Bilevel Programming Problem Research Articles

An elementary approach to solving the optimal taxation of a perfectly competitive firm with CES production function as a bilevel programming problem

An elementary approach to solving the optimal taxation of a perfectly competitive firm with CES production function as a bilevel programming problem

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.

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.

On Approximate Variational Inequalities and Bilevel Programming Problems

On Approximate Variational Inequalities and Bilevel Programming Problems

Distributed coordinated control of mixed expressway and urban roads: a sensitivity-based approach

The inherent interdependency between mixed expressway and urban road networks calls for an effective coordinated traffic control approach. This study introduces a sensitivity-based distributed method to coordinate ramp metering and traffic signal control. Firstly, the quantitative relationships between network nodes are established through a developed traffic flow model for mixed networks. The network-level sensitivity information is analyzed to assess the influence of controllers on the traffic state. Additionally, a dynamic subproblems decomposition algorithm is proposed, considering the network’s spatial structure and dynamic traffic characteristics. By employing community detection techniques, tightly connected controllers are clustered to enable efficient and adaptive distributed control. The optimization problem is then decomposed into several subproblems, each formulated as a bi-level programming problem within a model predictive control framework. Numerical experiments demonstrate that the proposed strategy enhances the traffic efficiency of mixed networks with a 14.5% reduction in total travel time during peak hours, which shows better performance compared with the other three existing control strategies.

A quadratic-linear bilevel programming approach to green supply chain management

Green Supply Chain Management requires coordinated decisions between the strategic and operational organization layers to address strict green goals. Furthermore, linking CO2 emissions to supply chain operations is not always easy. This study proposes a new mathematical model to minimize CO2 emissions in a three-layered supply chain. The model foresees using a financial budget to mitigate emissions contributions and optimize supply chain operations planning. The three-stage supply chain analyzed has inbound logistics and handling operations at the intermediate level. We assume that these operations contribute to emissions quadratically. The resulting bilevel programming problem is solved by transforming it into a nonlinear mixed-integer program by applying the Karush-Kuhn-Tucker conditions. We show, on different sets of synthetic data and on a case study, how our proposal produces solutions with a different flow of goods than a modified linear model version. This results in lower CO2 emissions and more efficient budget expenditure.

Iterative convex relaxation of unbalanced power distribution system integrated multi-energy systems

This study describes a multi-period, multi-energy operation of a three-phase unbalanced Electric Distribution System (EDS) that coordinates with a District Heating System (DHS) and a Natural Gas (NG) distribution system. The electrical subsystem problem is formulated as a bi-level programming problem, with levels 1 and 2 solving the subsystem’s linearized and relaxed nonlinear versions, respectively. Second-Order Cone Programming (SOCP) and polyhedral relaxations are availed to circumvent the non-convexities involved in the subsystems. The SOCP relaxation for an unbalanced power distribution network is inexact; also, the subsystems applying polyhedral relaxation may not generate a meaningful solution as the relaxations are not tight. Therefore, a solution recovery algorithm is developed for each subsystem to recover a feasible solution from their relaxed counterparts. The successive bound tightening algorithm employing a solution recovery procedure is proposed for each subsystem, improving solution quality and strengthening the relaxations with the desired computational efficiency. The proposed solution strategy to optimize the Multi-Energy System (MES) operation cost is verified on the three-phase IEEE-13 and IEEE-123 bus systems, each coordinating with a 30-node DHS and a 6-bus NG network. The results analyses demonstrate that the proposed solution strategy efficiently achieves an optimal solution, reducing maximum relaxation error below 0.1% for each subsystem. The proposed strategy delivered a 14.58% reduction in real power losses and a 12.73% decrease in phase voltage unbalance rate for the EDS in MES. Furthermore, a 1.96% decrease in operational cost demonstrates the techno-economic benefits of the proposed strategy.

A Problem based on Lagrange Relaxation Approach & Solving Optimization on Bi-level Model Programming Problem

Abstract: In this paper, a salutation method based on Lagrange relaxation for discrete-continuous bi-level problems, with binary variables in the leading problem, considering the optimistic approach in bi-level programming. Linearized taking advantage of the structure of the leading problem, an algorithm is solving two-dimensional bi-level linear programming problems, classification of constrains, algorithm removes all redundant constraints, cycling and solution of problem.

Utilization of neutrosophic Kuhn-Tucker’s optimality conditions for Solving Pythagorean fuzzy Two-Level Linear Programming Problems

This article considers a bi-level linear programming with single valued trapezoidal fuzzy neutrosophic cost coefficient matrix and Pythagorean fuzzy parameters in the set of constraints both in the right and left sides. Based on the score functions of the neutrosophic numbers and Pythagorean fuzzy numbers, the model is changed to the corresponding crisp bi-level linear programming (BLP) problem. This problem is designated as a Pythagorean fuzzy bi-level linear programming (PFBLP) problem under neutrosophic environment. Kuhn-Tucker's conditions for optimality are necessary and sufficient for the existence of the optimal solution to a BLP problem. Using the suggested methodology, the problem is formulated as a single-objective non-linear programming problem with several variables and constraints. Two typical numerical examples are examined to illustrate the proposed approach.

A nonmonton active interior point trust region algorithm based on CHKS smoothing function for solving nonlinear bilevel programming problems

<abstract><p>In this paper, an approach is suggested to solve nonlinear bilevel programming (NBLP) problems. In the suggested method, we convert the NBLP problem into a standard nonlinear programming problem with complementary constraints by applying the Karush-Kuhn-Tucker condition to the lower-level problem. By using the Chen-Harker-Kanzow-Smale (CHKS) smoothing function, the nonlinear programming problem is successively smoothed. A nonmonton active interior-point trust-region algorithm is introduced to solve the smoothed nonlinear programming problem to obtain an approximately optimal solution to the NBLP problem. Results from simulations on several benchmark problems and a real-world case about a watershed trading decision-making problem show how the effectiveness of the suggested approach in NBLP solution development.</p></abstract>

Fractional programming for Stackelberg game under type-2 fuzzy environment

This study intends to present a Stackelberg game model for design of the fractional type-2 fuzzy programming. In achieving this aspiration, we develop a probabilistic fuzzy multi-objective fractional linear programming where the entire parameters are of type-2 fuzzy numbers apart from the right-hand side of the constraints are follow Weibull distribution. In the projected approach, the membership function allied with each objective function is generated by using the first-order Taylor series approximation and converted into a single objective function by assuming the weights of the objective functions are equal. Type conversion is made in two ways by existing methods, and using stochastic programming, the probabilistic constraints are transformed into a deterministic form. The accessible model incorporates the non-linear programming viewpoint of the decision-maker and is solved with the help of intuitionistic fuzzy programming (IFS). A comparison study on the optimum results by genetic algorithm (GA) and particle swarm optimization (PSO) with the LINGO 15.0 iterative scheme is offered to resolve the created bi-level programming problem (BLPP) in the course of the Stackelberg game. To make obvious the feasibility of the projected representation and solution methodology, realistic data are measured and results are presented through several discussions.

Approximate Karush-Kuhn-Tucker condition for multi-objective optimistic bilevel programming problems

Approximate Karush-Kuhn-Tucker condition for multi-objective optimistic bilevel programming problems

Bi-level planning approach for incorporating the demand-side flexibility of cloud data centers under electricity-carbon markets

With the prevalence of cloud computing applications, cloud data centers (CDCs) are proliferating around the world. However, in the context of hybrid energy‑carbon markets environment, the high energy costs and environmental expenses caused by CDC operation are pressing CDC owners to restructure the future development of CDC in a more economical and low-carbon manner. The potential spatio-temporal transferability and reducibility of workloads provides CDCs with significant flexibility in their operations and thus may interact with the power grid as active demand users. Nonetheless, other than private data centers, public CDCs have no direct control over the workloads submitted by terminal cloud users. As such, this paper presents a bi-level model for CDC allocation planning, so as to incorporate cloud service-demand response (CS-DR) from the perspective of a hybrid electricity-carbon market. The upper level pertains to multi-domain resource collaborative planning model, which determines the optimum siting and sizing of CDCs as well as the incentive design for CS-DR program, with the objective of maximizing the total expected benefits of CDC. The lower level models correspond to the market clearing (electricity-carbon tariff model) of Independent system operator (ISO) and the decision-making of cloud users regarding CS-DR participation. The proposed model belongs to a bi-level mixed integer nonlinear programming problem with two non-convex lower levels, which can be intractable in mathematics. To solve such difficult problem, a hybrid solution method combining multiple linearization techniques and reformulation and decomposition (R&D) strategy based on column-and-constraint generation (C&CG) algorithm is developed. The proposed model is demonstrated on a modified IEEE 30-bus test case, and the simulation results verified the effectiveness of the proposed approach.

Sustainable road network design considering hydrogen fuel cell vehicles

Environmental pollution and energy shortages have brought about an increased focus on new energy vehicles. Hydrogen fuel cell vehicles (HFCVs) have experienced rapid development due to the potential to alleviate energy pressures and reduce pollution emissions. Near zero-carbon emissions offer a promising avenue for promoting sustainable transportation development. To evaluate the impact of HFCVs on the transportation environment, this paper investigates the problem of sustainable transportation network design including HFCVs. Specifically, the problem is formulated as a bi-level multi-objective programming problem, with the upper level aimed at determining the optimal network design scheme considering multiple objectives, while the lower level addresses the mixed traffic flow that comprises both HFCVs and fuel vehicles. To solve the multi-objective sustainable network design problem, an integrated solution framework that combines the particle swarm optimization (PSO) algorithm with the Frank-Wolfe algorithm (FW) is developed. Specifically, the PSO algorithm is utilized to solve the upper-level model and identify the optimal network design schemes, while FW algorithm is adopted to handle mixed traffic flow assignments. Finally, the proposed model and algorithm are implemented in two numerical experiment to demonstrate their effectiveness and efficiency.

Two regularized inertial Tseng methods for solving inclusion problems with applications to convex bilevel programming

Combining the inertial technique, regularization, and the Tseng method for approximating a zero of the sum of monotone operators in a real Hilbert space, we propose two regularized inertial Tseng methods for approximating a zero of the sum of two maximal monotone operators in the case where one of them is monotone and Lipschitz continuous. We establish strong convergence of the sequences generated by our algorithms and derive several interesting consequences of our results. We then apply our results to solving convex bilevel programming problems and illustrate the cost efficiency of our methods by numerically comparing them with other methods in the literature.

Sizing and energy management of standalone hybrid renewable energy systems based on economic predictive control

This paper aims to design and run a standalone hybrid renewable energy System (HRES) feeding residential building load. The suggested HRES encompasses photovoltaic panels and wind turbines as energy sources, combined with a hydrogen subsystem and a bank of batteries for energy storage. The meteorological input data includes irradiance, temperature, and wind speed for four seasonal weeks. These data sets are collected for a rural site situated in the town of Ras Munif, northern Jordan. The load profiles characterize a selected benchmark residential building. Consequently, a bi-level mixed integer nonlinear programming (BMINLP) problem is formulated to incorporate the designing problem with the energy management process. The upper level represents the sizing problem, while the lower level problem denotes the energy management strategy (EMS). Each problem has its independent objective function, constraints, and optimization variables. The EMS is an economic-based single objective optimization problem expressed as a mixed integer linear programming (MILP) problem. On the contrary, a mixed integer nonlinear programming (MINLP) approach frames the sizing problem to optimize a techno-economic-based multi-objective function. The interaction between the two problems is that the lower problem is an embedded constraint of the upper problem. Moreover, the candidate solutions from the upper problem are treated as fixed parameters in the lower problem. A multi-objective genetic algorithm is used to execute the overall problem by the MATLAB toolbox. The technical and economic performance parameters, including the investment cost, maintenance, operating costs, and others, are assessed based on the best findings. The key finding demonstrates that the summer week has the lowest total cost and energy loss factor, with around 132,000 $ and 0, respectively. Furthermore, the findings reveal that the entire system's energy loss factor for each sample period is less than 0.0009. This value is substantially less than the permitted limit of 0.01.

A novel framework for the day-ahead market clearing process featuring the participation of distribution system operators and a hybrid pricing mechanism

The electricity market liberalization propelled by the increased operational flexibility observed in distribution systems urges the development of mathematical models for competitive electricity markets accounting for the participation of distribution systems operators. Most existing approaches model the interaction between the agents and the market as a bilevel programming problem, wherein the market is characterized at the lower level. Hence, the upper-level agent’s objective function is favored over minimizing the costs of supplying the system’s demand. In this paper, a new bilevel formulation is proposed. In the upper level, the market operator determines the optimal energy price that ensures the demand supply and maximizes the overall social welfare. In the lower level, the generation companies and distribution systems operators determine the amount of energy injected/extracted into/from the transmission system to maximize profits/minimize operational costs. A new pricing method is proposed in this paper as an alternative to the transmission system’s dual variables, which are inaccessible under the proposed framework. The resulting nonlinear mixed-integer bilevel programming formulation is transformed into an equivalent single-level mixed-integer linear program. Numerical results illustrate the proposed approach’s effectiveness in maintaining the minimum operational costs while distributing the payoff amongst the agents.

War game problem considering the mobility of weapons and targets

War-Gaming is recognized as a valuable tool for commanders, leaders, and managers. Well-executed War-Games have delivered significant competitive advantages in numerous conflicts. The war-game confirmed the commanders’ knowledge of weapon systems and performance, as well as the time and space necessary to carry out battlefield maneuvers. One of the primary missions of each army on the battlefield is weapon target assignment. The weapon target assignment (WTA) is a critical problem to command to be solved in battlefield decisions. In a WTA problem, we should assign available weapons to determined targets appropriately to optimize the performance criteria. This study discusses a problem in relation to allocating and scheduling in WTA considering the mobility weapons and mobility targets. Bi-level linear programming problem is defined so that each level independently optimizes its own objective functions but is influenced by actions taken by another unit. To solve the under studied problem, three famous meta-heuristic algorithms including simulated annealing (SA), genetic algorithm (GA) and grey wolf optimizer (GWO) methods are proposed. Since the performance of meta-heuristic algorithms depends on setting the parameters, the Taguchi method has been used statistically for set parameters of the developed Algorithms. Performance evaluation of the presented algorithms is conducted through numerical experiments involving randomly generated test problems. To compare the results of proposed meta-heuristic algorithms, ANOVA and Tukey tests were used. The Computational results have shown that proposed GWO algorithm worked better than the SA and GA algorithms.

A note on optimality conditions in terms of convexificators for a bilevel multiobjective optimization problem

Necessary optimality conditions for a vectorial bilevel programming problem are given by Dempe et al. [Optimization 69, 1811–1830, 2020]. The main result, Theorem 1, is based on a result of Dutta and Chandra [J. Optim. Theory Appl. 113, 41–64, 2002] which turned out to be incorrect. In this work, we propose an improvement to Dempe et al.'s result that does not use any convexity-type restriction.

Optimizing the Growing Dual Credit Requirements for Automobile Manufacturers in China’s Dual Credit Policy

Dual credit policy (DCP) is a market-based mechanism introduced by the Chinese government to promote the new energy vehicle (NEV) industry and improve energy savings in China. To offer sufficient impetus for the NEV industry while providing sufficient transitional buffer time for automobile manufacturers (AMs), the government needs to scientifically design how to gradually increase its dual credit requirement for AMs year by year. To assist the multi-year DCP design, this paper proposes a generalized Nash equilibrium model to predict AMs’ short-term decisions (i.e., vehicle production and credit trading) and long-term decisions (i.e., investment in production capacity expansion and research and development) under any DCP, considering the interactions among AMs’ decisions, vehicle prices, and credit price. Based on the equilibrium model, we then develop a bi-level programming problem to optimize the multi-year DCP. With numerical experiments, we show that implementing the optimal DCP can effectively enhance the market share of NEVs. In the context of the optimal multi-year DCP, the credit requirements set by the government should maintain a relatively low threshold during the initial years, but rise rapidly after that. Such optimal DCP offers AMs sufficient transition time while compelling a quick shift in their developmental strategies.