Bilevel Problem Research Articles

An aspect of bilevel interval linear fractional transportation problem with disparate flows: a fuzzy programming approach

An aspect of bilevel interval linear fractional transportation problem with disparate flows: a fuzzy programming approach

Iterative Backpropagation Method for Efficient Gradient Estimation in Bilevel Network Equilibrium Optimization Problems

Network optimization or network design with an embedded traffic assignment (TA) to model user equilibrium principle, sometimes expressed as bilevel problems or mathematical programs with equilibrium constraints (MPEC), is at the heart of transportation planning and operations. For applications to large-scale multimodal networks with high dimensional decision variables, the problem is nontrivial, to say the least. General-purpose algorithms and problem-specific bilevel formulations have been proposed in the past to solve small problems for demonstration purposes. Research gap, however, exists in developing efficient solution methods for large-scale problems in both static and dynamic contexts. This paper proposes an efficient gradient estimation method called Iterative Backpropagation (IB) for network optimization problems with an embedded static TA model. IB exploits the iterative structure of the TA solution procedure and simultaneously calculates the gradients while the TA process converges. IB does not require any additional function evaluation and consequently scales very well with higher dimensions. We apply the proposed approach to origin-destination (OD) estimation, an MPEC problem, of the Hong Kong multimodal network with 49,806 decision variables, 8,797 nodes, 18,207 links, 2,684 transit routes, and 165,509 OD pairs. The calibrated model performs well in matching the link counts. Specifically, the IB-gradient based optimization technique reduces the link volume squared error by 98%, mean absolute percentage error (MAPE) from 95.29% to 21.23%, and the average GEH statistics from 24.18 to 6.09 compared with the noncalibrated case. The framework, even though applied to OD estimation in this paper, is applicable to a wide variety of optimization problems with an embedded TA model, opening up an efficient way to solve large-scale MPEC or bilevel problems. Funding: The study is supported by IVADO Postdoctoral Fellowship scheme 2021, HSBC 150th Anniversary Charity Programme HKBF17RG01, National Science Foundation of China (71890970, 71890974), General Research Fund (16212819, 16207920) of the HKSAR Government, and the Hong Kong PhD Fellowship.

Multilevel Thresholding of Color Image Segmentation Using Memory-based Grey Wolf Optimizer With Otsu Method, Kapur, and M.Masi Entropy

Determining the optimal threshold value for image segmentation has become more attention in recent years because of its varied uses. Otsu-based thresholding methods, minimum cross entropy, and Kapur entropy are efficient for solving bi-level thresholding image segmentation problems (BL-ISP), but not with multi-level thresholding image segmentation problems (ML-ISP). The main problem is exponentially increasing computational complexity. This study uses the memory-based Gray Wolf Optimizer (mGWO) to determine the optimal threshold value for solving ML-ISP on RGB images. The mGWO method is a variant of the standard grey wolf optimizer (GWO) that utilizes the best track record of each individual grey wolf for the global exploration and local exploitation phases of the problem solution space. The solution candidates are represented by each grey wolf using the image intensity values and optimized according to mGWO characteristics. Three objective functions, namely the Otsu method, Kapur Entropy, and M.Masi Entropy are used to evaluate the solutions generated in the optimization process. The GridSearch method is used to determine the optimal parameter combination of each method based on 10 training images. Evaluation of the performance of the mGWO method was measured using several benchmark images and compared with five standard swarm intelligence (SI) methods as benchmarks. Analysis of the results was carried out qualitatively and quantitatively based on the average PSNR, RMSE, SSIM, UQI, fitness value, and CPU processing time from 30 tests. The results were analyzed further with the Wilcoxon signed-rank test. The experimental results show that the performance of the mGWO method outperforms the benchmark method in most experiments and metrics. The mGWO variant also proved to be superior to the standard GWO in resolving multi-level color image segmentation problems. The mGWO performance results are also compared with other state-of-the-art SI methods in solving ML-ISP on grayscale images and was able to outperform those methods in most experiments.

Modelling a bi-level multi-objective post-disaster humanitarian relief logistics network design problem under uncertainty

Post-disaster humanitarian relief logistics focuses on managing the supply and shipment of relief items. Owing to the complexity arising from the hierarchical decision relationship and the uncertainties of demand and transportation, it is challenging to develop an effective humanitarian relief logistics network (HRLN). A globalized robust bi-level multi-objective HRLN model involving uncertainties in transportation time, transportation cost and demand is developed to address this problem, in which an upper-level decision-maker focuses on the allocation problem and a lower-level decision-maker considers the supply problem. For tractability, the goal programming approach is employed to trade off conflicting timeliness, satisfaction and operational cost objectives. The proposed globalized robust bi-level HRLN model is reformulated as its globalized robust counterpart formulations under two different perturbation structures. The primal–dual approach is adopted to transform the bi-level models into equivalent mixed-integer conic programs. A case study concerning an Iranian earthquake is conducted to prove the performance of the proposed model. Globalized robust models are more flexible and able to immunize against uncertainty by comparison with traditional robust optimization and deterministic solutions. The managerial insights of using the globalized robust bi-level optimization approach are reported for disaster response management. The design of an emergency aviation network to circumvent the effects of COVID-19 could be considered in future research.

Handling Imbalanced Classification Problems With Support Vector Machines via Evolutionary Bilevel Optimization.

Support vector machines (SVMs) are popular learning algorithms to deal with binary classification problems. They traditionally assume equal misclassification costs for each class; however, real-world problems may have an uneven class distribution. This article introduces EBCS-SVM: evolutionary bilevel cost-sensitive SVMs. EBCS-SVM handles imbalanced classification problems by simultaneously learning the support vectors and optimizing the SVM hyperparameters, which comprise the kernel parameter and misclassification costs. The resulting optimization problem is a bilevel problem, where the lower level determines the support vectors and the upper level the hyperparameters. This optimization problem is solved using an evolutionary algorithm (EA) at the upper level and sequential minimal optimization (SMO) at the lower level. These two methods work in a nested fashion, that is, the optimal support vectors help guide the search of the hyperparameters, and the lower level is initialized based on previous successful solutions. The proposed method is assessed using 70 datasets of imbalanced classification and compared with several state-of-the-art methods. The experimental results, supported by a Bayesian test, provided evidence of the effectiveness of EBCS-SVM when working with highly imbalanced datasets.

Simultaneous design and NMPC control under uncertainty and structural decisions: A discrete‐steepest descent algorithm

AbstractIn this article, we address the integration of design and nonlinear model‐based control under uncertainty and structural decisions for naturally ordered structures. We propose an algorithmic framework to determine the optimal location of process units or streams over an ordered discrete set that operates in closed‐loop with a model‐based controller. The formulation corresponds to a mixed‐integer bilevel problem (MIBLP) that is transformed into a single‐level mixed‐integer nonlinear problem (MINLP) using a KKT transformation strategy. In our methodology, the integer decisions are partitioned into subsets called external variables, such that the MINLP is decomposed into an integer‐based master problem and primal subproblems with fixed discrete variables. The master and primal problems are solved using a Discrete‐Steepest Descent Algorithm (D‐SDA). We illustrate the discrete‐based methodology in a case study for a binary distillation column. The D‐SDA showed an improved performance compared to a benchmark continuous‐based formulation using differentiable distribution functions (DDFs).

Quantum computing for transport network design problems

Transport network design problem (TNDP) is a well-studied problem for planning and operations of transportation systems. They are widely used to determine links for capacity enhancement, link closures to schedule maintenance, identify new road or transit links and more generally network enhancements under resource constraints. As changes in network capacities result in a redistribution of demand on the network, resulting in changes in the congestion patterns, TNDP is generally modelled as a bi-level problem, which is known to be NP-hard. Meta-heuristic methods, such as Tabu Search Method are relied upon to solve these problems, which have been demonstrated to achieve near optimality in reasonable time. The advent of quantum computing has afforded an opportunity to solve these problems faster. We formulate the TNDP problem as a bi-level problem, with the upper level formulated as a Quadratic Unconstrained Binary Optimization (QUBO) problem that is solved using quantum annealing on a D-Wave quantum computer. We compare the results with Tabu Search. We find that quantum annealing provides significant computational benefit. The proposed solution has implications for networks across different contexts including communications, traffic, industrial operations, electricity, water, broader supply chains and epidemiology.

Optimal scheduling of ethylene plants under uncertainty: An unsupervised learning-based data-driven strategy

Ethylene cracking furnaces are core units that determine the yield of key products, and its cycle scheduling optimization is of great significance to improve the efficiency of the whole plant. However, uncertainties in cracking batch durations may lead to sub-optimal or ineffective scheduling arrangements. This paper develops a novel data-driven stochastic robust optimization strategy to achieve the optimal scheduling of cracking furnaces under uncertainty. The proposed stochastic robust optimization model applies machine learning methods to extract data information and is represented as a bilevel problem: the outer layer is a two-stage stochastic programming, where typical scenarios are clustered by a three-level data mining approach; the inner layer is robust optimization, whose uncertainty set is constructed by an efficient kernel-based approach. An actual ethylene cracking system is investigated and the results show that the proposed method achieves the highest daily profit, performing well in data coverage (92.3%) and change rate (0.826%). Moreover, the sensitivity analysis for the number of clusters and regularization parameter values is performed.

Exact methods for discrete {varGamma }-robust interdiction problems with an application to the bilevel knapsack problem

Developing solution methods for discrete bilevel problems is known to be a challenging task—even if all parameters of the problem are exactly known. Many real-world applications of bilevel optimization, however, involve data uncertainty. We study discrete min-max problems with a follower who faces uncertainties regarding the parameters of the lower-level problem. Adopting a varGamma -robust approach, we present an extended formulation and a multi-follower formulation to model this type of problem. For both settings, we provide a generic branch-and-cut framework. Specifically, we investigate interdiction problems with a monotone varGamma -robust follower and we derive problem-tailored cuts, which extend existing techniques that have been proposed for the deterministic case. For the varGamma -robust knapsack interdiction problem, we computationally evaluate and compare the performance of the proposed algorithms for both modeling approaches.

An inertial projection and contraction method for solving bilevel quasimonotone variational inequality problems

An inertial projection and contraction method for solving bilevel quasimonotone variational inequality problems

Algorithmic schemes for non-smooth simple bilevel programming problem and simple MPEC problem

ABSTRACT This article continues our study on simple bilevel and simple MPEC problems. In this article, we focus on developing algorithms. We show how using the idea of a gap function one can represent a simple MPEC as a simple bilevel problem with non-smooth data. This motivates us first to develop an algorithm for a simple bilevel problem with non-smooth data and modify the scheme for the same to develop an algorithm for the simple MPEC problem. We also discuss how the simple bilevel formulation of a simple MPEC can help us in formulating a stopping criteria for the simple MPEC problem.

Solution techniques for Bi-level Knapsack Problems

Traditional funding mechanisms for healthcare projects involve ranking the projects and awarding funds based on their cost to benefit ratio. An alternative funding mechanism based on Bi-level programming was proposed in the literature. We refer to this as Donor-Recipient Bi-level Knapsack Problem (DR-BKP), which we explore further in this work. There are two participants, a leader (a donor agency) and a follower (recipient country) in this problem. Both the participants have their individual budgets. There is a set of projects, each having a certain cost and profit associated. The cost of projects are common to both the participants however the profits can be different for them. There is an external project that is of exclusive interest to the follower. The leader decides on cost subsidies to provide for the projects that is within her budget, while the follower solves a knapsack problem with the cost subsidised projects and the external project. Two enumerative algorithms were proposed in the literature for Bi-level problems with discrete upper level variables. We adapt them for DR-BKP that has continuous upper level variables having non-linear interaction with lower level variables. We first show the existence of a solution for DR-BKP and show the convergence of these algorithms. We provide evidence for Σ2P-hardness by showing that the problem is both NP-hard and Co-NP hard. Finally, we have implemented these two enumerative algorithms and shared the results and analyses of the computational experiments. A set of fifteen differing data sets each having randomly generated 10 instances have been solved to evaluate the performance of the proposed algorithms.

Network-Constrained Transactive Control for Multi-Microgrids-Based Distribution Networks With Soft Open Points

Different from most transactive control studies only focusing on economic aspect, this paper develops a novel network-constrained transactive control (NTC) framework that can address both economic and secure issues for a multi-microgrids-based distribution network considering uncertainties. In particular, we innovatively integrate a transactive energy market with the novel power-electronics device (i.e., soft open point) based AC power flow regulation technique to improve economic benefits for each individual microgrid and meanwhile ensure the voltage security of the entire distribution network. In this framework, a dynamic two-timescale NTC model consisting of slow-timescale pre-scheduling and real-time corrective scheduling stages is formulated to work against multiple system uncertainties. Moreover, the original bilevel game problems are transformed into a single-level mixed-integer second-order cone programming problem through KKT conditions, duality, linearization and relaxation techniques to avoid iterations of transitional methods, so as to improve the solving efficiency. Finally, numerical simulations on modified 33-bus and 123-bus test systems with multi-microgrids verify the effectiveness of the proposed framework.

Blockchain-Enabled Carbon and Energy Trading for Network-Constrained Coal Mines With Uncertainties

In this paper, a blockchain-enabled distributed market framework is proposed for the bi-level carbon and energy trading between coal mine integrated energy systems (CMIESs) and a virtual power plant (VPP) with network constraints. To maximize the profits of these two entities and describe their complicated interactions in the market, the bi-level trading problem is formulated as a Stackelberg game considering integrating the energy market and the “cap-and-trade” carbon market mechanism. Meanwhile, in the CMIES, energy recovery units and belt conveyors can be flexibly scheduled and the pumped hydroelectric storage in the VPP is scheduled for energy management. To tackle uncertainties from PV outputs, the joint trading, and the energy management is solved by the distributionally robust optimization (DRO) method. In addition, for participants' privacy, the alternating direction method of multipliers (ADMM) - based DRO algorithm is applied to solve the trading problem in a distributed framework. Further, the Proof-of-Authority (PoA) blockchain is deployed to develop a safe and anonymous market platform. Finally, case studies along with numerous comparison cases are conducted to verify the effectiveness of the proposed method. Simulation results indicate that the proposed method can effectively reduce the system operation cost and regional carbon emission, reduce the conservativeness and protect the privacy of each participant.

Investing in Wind Energy Using Bi-Level Linear Fractional Programming

Investing in wind energy is a tool to reduce greenhouse gas emissions without negatively impacting the environment to accelerate progress towards global net zero. The objective of this study is to present a methodology for efficiently solving the wind energy investment problem, which aims to identify an optimal wind farm placement and capacity based on fractional programming (FP). This study adopts a bi-level approach whereby a private price-taker investor seeks to maximize its profit at the upper level. Given the optimal placement and capacity of the wind farm, the lower level aims to optimize a fractional objective function defined as the ratio of total generation cost to total wind power output. To solve this problem, the Charnes-Cooper transformation is applied to reformulate the initial bi-level problem with a fractional objective function in the lower-level problem as a bi-level problem with a fractional objective function in the upper-level problem. Afterward, using the primal-dual formulation, a single-level linear FP model is created, which can be solved via a sequence of mixed-integer linear programming (MILP). The presented technique is implemented on the IEEE 118-bus power system, where the results show the model can achieve the best performance in terms of wind power output.

Finding global solutions of some inverse optimal control problems using penalization and semismooth Newton methods

We present a method to solve a special class of parameter identification problems for an elliptic optimal control problem to global optimality. The bilevel problem is reformulated via the optimal-value function of the lower-level problem. The reformulated problem is nonconvex and standard regularity conditions like Robinson’s CQ are violated. Via a relaxation of the constraints, the problem can be decomposed into a family of convex problems and this is the basis for a solution algorithm. The convergence properties are analyzed. It is shown that a penalty method can be employed to solve this family of problems while maintaining convergence speed. For an example problem, the use of the identity as penalty function allows for the solution by a semismooth Newton method. Numerical results are presented. Difficulties and limitations of our approach to solve a nonconvex problem to global optimality are discussed.

Optimal bidding strategy of multi-carrier systems in electricity markets using information gap decision theory

This paper proposes a novel optimization framework based on information gap decision theory (IGDT) for strategic participation of multi-carrier systems (MCS) in the electricity market under price uncertainty. The proposed framework helps the operator of MCS to take advantage of flexible generation units such as combined heat and power (CHP) systems, boilers, and battery energy storage systems (BESS) and participate in the electricity market considering the strategies presented by IGDT against the uncertainty in the market clearing price. The proposed optimization framework is developed as a bilevel problem, in which the risk-based bidding strategy of the MCS operator is modeled in the upper-level problem, and the energy market is cleared by the bulk power system operator in the lower-level problem. Karush–Kuhn–Tucker conditions are used to transform the bilevel problem into a single-level optimization problem. Duality theory is later used to approximate the product of power and market clearing price in the MCS operation problem, and the non-linearity enforced by the implementation of IGDT is handled by a linearization method called Triangle method. The studied optimization framework is modeled as a single-level mixed-integer linear programming problem, and CPLEX is used to solve it. The simulations are carried out on the modified IEEE 6-bus transmission network, and the results are analyzed to verify the economic and computation efficiency of the studied model. According to the simulation results, by taking a risk-averse bidding strategy provided by the IGDT, the operation cost of MCS increases 16.6% in order to increase the robustness level of MCS up to 36%, indicating that MCS becomes robust against 36% of the increase in the electricity market price, compared to the case with a risk-neutral bidding strategy being taken.

Charging Station Location and Sizing for Electric Vehicles Under Congestion

This paper studies the problem of determining the strategic location of charging stations and their capacity levels under stochastic electric vehicle flows and charging times taking into account the route choice response of users. The problem is modeled using bilevel optimization, where the network planner or leader minimizes the total infrastructure cost of locating and sizing charging stations while ensuring a probabilistic service requirement on the waiting time to charge. Electric vehicle users or followers, on the other hand, minimize route length and may be cooperative or noncooperative. Their choice of route in turn determines the charging demand and waiting times at the charging stations and hence, the need to account for their decisions by the leader. The bilevel problem reduces to a single-level mixed-integer model using the optimality conditions of the follower’s problem when the charging stations operate as M/M/c queues and the followers are cooperative. To solve the bilevel model, a decomposition-based solution methodology is developed that uses a new logic-based Benders algorithm for the location-only problem. Computational experiments are performed on benchmark and real-life highway networks, including a new eastern U.S. network. The impact of route choice response, service requirements, and deviation tolerance on the location and sizing decisions are analyzed. The analysis demonstrates that stringent service requirements increase the capacity levels at open charging stations rather than their number and that solutions allowing higher deviations are less costly. Moreover, the difference between solutions under cooperative and uncooperative route choices is more significant when the deviation tolerance is lower. History: This paper has been accepted for the Transportation Science Special Issue on 2021 TSL Workshop: Supply and Demand Interplay in Transport and Logistics. Funding: This research was supported by the Ontario Graduate Scholarship when Ö. B. Kınay was a PhD candidate at the University of Waterloo, and this support is acknowledged. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2021.0494 .

Inexact simultaneous projection method for solving bilevel equilibrium problems

Inexact simultaneous projection method for solving bilevel equilibrium problems

Optimal Defensive Strategy for Power Distribution Systems Against Relay Setting Attacks

Improving the cyber security of critical protection devices is of paramount importance for reducing the impact of cyber attacks on power distribution system. This paper proposes a game-theoretic approach to identify the optimal defensive strategy and scheme to minimize the impact of relay setting attack on power distribution systems. To this end, two types of attacks, namely active and passive relay setting attacks (RSA), are introduced and modeled as the potential threats to the protection relays. The active RSA impacts the distribution system immediately, while passive RSA disrupts the relay coordination and escalates the outages resulted from future system faults. The optimal budget allocation problem for strengthening the security of relays is modeled by formulating the interaction between the defender, the attacker and the distribution system operation as a tri-level optimization problem, which is then cast into a single-level operation and a bi-level defender-attacker problem. The bi-level problem is further reduced to an expected energy loss minimization problem using an auxiliary constrained parameter. The proposed approach is tested on the IEEE 123-node test feeder, the results of which exhibit the effectiveness of the proposed model on determining the most vulnerable relays and reducing the expected energy not served.