approach allows for adjusting haulage demand in both temporal and spatial dimensions, effectively preventing road congestion. This study advances rational mining processes and enhances the efficiency of open-pit mining haulage systems.

A Knee Point Driven Evolutionary Algorithm for Multiobjective Bilevel Optimization.

Bilevel optimization is a special type of optimization in which one problem is embedded within another. The bilevel optimization problem (BLOP) of which both levels are multiobjective functions is usually called the multiobjective BLOP (MBLOP). The expensive computation and nested features make it challenging to solve. Most existing studies look for complete lower-level solutions for every upper-level variable. However, not every lower-level solution will participate in the bilevel Pareto-optimal front. Under a limited computational budget, instead of wasting resources to find complete lower-level solutions that may not be in the feasible region or inducible region of the MBLOP, it is better to concentrate on finding the solutions with better performance. Bearing these considerations in mind, we propose a multiobjective bilevel optimization solving routine combined with a knee point driven algorithm. Specifically, the proposed algorithm aims to quickly find feasible solutions considering the lower-level constraints in the first stage and then concentrates the computational resources on finding solutions with better performance. Besides, we develop several multiobjective bilevel test problems with different properties, such as scalable, deceptive, convexity, and (dis)continuous. Finally, the performance of the algorithm is validated on a practical petroleum refining bilevel problem, which involves a multiobjective environmental regulation problem and a petroleum refining operational problem. Comprehensive experiments fully demonstrate the effectiveness of our presented algorithm in solving MBLOPs.

Estimation of schedule preference and crowding perception in urban rail corridor commuting: An inverse optimization method

This paper introduces an inverse optimization method to uncover commuters’ schedule preference and crowding perception based on aggregated observations from smart card data for an urban rail corridor system. The assessment of time-of-use preferences typically involves the use of econometric models of discrete choice based on detailed travel survey data. However, discrete choice models often struggle with potential endogeneity issues in behavioral observations when estimating individual samples from massive transit data with limited exogenous identifying information. This motivates us to employ an equilibrium modeling approach to capture the dynamism hidden in commuters’ departure time decision-making from aggregations. Assuming user optimality in observed choices, an inverse optimization method is proposed to find a set of preference parameters in the stochastic user equilibrium-based morning commuting model with heterogeneous commuters so that the resulting equilibrium pattern best approximates the observed departure rate distribution over time. The proposed inverse optimization problem can be formulated by a bi-level programming model and a sensitivity analysis-based solution framework is further designed for model estimation. Lastly, the smart card data and train timetable data from the rail corridor along the Beijing Subway Batong Line are synthesized for a case study to estimate commuters’ departure time choice preferences during morning peak periods, as well as to validate the robustness and practicality of the proposed method.

Necessary optimality conditions for strictly robust bilevel optimization problems

The study of robust bilevel programming problems is a relatively new area of optimization theory. In this work, we investigate a bilevel optimization problem where the upper-level and the lower-level constraints incorporate uncertainty. Reducing the problem into a single-level nonlinear and nonsmooth program, necessary optimality conditions are then developed in terms of Clarke subdifferentials. Our approach consists of using the optimal value reformulation together with a partial calmness condition for the robust counterpart of the initial problem. To aid in the detection of Karush-Kuhn-Tucker (KKT) multipliers, an appropriate nonsmooth Mangasarian-Fromovitz constraint qualification is introduced. There are examples highlighting both our results and the limits of certain past studies.

Multi-level and multi-objective structural optimization for hypersonic vehicle design

This research introduces a novel methodology for optimizing the structural design of a full-scale hypersonic aircraft, integrating a multifaceted approach across different levels of modelling detail to enhance a variety of performance metrics. The proposed approach is capable of reduce the vehicle mass, while meeting the necessary operational requirements and maintaining an acceptable computational cost. The methodology, based on a coupled bi-level size optimization and denoted as single-objective bi-level optimization (SOBLO), is applied to the passenger cabin of the STRATOFLY MR3 hypersonic cruiser vehicle. This results in a substantial reduction in the baseline concept design weight, estimated to be more than half on average. The procedure is extended to incorporate a multi-objective optimization approach (MOBLO), which generates a Pareto frontier that provides significant information for selecting the optimal trade-off design, considering manufacturability constraints. The outcomes underscore the efficacy of the proposed methodology and highlight its usefulness in sizing complex aircraft configurations, particularly under the demanding loads imposed by a hypersonic flight regime. This approach has the potential to improve the overall production process and enable designers to attain feasible structural design solutions during early stages of development.

Developing a Bi-Level Optimization Model for the Coupled Street Network and Land Subdivision Design Problem with Various Lot Areas in Irregular Blocks

Street design and land subdivision are significant tasks in the development and redevelopment planning process. Optimizing street and land subdivision layouts within a unified framework to achieve solutions that meet a set of objectives and constraints (e.g., minimizing parcel area deviation from standard values, minimizing land consumption for street construction, etc.) is a critical concern for planners, particularly in complex contexts such as blocks with irregular shapes and parcels of varying sizes and requirements. To address this challenge, a mathematical formulation is presented for the bi-level street network and land subdivision optimization problem. Subsequently, the solution procedure is outlined, which utilizes a genetic-based algorithm for street design and a memetic–genetic-based algorithm for land subdivision. Finally, two cases are presented, solved, and discussed to analyze and verify the proposed mathematical model and solution procedures. The results suggest that the formulated problem is suitable for addressing the coupled street network and land subdivision design problem, and it can be adapted and extended to other case studies. Additionally, the introduced ideas and algorithms satisfactorily solved the stated problem.

Load redistribution attack for power systems with high penetration of EVs

This paper provides insights for cyber defenders of power systems with high penetration of electric vehicles (EVs) by proposing a novel consecutive attack model based on load redistribution for price control in both transmission networks (TNs) and distribution networks (DNs). The target of the attack is to induce massive EV users in DNs to charge simultaneously and cause a spike of loads in the TN at peak hours. The problem is formulated as a multi-slot bi-level optimization problem, where the upper level describes the hacker behavior and attacking constraints. The lower level explains the TN operator and DN operators’ behaviors and the relationship between the local marginal price and retail charging price. The bi-level problem is converted into an equivalent single-level mix-integer linear problem and is solved based on greedy algorithm. Simulations on IEEE 30-bus system prove the effectiveness of the attack strategy.

A two-level game theoretic approach for task offloading in mobile edge computing

In the mobile edge computing system subject to wireless interference, the Edge Server Provider (ESP) aims to offer profitable computing resources to Device Managers (DMs), who make optimal strategies based on the provided prices. However, the existence of mixed variables typically constitutes an NP-hard problem, posing a significant challenge for optimization. In response to address this issue, we formulate a bi-level optimization problem, where the upper level is devoted to optimizing the pricing of computing resources. The lower level optimizes DM migration strategies and resource allocation at specified prices. Leveraging game theory, we achieve distributed and efficient computation offloading by formulating the distributed computing offloading strategy among lower-level DMs as a task offloading game. Our analysis identifies Nash equilibrium and finite improvement characteristics within the game. Based on these insights, we present a bi-level distributed computation offloading algorithm capable of reaching Nash equilibrium, thus optimizing the profit for both DMs and ESP. The experiments have demonstrated the algorithm’s effectiveness in reducing system costs and maximizing the profits of ESP with DMs across diverse scenarios.

A decision support model for planning of spatial large extent biomass to bioenergy procurement network

This study presents a strategic-tactical decision model with two main objectives: (i) identifying the spatial distribution and estimating the harvestable biomass potential of agricultural farms within a complex landscape; and (ii) optimizing the placement of bioenergy facilities to minimize total transportation costs. Remote sensing techniques combined with spatial analyses were utilized to discriminate agricultural lands from other land-cover types. Field-data measurements were then employed to estimate the quantity of available biomass. Subsequently, a bi-level optimization model was developed to determine optimal bioenergy facility locations and calculate transportation costs. The model was applied to a 20000 km2 region in northeastern Iran. The results revealed an average annual biomass density of 4.20 Mg ha−1 from agricultural lands, corresponding to 1867 Gg of biomass across the region. The optimal solution involved placing two bioenergy facilities within a 75 km radius to collect residues biomass, resulting in a delivery cost of $16 kWh−1.

A bilevel programming approach to price decoupling in Pay-as-Clear markets, with application to day-ahead electricity markets

Motivated by the recent crisis of the European electricity markets, we propose the concept of Segmented Pay-as-Clear (SPaC) market, introducing a new family of market clearing problems that achieve a degree of decoupling between groups of participants. This requires a relatively straightforward modification of the standard PaC model and retains its crucial features by providing both long- and short-term sound price signals. The approach is based on dynamically partitioning demand across the segmented markets, where the partitioning is endogenous, i.e., controlled by the model variables, and is chosen to minimise the total system cost. The thusly modified model leads to solving Bilevel Programming problems, or more generally Mathematical Programs with Complementarity Constraints; these have a higher computational complexity than those corresponding to the standard PaC, but in the same ballpark as the models routinely used in real-world Day Ahead Markets (DAMs) to represent “nonstandard” requirements, e.g., the unique buying price in the Italian DAM. Thus, SPaC models should still be solvable in a time compatible with market operation with appropriate algorithmic tools. Like all market models, SPaC is not immune to strategic bidding techniques, but some theoretical results indicate that, under the right conditions, the effect of these could be limited. An initial experimental analysis of the proposed models, carried out through Agent Based simulations, seems to indicate a good potential for significant system cost reductions and an effective decoupling of the two markets.

Using Bi-Level Programming to Evaluate Banking Operations

The importance of performance evaluation in diagnosing the positive and negative cases of excellence of banks appears as it is a necessary means to achieve the extent to which banks can achieve the required level of performance. The research sought to evaluate the performance of banks through the use and application of one of the programming models that is rarely used in banking literature, which is the two-level programming model (quadratic - linear programming). The efficiency index for each banking process is either (1) or (zero). The process that... Achieving an efficiency score of (1) means that it is 100% efficient and falls within the limits of efficiency and achieves the highest returns of profits. As for the banking operation that has an efficiency index of (zero), this means that it does not achieve any return of profits. Thus, applying the two-level programming model determines the extent to which Each bank in the sample, which was taken as a random sample, reached (19) banks out of (65) banks, represented by Al-Rafidain Bank in the city of Baghdad, which was considered as a statistical population for the research for the year 2022. After addressing the theoretical aspect of two-level programming, which is considered one of the improvement issues. It is nested on two levels, one of which is the upper level, called the independent variable (Leader), and the other is represented by the lower level, called the dependent variable (Follower). Each level has a special function and has restrictions. In order to evaluate the upper and lower bounds, the Karch-Cohen-Tucker conditions were used, thus converting two-level programming into single-level programming, and thus applying Program (). The results of the analysis showed that there is a variation in performance between the banks in the research sample, as well as a variation in performance at the level of a single bank depending on the variation in percentages and quantities of profits achieved to which the research sample contributed, and through the results achieved, Al-Mansour Bank obtained first place among the sample banks. Whereas, the value of the objective function achieved for the bank reached (339,636,731), and Al-Mustansir Street Bank ranked (last) among the sample banks, as the value of the objective function achieved for the bank reached (337,791,490). The highest return on profits was achieved by the banks in the research sample from the banking operation (long-term loans) and its value reached (190,193,543), followed by the banking operation (personal advances), which amounted to (15,738,985), then the banking operation (short-term loans), whose value amounted to (8,104,846), and then the banking operation ( Debit current accounts, amounting to (1,847,239), and finally the banking transaction (discounted commercial papers), amounting to (43,720). The research concluded with a set of conclusions and recommendations

A robust bi-level programming model to enhance resilience in supply chain network considering government role, taxing, and raw sale: An iron ore case study

The mining industry plays a substantial role in any economy because of its enormous influence on other sections of a country. Despite the significant challenges in this industry, such as government regulations, taxing policies, raw material sales, and disruptions, there are rare studies that have considered mining supply chain network design. Thus, this paper proposes a mixed-integer linear bi-level programming model under operational and disruption risks to design a robust and resilient mining supply chain network. The government is considered as the leader that aims to decrease the sales of raw materials, boost the export of final products, and maximize its profit. Moreover, the closed-loop mining supply chain is introduced as the follower level that maximizes the total profit. In other words, this study presents a bi-level model that balances the export of raw materials and final products, for the first time. Additionally, a data envelopment analysis approach is applied to rank the candidate mine areas and reduce the model’s complexity. Finally, a real-world case study of the Iranian iron ore supply chain is applied to demonstrate the efficiency and applicability of the addressed model. The results indicate that the effect of raw material price on the upper level is 22% higher than the lower level under the same conditions. Moreover, the exchange rate has a great impact on the objective functions of both levels. Furthermore, Yazd province has great potential to be introduced as a steel pole in Iran.

Optimizing local electricity markets: A bi-level primal-dual approach for integrating price-based demand response and distribution locational marginal pricing

This paper addresses the conundrum of optimizing electricity consumption patterns in response to fluctuations in demand and price, a task managed by both load aggregators and the distribution system operator (DSO). The conventional approaches in the literature to integrate demand response (DR) into optimal power flow (OPF) problems typically overlook the price responsiveness of consumers or simplify power flow equations to account for price-elastic demand.In this paper, we strive to close this gap by introducing a bi-level primal-dual optimization framework that incorporates aggregators' objectives into the market-clearing process while preserving the precision of the OPF equations. At the upper level, the model seeks to minimize both the total payment and peak load. The lower level, structured as a second-order conic (SOC) problem, aims to reduce overall generation costs subject to constraints of the second-order conic (SOC) branch flow model (BFM). The two levels interact through the optimal demand response and distribution locational marginal prices (DLMPs). The principles of convex duality principles together with the strong duality constraint are leveraged to transform the market-clearing problem into a single primal-dual problem. We also circumvent the non-linearity that stems from the DR payment term by incorporating discretizing loads and the big-M method, thus converting the problem into a mixed-integer SOC (MI-SOC) formulation. The merits of the proposed MI-SOC framework are validated through case studies conducted on the IEEE 33-bus test system, showcasing its potential to enforce price-elasticity demand response in distribution system's electricity markets.