Bilevel Multiobjective Optimization Problem Research Articles

Multi-Objective BiLevel Optimization by Bayesian Optimization

In a multi-objective optimization problem, a decision maker has more than one objective to optimize. In a bilevel optimization problem, there are the following two decision-makers in a hierarchy: a leader who makes the first decision and a follower who reacts, each aiming to optimize their own objective. Many real-world decision-making processes have various objectives to optimize at the same time while considering how the decision-makers affect each other. When both features are combined, we have a multi-objective bilevel optimization problem, which arises in manufacturing, logistics, environmental economics, defence applications and many other areas. Many exact and approximation-based techniques have been proposed, but because of the intrinsic nonconvexity and conflicting multiple objectives, their computational cost is high. We propose a hybrid algorithm based on batch Bayesian optimization to approximate the upper-level Pareto-optimal solution set. We also extend our approach to handle uncertainty in the leader’s objectives via a hypervolume improvement-based acquisition function. Experiments show that our algorithm is more efficient than other current methods while successfully approximating Pareto-fronts.

An Optimization Control Method of IEH Considering User Thermal Comfort

In this paper, a user thermal comfort criterion based on predicted mean vote (PMV) values is introduced to realize the optimal operation of an improved energy hub (IEH) while considering thermal inertia and user thermal behavior. A three-layer optimization model based on user thermal comfort is constructed which fully considers user thermal comfort demand, IEH operating costs, and energy network constraints. Moreover, since IEH optimization considering user thermal comfort is a multi-objective bilevel optimization (MNBO) problem, this paper proposes an improved multilayer nested quantum genetic algorithm (IMNQGA) to solve it. Finally, the effectiveness of the proposed optimization model and algorithm is verified through the analysis of the four modes. The examples show that the proposed optimal control method can reduce the system’s operating costs and improve energy efficiency while satisfying user thermal comfort demand.

Bi-Level Multiobjective Evolutionary Learning: A Case Study on Multitask Graph Neural Topology Search

The construction of machine learning models involves many bi-level multi-objective optimization problems (BL-MOPs), where upper level (UL) candidate solutions must be evaluated via training weights of a model in the lower level (LL). Due to the Pareto optimality of sub-problems and the complex dependency across UL solutions and LL weights, an UL solution is feasible if and only if the LL weight is Pareto optimal. It is computationally expensive to determine which LL Pareto weight in the LL Pareto weight set is the most appropriate for each UL solution. This paper proposes a bi-level multi-objective learning framework (BLMOL), coupling the above decision-making process with the optimization process of the UL-MOP by introducing LL preference r. Specifically, the UL variable and r are simultaneously searched to minimize multiple UL objectives by evolutionary multi-objective algorithms. The LL weight with respect to r is trained to minimize multiple LL objectives via gradient-based preference multi-objective algorithms. In addition, the preference surrogate model is constructed to replace the expensive evaluation process of the UL-MOP. We consider a novel case study on multi-task graph neural topology search. It aims to find a set of Pareto topologies and their Pareto weights, representing different trade-offs across tasks at UL and LL, respectively. The found graph neural network is employed to solve multiple tasks simultaneously, including graph classification, node classification, and link prediction. Experimental results demonstrate that BLMOL can outperform some state-of-the-art algorithms and generate well-representative UL solutions and LL weights.

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.

Hyperparameter Learning Under Data Poisoning: Analysis of the Influence of Regularization via Multiobjective Bilevel Optimization.

Machine learning (ML) algorithms are vulnerable to poisoning attacks, where a fraction of the training data is manipulated to deliberately degrade the algorithms' performance. Optimal attacks can be formulated as bilevel optimization problems and help to assess their robustness in worst case scenarios. We show that current approaches, which typically assume that hyperparameters remain constant, lead to an overly pessimistic view of the algorithms' robustness and of the impact of regularization. We propose a novel optimal attack formulation that considers the effect of the attack on the hyperparameters and models the attack as a multiobjective bilevel optimization problem. This allows us to formulate optimal attacks, learn hyperparameters, and evaluate robustness under worst case conditions. We apply this attack formulation to several ML classifiers using L2 and L1 regularization. Our evaluation on multiple datasets shows that choosing an "a priori" constant value for the regularization hyperparameter can be detrimental to the performance of the algorithms. This confirms the limitations of previous strategies and evidences the benefits of using L2 and L1 regularization to dampen the effect of poisoning attacks, when hyperparameters are learned using a small trusted dataset. Additionally, our results show that the use of regularization plays an important robustness and stability role in complex models, such as deep neural networks (DNNs), where the attacker can have more flexibility to manipulate the decision boundary.

Cyber-secure energy and flexibility scheduling of interconnected local energy networks with introducing an XGBoost-assisted false data detection and correction method

The growing penetration of renewable energy resources (RERs) and the ever-increasing implementation of information and communication technologies in smart grids necessitate the utilization of fast and safe operation methods. These schemes should be able to cope with the intermittency of uncertain RERs and fully explore the capabilities of new technologies while considering the vulnerabilities as a result of digitalization. Here, the integrity of the energy and flexibility scheduling and trading heavily depends on the security of the cyber-physical system. Local energy networks (LENs) are introduced to easily incorporate and utilize distributed energy resources, improving electrical power reliability and efficiency. This study develops a cyber-secure energy and flexibility scheduling platform for interconnected local energy networks. Hence, it introduces a novel false data detection and correction method based on the XGBoost machine learning method and investigates the performance of the proposed algorithm in the context of an interconnected local energy network and the corresponding energy and flexibility transactions. In this regard, this paper initially presents a bilevel energy and flexibility scheduling problem for interconnected local energy networks (ILENs) considering cyber-attacks (BLEFSCAILEN), which tries to satisfy the energy and flexibility requirements of the ILEN guaranteeing a resilient decision against false data injection cyber-attacks. Within the proposed BLEFSCAILEN a bi-level multi-objective optimization problem is defined where the operator of ILEN optimizes both energy and flexibility trading among LENs in the upper level to maximize its revenue. In the proposed structure, the operator of each LEN is situated at the lower level and tries to minimize the scheduling cost and simultaneously maximize the local flexibility index. Furthermore, in order to evaluate and improve the proposed system’s cyber-security, a false data injection attack is constructed using the Gaussian probability distribution function. Herein, the scheduling of ILEN is optimized considering vulnerabilities of electricity consumption data as a consequence of cyber-attack. Finally, the novel XGBoost-assisted deviation bound-based false data detection/correction (XGBFDD) algorithm is suggested to mitigate the above-mentioned cyber-attack and attain near real-world conditions. Simulation results demonstrate that the proposed false data detection/correction strategy exhibits a maximum accuracy of 91.67 precent in mitigating the attack effects. This is because the acquired post-correction results exhibit minor deviations in comparison to the original optimization output that is calculated with the pre-attack consumption data.

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.

Necessary optimality conditions for a semivectorial bilevel optimization problem via a revisited Charnes–Cooper scalarization

In this paper, we investigate a multiobjective bilevel optimization problem with a vector-valued lower-level objective function. We revisit the Charnes–Cooper scalarization technique for multi-objective programs and use optimal value reformulation to transform the scalar problem into a one-level optimization problem. Using Mordukhovich's generalized differentiation calculus, a new subdifferential estimate and Lipschitz condition for the optimal value function of this problem are developed as a result of this reformulation. In addition, we construct first-order necessary optimality conditions in the smooth setting.

Erratum to: optimality conditions in terms of convexificators for a bilevel multiobjective optimization problem

Erratum to: optimality conditions in terms of convexificators for a bilevel multiobjective optimization problem

Hierarchical optimal allocation of BESS using APT-FPSO based on stochastic programming model considering voltage sensitivity and eigenvalues analyses

Integration of energy storage systems (ESSs) to microgrids (MGs) due to the growing penetration of intermittent energy resources is a great motivation to develop studies in the context of optimal sizing and placement of these systems. In this paper, a multi-objective bi-level optimization problem is addressed based on the stochastic nature and dynamic model of power system elements.As a preliminary step, the optimal size and capacity of the battery energy storage system (BESS) are determined to minimize frequency fluctuations. This level has been performed based on three approaches considering the daily intra-hour stochastic data. In the secondary level, the dynamic behaviour of all MG elements has been considered and correspondingly a dynamic objective function based on system eigenvalues and voltage sensitivity index (VSI) has been developed. According to the proposed objective function, the secondary step for sizing and placement of the battery units has been performed by adaptive particularly tuneable fuzzy particle swarm optimization (APT-FPSO) algorithm to reduce power losses, improve the voltage profile/ stability, and mitigate low-frequency fluctuations. Finally, the effectiveness of the proposed approach is evaluated using time-domain simulations, voltage sensitivity studies, and eigenvalue analysis.

6G shared base station planning using an evolutionary bi-level multi-objective optimization algorithm

To improve the utilization of infrastructure resources and reduce the cost of operators in the future 6G network construction, a 6G shared base stations optimization model is proposed in this paper, which is a bi-level multiobjective optimization problem (BLMOP). In such a BLMOP, the tower company is responsible for the construction of base stations at the upper level, while operators share the base station resources of the tower company at the lower level. In addition, we also propose two strategies to solve the optimization efficiently. First, we use surrogate models to fit lower-level Pareto fronts (PF), then the degree of lower-level optimality constraint violation is converted to distance between the candidate solutions and the approximate lower level PF. So the BLMOP is transformed to a single-level constrained multi-objective optimization problem. Second, to accelerate the current lower-level optimization, we migrate the modified population from the adjacent lower-level optimization tasks. These two strategies effectively reduce the computational overhead. Compared with three existing works, the proposed method has achieved the best or comparable results on 7 benchmark problems and 5 generated test instances with less computation overhead, whose efficiency has been confirmed.

On multiobjective bilevel optimization using tangential subdifferentials

<p style='text-indent:20px;'>Multiobjective optimization problems typically have conflicting objectives, and a gain in one objective very often is an expense in another. In this paper, we investigate a multiobjective bilevel optimization problem. Using the concept of efficiency together with the optimal value reformulation, we give necessary optimality conditions in terms of tangential subdifferentials. Completely detailed first-order necessary optimality conditions are then derived in the special case where the lower level problem is convex. An example that illustrates our findings is also given.</p>

Cooperative Coevolution With Knowledge-Based Dynamic Variable Decomposition for Bilevel Multiobjective Optimization

Many practical multiobjective optimization problems have a nested bilevel structure in variables, which can be modeled as bilevel multiobjective optimization problems (BLMOPs). In this article, a cooperative coevolution (CC) with knowledge-based variable decomposition, called bilevel multiobjective CC (BLMOCC), is proposed for BLMOPs. In BLMOCC, the variable interactions are represented by an interaction matrix. The perturbation-based variable decomposition combined with the matrix completion approach has been designed for dynamically discovering the correlation among the bilevel variables, based on which the variables are divided into different groups. To further handle possible weak correlations among various groups of variables, a CC has been adopted for optimizing them in a collaborative way. In experimental studies, BLMOCC is compared with a nested method (NS) and a state-of-the-art algorithm (H-BLEMO) on a set of benchmark problems. The effects of each component in BLMOCC have also been verified by comparing it with its three variants. The experimental results demonstrate that BLMOCC has the best performance among all the compared algorithms. In addition, BLMOCC has also been applied to a real-world management decision-making problem, which further validates its efficiency and effectiveness.

Extension of the value function reformulation to multiobjective bilevel optimization

We consider a multiobjective bilevel optimization problem with vector-valued upper- and lower-level objective functions. Such problems have attracted a lot of interest in recent years. However, so far, scalarization has appeared to be the main approach used to deal with the lower-level problem. Here, we utilize the concept of frontier map that extends the notion of optimal value function to our parametric multiobjective lower-level problem. Based on this, we build a tractable constraint qualification that we use to derive necessary optimality conditions for the problem. Subsequently, we show that our resulting necessary optimality conditions represent a natural extension from standard optimistic bilevel programs with scalar objective functions.

Necessary optimality conditions for nonsmooth multi-objective bilevel optimization problem under the optimistic perspective

Necessary optimality conditions for nonsmooth multi-objective bilevel optimization problem under the optimistic perspective

Reducing fuel consumption and related emissions through optimal sizing of energy storage systems for diesel-electric trains

Hybridization of diesel multiple unit railway vehicles is an effective approach to reduce fuel consumption and related emissions in regional non-electrified networks. This paper is part of a bigger project realized in collaboration with Arriva, the largest regional railway undertaking in the Netherlands, to identify optimal solutions in improving trains’ energy and environmental performance. A significant problem in vehicle hybridization is determining the optimal size for the energy storage system, while incorporating an energy management strategy as well as technical and operational requirements. With the primary requirement imposed by the railway undertaking to achieve emission-free and noise-free operation within railway stations, we formalize this as a bi-level multi-objective optimization problem, including vehicle performance, the trade-off between fuel savings and hybridization cost, influence of the energy management strategy, and other constraints. By deriving a Li-ion battery parameters at the cell level, a nested coordination framework is employed, where a brute force search finds the optimal battery size using dynamic programming for full controller optimization for each feasible solution. In this way, the global minimum for fuel consumption for each battery configuration is achieved. The results from a Dutch case study demonstrated fuel savings and CO2 emission reduction of more than 34% compared to a standard vehicle. Additionally, benefits in terms of local pollutants (NOx and PM) emissions are observed. Using an alternative sub-optimal rule-based control demonstrated a significant impact of the energy management on the results, reflected in higher fuel consumption and increased battery size together with corresponding costs.

Bilevel optimization of conservation practices for agricultural production

Uncertainty surrounding the sources of nonpoint pollution and producer response to prospective policy incentives complicates the nonpoint policy problem. To explain, neither policy-makers nor producers know the exact effect of current and alternative farming practices on the contributions of specific cropped fields to nutrient pollution. Spatial heterogeneity of the production technology and environmental damage of runoff also precludes the formulation of an analytic solution, so that producer response to prospective policies is unknown a priori to the policy maker. To address the complexity arising from point source uncertainty and spatial heterogeneity, we draw on recent computational advances to reformulate the classic nonpoint source pollution problem as a multi-objective bilevel optimization problem, employing genetic algorithm (GA) solution methods. This computational framework explicitly accounts for the nested nature of farm-level management decisions in response to prospective agri-environmental policy incentives, and spatial heterogeneity of both production and pollution effects. Our application considers the optimal spatial targeting of multiple management practices in the Iowa Raccoon River Watershed, an intensive corn and soybean production region of the Upper Mississippi River Basin (UMRB). Consistent with theory and previous empirical results, we find that combining multiple management practices, versus relying on single or one-size policies, lowers the total cost for a given level of nitrogen reduction. But we also find overall limited potential nitrogen reduction via implementing these practices on working land, suggesting the continued need for land retirement in meeting current nonpoint policy goals for the UMRB. We believe that the main contribution of this study lies in the novelty of the bilevel approach, which explicitly accounts for feedbacks between policy makers and agricultural producers, while the associated GA computational methods allow for better handling of large scale and complex spatial heterogeneity over the agricultural watershed.

An efficient solution strategy for bilevel multiobjective optimization problems using multiobjective evolutionary algorithm

An efficient solution strategy is proposed for bilevel multiobjective optimization problem (BLMOP) with multiple objectives at both levels when multiobjective optimization problem (MOP) at the lower level satisfies the convexity and differentiability for the lower-level decision variables. In the proposed strategy, the MOP at the lower level is first converted into a single-objective optimization formulation through adopting adaptive weighted sum scalarization, in which the lower-level weight vector is adjusted adaptively while the iteration progressing. The Karush-Kuhn-Tucker optimality conditions are used to the lower-level single-objective scalarization problem, thus the original BLMOP can be converted into a single-level MOP with complementarity constraints. Then an effective smoothing technique is suggested to cope with the complementarity constraints. In such a way, the BLMOP is finally formalized as a single-level constrained nonlinear MOP. A decomposition-based constrained multiobjective differential evolution is developed to solve this transformed MOP and some instances are tested to illustrate the feasibility and effectiveness of the solution methodology. The experimental results show that the proposed solution method possesses favorite convergence and diversity.

An efficient chemical reaction algorithm for multi-objective combinatorial bi-level optimization

ABSTRACT The Bi-Level Optimization Problem (BLOP) is defined as a mathematical program with two nested optimization tasks. Although many applications fit the bi-level framework, however, existing resolution methods were most proposed to solve single-objective bi-level problems. Regarding Multi-objective BLOPs (MBLOPs), there do not exist too many previous studies because of the difficulties associated with solving these complex problems. Additionally, a recently proposed metaheuristic, called Non-dominated sorting Chemical Reaction Optimization (NCRO), has been successfully applied to solve single-level Multi-Objective Problems (MOPs). NCRO applies a quick-non-dominated sorting technique that makes it one of the most powerful search algorithms in solving MOPs. Based on these observations, a new Bi-level Multi-objective CRO method, called BMCRO, is proposed in this article for solving MBLOPs. The main idea behind BMCRO is to come up with good solutions in an acceptable execution time within the bi-level framework. Experimental results on well-established benchmarks reveal the outperformance of the proposed algorithm against a bi-level variant of the Non-dominated Sorting Genetic Algorithm (NSGA-II) which is developed for this purpose.

Scenario-Based Network Reconfiguration and Renewable Energy Resources Integration in Large-Scale Distribution Systems Considering Parameters Uncertainty

Renewable energy integration has been recently promoted by many countries as a cleaner alternative to fossil fuels. In many research works, the optimal allocation of distributed generations (DGs) has been modeled mathematically as a DG injecting power without considering its intermittent nature. In this work, a novel probabilistic bilevel multi-objective nonlinear programming optimization problem is formulated to maximize the penetration of renewable distributed generations via distribution network reconfiguration while ensuring the thermal line and voltage limits. Moreover, solar, wind, and load uncertainties are considered in this paper to provide a more realistic mathematical programming model for the optimization problem under study. Case studies are conducted on the 16-, 59-, 69-, 83-, 415-, and 880-node distribution networks, where the 59- and 83-node distribution networks are real distribution networks in Cairo and Taiwan, respectively. The obtained results validate the effectiveness of the proposed optimization approach in maximizing the hosting capacity of DGs and power loss reduction by greater than 17% and 74%, respectively, for the studied distribution networks.