Bilevel Programming Research Articles

Optimal scheduling model using the IGDT method for park integrated energy systems considering P2G–CCS and cloud energy storage

To enhance the energy efficiency and financial gains of the park integrated energy system (PIES). This paper constructs a bi-level optimization model of PIES-cloud energy storage (CES) based on source-load uncertainty. Firstly, the scheduling framework of PIES with refined power-to-gas (P2G), carbon capture and storage (CCS) and CES coupling is constructed. Moreover, a bi-level optimization model with the upper tier subject being the PIES operator and the lower tier subject being the CES operator is established under the ladder-type carbon price mechanism with reward and punishment (LCPMRP). Then a proposed entropy weight adaptive information gap decision theory method (EAIGDT) is proposed to eliminate the subjectivity factor and retain its non-probabilistic features while dealing with multiple source-load uncertainties, and according to the operator’s risk preference to build risk-averse (RA) and risk-seeking (RS) strategies, respectively. Finally, the measured data in a certain area of Xinjiang verifies the proposed optimal scheduling method. The results show that the method can effectively take into account the interests of various subjects and realise PIES low-carbon economic operation.

Service product codesign with digital platform operators based on hierarchical interactive optimization

This paper proposes a novel approach to address the evolving landscape of service product design, particularly in collaboration with digital platform operators. By formulating a nonlinear hierarchical interactive optimization model (HIO) based on bilevel programming (BLP), the paper tackles the complexities of joint decision-making between service providers and digital platforms. At the upper level, the model optimizes digital platform selection, product attribute configuration, and pricing to maximize provider profit, while the lower level determines platform participation in codesign to maximize their own profits. Furthermore, a codesign incentive mechanism, encompassing revenue-sharing proportions and platform promotion efforts, is designed to incentivize collaboration.Applied to the tourism industry, the HIO model is solved using a nested genetic algorithm (NGA), allowing for sensitivity analyses on platform contribution and promotion efforts. Results indicate that increased platform participation in codesign may raise configuration complexity and costs for the service agency, potentially impacting profitability. Additionally, higher platform promotion efforts can drive profit growth for both parties, supporting long-term development. Ultimately, the HIO model demonstrates superior performance compared to traditional two-stage method (TSM), underscoring its applicability and effectiveness in addressing codesign challenges between service agencies and digital platforms.

Solar–Hydrogen-Storage Integrated Electric Vehicle Charging Stations with Demand-Side Management and Social Welfare Maximization

The reliable operation of a power system requires a real-time balance between supply and demand. However, it is difficult to achieve this balance solely by relying on supply-side regulation. Therefore, it is necessary to cooperate with effective demand-side management, which is a key strategy within smart grid systems, encouraging end-users to actively engage and optimize their electricity usage. This paper proposes a novel bi-level optimization model for integrating solar, hydrogen, and battery storage systems with charging stations (SHS-EVCSs) to maximize social welfare. The first level employs a non-cooperative game theory model for each individual EVCS to minimize capital and operational costs. The second level uses a cooperative game framework with an internal management system to optimize energy transactions among multiple EVCSs while considering EV owners’ economic interests. A Markov decision process models uncertainties in EV charging times, and Monte Carlo simulations predict charging demand. Real-time electricity pricing based on the dual theory enables demand-side management strategies like peak shaving and valley filling. Case studies demonstrate the model’s effectiveness in reducing peak loads, balancing energy utilization, and enhancing overall system efficiency and sustainability through optimized renewable integration, energy storage, EV charging coordination, social welfare maximization, and cost minimization. The proposed approach offers a promising pathway toward sustainable energy infrastructure by harmonizing renewable sources, storage technologies, EV charging demands, and societal benefits.

Bi-level capacity optimization of electricity-hydrogen coupled energy system considering power curtailment constraint and technological advancement

The instability of renewable energy generation seriously undermines the safe and stable operation of the power system and contributes to the wastage of power resources. Against this background, this paper proposes a hybrid renewable energy system (HRES) coupled with photovoltaic/wind/battery/hydrogen. By considering the minimization of average annual cost as the upper-level objective and the minimization of power shortage and excess generation as the lower-level objective, a bi-level optimization model is constructed. Based on the scenario simulation, this paper analyzes the optimal configuration of HRES under different technology development levels and different power curtailment rate (PCR) constraints. The results indicate the following: (1) At the current technology level, the high installed proportion of wind power and the low cost-effectiveness of energy storage systems enable the high PCR. (2) Increasing the proportion of photovoltaic power generation and expanding the installed capacity of battery storage and hydrogen storage can effectively reduce the PCR of the power system. (3) As technology advances, the growth in levelized cost of electricity (LCOE) driven by declining PCR constraint will diminish, making it possible to fully utilize renewable energy sources.

Multi-objective meta-learning

Meta-learning has arisen as a powerful tool for many machine learning problems. With multiple factors to be considered when designing learning models for real-world applications, meta-learning with multiple objectives has attracted much attention recently. However, existing works either linearly combine multiple objectives into one objective or adopt evolutionary algorithms to handle it, where the former approach needs to pay high computational cost to tune the combination coefficients while the latter approach is computationally heavy and incapable to be integrated into gradient-based optimization. To alleviate those limitations, in this paper, we aim to propose a generic gradient-based Multi-Objective Meta-Learning (MOML) framework with applications in many machine learning problems. Specifically, the MOML framework formulates the objective function of meta-learning with multiple objectives as a Multi-Objective Bi-Level optimization Problem (MOBLP) where the upper-level subproblem is to solve several possibly conflicting objectives for the meta-learner. Different from those existing works, in this paper, we propose a gradient-based algorithm to solve the MOBLP. Specifically, we devise the first gradient-based optimization algorithm by alternately solving the lower-level and upper-level subproblems via the gradient descent method and the gradient-based multi-objective optimization method, respectively. Theoretically, we prove the convergence property and provide a non-asymptotic analysis of the proposed gradient-based optimization algorithm. Empirically, extensive experiments justify our theoretical results and demonstrate the superiority of the proposed MOML framework for different learning problems, including few-shot learning, domain adaptation, multi-task learning, neural architecture search, and reinforcement learning. The source code of MOML is available at https://github.com/Baijiong-Lin/MOML.

Enhancing molecular property prediction with auxiliary learning and task-specific adaptation

Pretrained Graph Neural Networks have been widely adopted for various molecular property prediction tasks. Despite their ability to encode structural and relational features of molecules, traditional fine-tuning of such pretrained GNNs on the target task can lead to poor generalization. To address this, we explore the adaptation of pretrained GNNs to the target task by jointly training them with multiple auxiliary tasks. This could enable the GNNs to learn both general and task-specific features, which may benefit the target task. However, a major challenge is to determine the relatedness of auxiliary tasks with the target task. To address this, we investigate multiple strategies to measure the relevance of auxiliary tasks and integrate such tasks by adaptively combining task gradients or by learning task weights via bi-level optimization. Additionally, we propose a novel gradient surgery-based approach, Rotation of Conflicting Gradients (RCGrad\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathop {\ exttt{RCGrad}}\\limits$$\\end{document}), that learns to align conflicting auxiliary task gradients through rotation. Our experiments with state-of-the-art pretrained GNNs demonstrate the efficacy of our proposed methods, with improvements of up to 7.7% over fine-tuning. This suggests that incorporating auxiliary tasks along with target task fine-tuning can be an effective way to improve the generalizability of pretrained GNNs for molecular property prediction.Scientific contributionWe introduce a novel framework for adapting pretrained GNNs to molecular tasks using auxiliary learning to address the critical issue of negative transfer. Leveraging novel gradient surgery techniques such as RCGrad\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathop {\ exttt{RCGrad}}\\limits$$\\end{document}, the proposed adaptation framework represents a significant departure from the dominant pretraining fine-tuning approach for molecular GNNs. Our contributions are significant for drug discovery research, especially for tasks with limited data, filling a notable gap in the efficient adaptation of pretrained models for molecular GNNs.

Carbon emission allocation policy making in liner shipping: A novel approach toward equitable and efficient maritime sustainability

Maritime shipping, an integral component of global trade, accounts for nearly 3% of global carbon dioxide emissions and faces significant challenges in mitigating its carbon footprints. This study addresses the shortcomings of current carbon emission surcharges levied by shipping companies and the deficiencies of allocation methods utilized by the government, such as the inequity of uniform carbon emission charges across diverse origin–destination (od) pairs and the reliance on potentially inaccurate self-reported emissions data to assign carbon emission allowances. We propose a novel approach to carbon emission allocation per twenty-foot equivalent unit (TEU) container per od pair in liner shipping from the perspective of the government, aiming to enhance government oversight and minimize social costs. Leveraging a bi-level programming model, our research navigates the complex interplay between government regulatory objectives and shipping companies’ operational decisions, identifying a socially optimal carbon allocation policy. Our approach introduces three innovative carbon allocation methods that advance beyond the traditional metric that typically focuses solely on emissions and cargo volume. Our first two innovative methods integrate shipping distances into the carbon allocation process, specifically considering both the actual traveling distance and the shortest sailing distance. Furthermore, we propose the third innovative “harmonic distance,” which is a convex combination of the actual traveling distance and the shortest sailing distance. This measure effectively reflects both the operational realities and equity concerns associated with shipping services. Our findings reveal the effectiveness of utilizing the shortest sailing distance for carbon allocation per TEU per od pair, offering a straightforward, route-independent, and equitable solution. This approach diverges from current industrial practices, favoring a fairer emission allocation basis and balancing the interests of governments, shipping companies, and consumers. Through comprehensive experiments, we demonstrate the superiority of this method in minimizing total social costs and providing a viable path toward environmental sustainability and economic efficiency in liner shipping.

Optimization of Charging Station Capacity Based on Energy Storage Scheduling and Bi-Level Planning Model

With the government’s strong promotion of the transformation of new and old driving forces, the electrification of buses has developed rapidly. In order to improve resource utilization, many cities have decided to open bus charging stations (CSs) to private vehicles, thus leading to the problems of high electricity costs, long waiting times, and increased grid load during peak hours. To address these issues, a dual-layer optimization model was constructed and solved using the Golden Sine Algorithm, balancing the construction cost of CSs and user costs. In addition, the problem was alleviated by combining energy storage scheduling and the M/M/c queue model to reduce grid pressure and shorten waiting times. The study shows that energy storage scheduling effectively reduces grid load, and the electricity cost is reduced by 6.0007%. The average waiting time is reduced to 2.1 min through the queue model, reducing the electric vehicles user’s time cost. The bi-level programming model and energy storage scheduling strategy have positive implications for the operation and development of bus CSs.

Improving capacity configuration and load scheduling for an integrated multi-sourced cogeneration system

The integration of renewable energy sources and fossil fuel e.g. coal in a heat and power cogeneration system has been proven to be an effective way to reduce CO2 emissions. An integrated multi-sourced cogeneration (IMC) system consisting of wind, solar photovoltaic (PV), and coal-fired combined heat and power (CHP) plant on the energy supply side, is studied in this paper. A bi-level collaborative optimization model for capacity configuration and load scheduling of the wind-PV-coal IMC system is developed. In the upper level, the capacity of wind farm and PV farm are optimized with the enumeration method, while the load scheduling under typical annual conditions is optimized with adaptive particle swarm optimization algorithm in the lower level. The results show that the optimal capacities of wind and PV are 120 MW and 280 MW for the IMC system, respectively. The annual and typical day load scheduling with optimal wind and PV capacities are analyzed. The result of the load scheduling shows that heating and power load distribution unevenly between the two CHP units contributes to reducing total standard coal consumption, which saves 34.96 t in a typical day during the heating season. This study can provide a reference for the capacity configuration and operation scheduling of the IMC system.

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.

A new bi-level model for the false data injection attack on real-time electricity market considering uncertainties

State Estimation (SE) plays a critical role in various applications of power systems including the electricity market. False Data Injection Attacks (FDIAs) are capable of manipulating the power system SE process aiming to gain financial profits by the players. In this paper, the potential economic influences of FDIA on SE and consequently the Real-Time (RT) electricity market are examined. Therefore, a new bi-level optimization framework is proposed to realistically model the FDIAs from the attacker perspective. At the upper-level, the attacker intends to intelligently choose the most critical congested transmission lines and manipulate their associated meter readings to maximize the expected financial profitability. At the lower-level, the RT market-clearing model is formulated under the influence of FDIA. Considering the incomplete attacker's prior knowledge of the network topology i.e., connection/disconnection of transmission lines, the angles of the phase-shifting transformers, and the real-time load uncertainty, the proposed formulation is extended to a new bi-level scenario-driven stochastic model. The developed attack model, which must remain undetectable by the system operator in all scenarios, is solved using the Karush–Kuhn–Tucker (KKT) approach. The correctness and effectiveness of the proposed optimization scheme have been validated in GAMS software using CONOPT and OQNLP solvers based on the IEEE 14-bus test system and also comparative results are presented to demonstrate the merits of the proposed formulation.

Aggregator-supported strategy for electric bus fleet charging: A hierarchical optimisation approach

The increasing adoption of electric buses in cities presents an opportunity for these vehicles to serve as mobile energy storage systems, providing valuable grid services through their substantial battery capacity. However, the complex interplay of fleet management and energy market participation requires that public transportation operators (PTOs) use adequate optimisation models and algorithms to make the best operational decisions. To address these issues, our research proposes an aggregator-supported charging strategy using a bi-level optimisation framework for electric bus fleets. The main goal is to achieve a balance between reducing PTO operational costs and maximising aggregator revenues, while ensuring reliable bus system operation. The results indicate that a coordinated charging strategy, including energy arbitrage and reserve provision, can significantly reduce operational costs, although potential limitations arise from associated battery degradation costs. Furthermore, managing fleets for reserve services appears to be more lucrative than engaging in energy arbitrage schemes. In this context, this study emphasizes the role of aggregator entities in managing the energy flexibility of electric bus fleets and enhancing the sustainability and cost-effectiveness of urban public transportation.

Research on Risk-Averse Procurement Optimization of Emergency Supplies for Mine Thermodynamic Disasters

Reducing procurement risks to ensure the supply of emergency supplies is crucial for mitigating the losses caused by mine thermodynamic disasters. The risk preference of decision-makers and supply chain collaboration are the important aspects for this reductiom. In this study, a novel P-CVaR (Piecewise conditional risk value) distributionally robust optimization model is proposed to accurately assist the decision-makers’ decision of risk preference for reducing procurement risks. Meanwhile, the role of cooperation between procurement and reserves are considered for the weakening procurement risks. A risk-averse bi-level optimization model is proposed to obtain the optimal procurement strategy. Furthermore, by applying the Lagrange duality theorem, the complexity of the bi-level optimization model is simplified then solved using a PSO algorithm. Using empirical analysis, it has been verified that the model presented in this paper serves as a valuable guideline for mine thermodynamic pre-disaster emergency material procurement strategies for the prevention of thermodynamic disasters.

A bilevel optimization approach of energy transition in freight transport: SOS1 method and application to the Ecuadorian case

A bilevel optimization approach of energy transition in freight transport: SOS1 method and application to the Ecuadorian case

Bilevel Optimization of the Kantorovich Problem and Its Quadratic Regularization

This paper is concerned with an optimization problem which is governed by the Kantorovich problem of optimal transport. More precisely, we consider a bilevel optimization problem with the underlying problem being the Kantorovich problem. This task can be reformulated as a mathematical problem with complementarity constraints in the space of regular Borel measures. Because of the non-smoothness that is induced by the complementarity constraints, problems of this type are often regularized, e.g., by an entropic regularization. However, in this paper we apply a quadratic regularization to the Kantorovich problem. By doing so, we are able to drastically reduce its dimension while preserving the sparsity structure of the optimal transportation plan as much as possible. As the title indicates, this is the second part in a series of three papers. While the existence of optimal solutions to both the bilevel Kantorovich problem and its regularized counterpart were shown in the first part, this paper deals with the (weak-∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$*$$\\end{document}) convergence of solutions to the regularized bilevel problem to solutions of the original bilevel Kantorovich problem for vanishing regularization parameters.

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.