Bilevel Problem Research Articles

Bilevel optimization based on iterative approximation of multiple mappings

A large number of application problems involve two levels of optimization, where one optimization task is nested inside the other. These problems are known as bilevel optimization problems and have been studied by both classical optimization community and evolutionary optimization community. Most of the solution procedures proposed until now are either computationally very expensive or applicable to only small classes of bilevel optimization problems adhering to mathematically simplifying assumptions. In this paper, we propose an evolutionary optimization method that tries to reduce the computational expense by iteratively approximating two important mappings in bilevel optimization; namely, the lower level rational reaction mapping and the lower level optimal value function mapping. The algorithm has been tested on a large number of test problems and comparisons have been performed with other algorithms. The results show the performance gain to be quite significant. To the best knowledge of the authors, a combined theory-based and population-based solution procedure utilizing mappings has not been suggested yet for bilevel problems.

Profitable Emissions-Reducing Energy Storage

While energy arbitrage from energy storage can lower power system operating costs, it can also increase greenhouse gas emissions. If power system operations are conducted with the constraint that energy storage operation must not increase emissions, how does this constraint affect energy storage investment decisions? Two bi-level energy storage investment problems are considered, representing `philanthropic' (profitability-constrained) and profitmaximizing storage investors (PhSI, PMSI). A MILP heuristic is developed to obtain good candidate solutions to these inherently MINLP bi-level problems. A case study is conducted on a 30% renewable system, with sensitivity analyses on the price of storage and the price of carbon emissions. Regardless of the emissions-neutrality constraint, a PhSI installs significantly more energy storage than a PMSI, increasing system flexibility. The effect of the emissions-neutrality constraint in the absence of a carbon price is to reduce the quantity of storage purchased and reduce annual system emissions (~3%), with only minor increases in overall cost (~0.1%). In cases with a carbon price, storage does not tend to increase emissions and the emissions constraint does not tend to decrease storage investment. The emissions-neutrality constraint is seen to deliver similar emissions reductions even in a system with much higher renewable penetration (46%).

A co-evolutionary hybrid decomposition-based algorithm for bi-level combinatorial optimization problems

Bi-level programming problems are a special class of optimization problems with two levels of optimization tasks. These problems have been widely studied in the literature and often appear in many practical problem solving tasks. Although many applications fit the bi-level framework, however, real-life implementations are scarce, due mainly to the lack of efficient algorithms able to handle effectively this NP-hard problem. Several solution approaches have been proposed to solve these problems; however, most of them are restricted to the continuous case. Motivated by this observation, we have recently proposed a Co-evolutionary Decomposition-based Algorithm (CODBA) to solve bi-level combinatorial problems. CODBA scheme has been able to bring down the computational expense significantly as compared to other competitive approaches within this research area. In this paper, we further improve CODBA approach by incorporating a local search procedure to make the search process more efficient. The proposed extension called CODBA-LS includes a variable neighborhood search to the lower-level task to help in faster convergence of the algorithm. Further experimental tests based on the bi-level production–distribution problems in supply chain management model on a set of artificial and real-life data turned out to be effective on both computation time and solution quality.

Tabu search based on exact approach for protecting hubs against jamming attacks

In this paper, we consider an interdiction/protection problem on a wireless network where a defender places a set of hubs and detects the locations for deploying anti-jamming devices in order to minimize the cost of installing hubs and the energy consumption before and after jamming attacks. Moreover, the attacker's objective is to detect the locations for deploying jamming devices and disrupting the operation of a wireless network such that the energy consumption is maximized. We formulate this problem as a quadratic bi-level problem. A tabu search algorithm is implemented and tested on 48 randomly generated networks. The experimental results indicate that deploying anti-jamming devices can decrease the impact of jamming attacks. We also analyze the results in comparison with a decomposition method and two metaheuristics called the genetic algorithm and teaching learning. Numerical results show the superiority of the proposed algorithm against above-mentioned approaches regarding the execution time and accuracy.

Inertial Method for Bilevel Variational Inequality Problems with Fixed Point and Minimizer Point Constraints

In this paper, we introduce an iterative scheme with inertial effect using Mann iterative scheme and gradient-projection for solving the bilevel variational inequality problem over the intersection of the set of common fixed points of a finite number of nonexpansive mappings and the set of solution points of the constrained optimization problem. Under some mild conditions we obtain strong convergence of the proposed algorithm. Two examples of the proposed bilevel variational inequality problem are also shown through numerical results.

Real-Time Feedback-Based Optimization of Distribution Grids: A Unified Approach

This paper develops an algorithmic framework for real-time optimization of distribution-level distributed energy resources (DERs). The framework optimizes the operation of DERs that are individually controllable as well as groups of DERs that are jointly controlled at an electrical point of connection. From an electrical standpoint, wye and delta single-phase and multiphase connections are accounted for. The algorithm enables DERs to pursue given performance objectives, while adjusting their powers to respond to services requested by grid operators and to maintain electrical quantities within engineering limits. The design of the algorithm leverages a time-varying bilevel problem formulation, capturing various performance objectives and engineering constraints, and an online implementation of primal-dual projected-gradient methods. The gradient steps are suitably modified to accommodate appropriate measurements from the distribution network and the DERs. The resulting algorithm can cope with inaccuracies in the distribution-system modeling; moreover, it avoids pervasive metering to gather the state of noncontrollable resources, and it naturally lends itself to a distributed implementation. Analytical stability and convergence claims are established in terms of tracking the solution of the formulated time-varying optimization problem. The proposed method is tested in a realistic distribution system by using real data.

Mixed-integer bilevel representability

We study the representability of sets by extended formulations using mixed-integer bilevel programs. We show that feasible regions modeled by continuous bilevel constraints (with no integer variables), complementarity constraints, and polyhedral reverse convex constraints are all finite unions of polyhedra. Conversely, any finite union of polyhedra can be represented using any one of these three paradigms. We then prove that the feasible region of bilevel problems with integer variables exclusively in the upper level is a finite union of sets representable by mixed-integer programs and vice versa. Further, we prove that, up to topological closures, we do not get additional modeling power by allowing integer variables in the lower level as well. To establish the last statement, we prove that the family of sets that are finite unions of mixed-integer representable sets (up to topological closures) forms an algebra of sets; i.e., this family is closed under finite unions, intersections and complementation.

A market regulation bilevel problem: A case study of the Mexican petrochemical industry

In this paper, a bilevel programming model is proposed to study a problem of market regulation through government intervention. One of the main characteristics of the problem herein analyzed is that the government monopolizes the raw material in one industry, and competes in another industry with private firms for the production of commodities. Under this scheme, the government controls a state-owned firm to balance the market; that is, to minimize the difference between the produced and demanded commodities. On the other hand, a regulatory organization that coordinates private firms aims to maximize the total profit by deciding the amount of raw material bought from the a state-owned firm. Two equivalent single-level reformulations are proposed to solve the problem. The first reformulation is based on the strong duality condition of the lower level and results in a continuous non-linear model. The second reformulation resorts to the complementarity slackness optimality constraints yielding a mixed-integer linear model. Additionally, three heuristic algorithms are designed to obtain good-quality solutions with low computational effort. In this problem, the feasible region of the dual problem associated to the follower is independent from the leader’s decision. Therefore, the proposed heuristics exploit this particular characteristic of the bilevel model. Moreover, the third heuristic hybridizes the other two algorithms to enhance its performance. Extensive computational experimentation is carried out to measure the efficiency of the proposed solution methodologies. A case study based on the Mexican petrochemical industry is presented. Additional instances generated from the case study are considered to validate the robustness of the proposed heuristic algorithms. Numerical results indicate that the hybrid algorithm outperforms the other two heuristics. However, all of them demonstrate to be good alternatives for solving the problem. Additionally, optimal solutions of all the instances are obtained by using good quality solutions (given by the hybrid algorithm) as initial solutions when solving the second reformulation via a general purpose solver.

Multi-period Market Operation of Transmission-Distribution Systems Based on Heterogeneous Decomposition and Coordination

The integration of shiftable/curtailment distribution generators (DGs) along with quick-response storage has not only increased the transaction’s flexibility but also puzzled the bidding willingness of transmission-connected market players (TMPs). In this paper, the method of heterogeneous decomposition and coordination (HGDC) is applied to decompose the integrated transmission-distribution market framework into a bi-level problem with a transmission wholesale market master problem and several distribution retail market subproblems in a decentralized organization structure. The price-based bidding willingness of demand-side resources’ (DSRs’) aggregator is simulated considering the relation between distribution system operator’s (DSO’s) operation modes and its equivalent market transactive price. Besides the traditional mixed-integer linear programming (MILP) model, the active reconfiguration model of DSOs based on mixed-integer second-order conic programming (MI-SOCP) is proposed to rearrange its operation switch status and elaborate its operation cost according to the market transaction. Multi-period optimal operation modes could be obtained through an HGDC-based iteration process by coordinating the transmission system operator (TSO) with DSOs and considering the market energy equilibrium and reserve requirements for security considerations. Karush-Kuhn-Tucker (KKT) conditions are used to testify the optimality and convergence of the bi-level model in theory. The T5-3D33 case is employed to illustrate the efficiency of the proposed model and method.

Solving the Bi-level Problem of a Closed Optimization of Electricity Price Zone Configurations using a Genetic Algorithm

The topic of alternative price zone configurations is frequently discussed in Central Western Europe where – so far – national borders coincide with borders of price zones. Reconfiguring these price zones is one option in order to improve congestion management, foster trading across borders of price zones and, thus, to increase welfare. In view of the significant increase in redispatch volumes and costs over the last years due to increasing feed-in from renewable energy sources in conjunction with delayed grid expansion, this topic has gained in importance. To determine these improved price zone configurations for a large-scale system like Central Western Europe, often either configurations based on expert guesses are considered or heuristics using approximate criteria like locational marginal prices are used to obtain price zones through clustering. In contrast, the present paper formulates a bi-level optimization problem of how to determine optimal configurations in terms of system costs and – given the size and nature of the problem – solves it with a specially developed genetic algorithm. Resulting price zone configurations are compared to both exogenously given, expert-based price zone configurations from the Entso-E bidding zone study and endogenously assessed configurations from a hierarchical cluster algorithm. Results show that the genetic algorithm achieves best results in terms of system costs. Moreover, the comparison with solutions from a hierarchical cluster analysis reveals important drawbacks of the latter methodology.

Optimal wind energy bidding strategies in real‐time electricity market with multi‐energy sources

This study investigates optimal wind power generator bidding strategies in the real-time electricity market. The goal is to maximise its operating profit by determining the optimal amount of wind power to bid in the real-time market. A bi-level stochastic optimisation model is proposed in which the upper-level problem is minimising the negative profit of wind power producers, while the low-level problem clears the real-time market. The uncertainties in the wind power production, thermal power, hydro power, demand, and energy storage are considered in the stochastic model. This study utilises a mathematical programming problem with equilibrium constraints (MPEC) and Karush-Kuhn-Tucker (KKT) conditions to transform the bi-level problem into an equivalent single-level mixed-integer linear problem (MILP). Case studies obtained with the IEEE RTS-24 Bus system demonstrate the effectiveness of the proposed model and the effect of scenarios on the wind power producer's profit.

Optimal sectional fare and frequency settings for transit networks with elastic demand

Sectional fares have been used in transit services in practice but are rarely examined analytically and compared with flat and distance-based fares, especially under the considerations of path choice, elastic demand, service frequency, and profitability. This paper proposes a bilevel programming model to jointly determine the fare and frequency setting to maximize transit operator's profit. The preceding three fare structures can be incorporated into the bilevel model. To consider the path choice and elastic demand in the bilevel model, the existing approach-based stochastic user equilibrium transit assignment model for the fixed demand was extended to the elastic demand case and the resultant model was used in the lower level model. To solve the bilevel model, the sensitivity-based descent search method that takes into account the approach-based formulation for the elastic demand transit assignment is proposed, in which the approach-based formulation was solved by the cost-averaging self-regulated averaging method. Numerical studies and mathematical analyses were performed to examine the model properties and compare the three fare structures. The result of the Tin Shui Wai network instance is also provided to illustrate the performance of the solution method.It is proven that when all passengers’ destinations are located at transit terminals, the sectional fare structure is always better than the other two fare structures in terms of profitability. For more general networks, the sectional fare structure is always better than the flat fare structure, but the choice between sectional and distance-based fare structures depends on the geometry of the network (e.g., the route structure and the distance between stops), the demand distribution, and the maximum allowable fares. It is also proven that the optimal profit (total vehicle mileage) is strictly monotonically decreasing (monotonically decreasing) with respect to the unit operating cost. Moreover, it is proven that the lower level approach-based assignment problem with elastic demand has exactly one solution. However, the bi-level problem can have multiple optimal solutions. Interestingly, it is found that from the operator's profitability point of view, providing better information to the passengers may not be good.

BIVAS: A Scalable Bayesian Method for Bi-Level Variable Selection With Applications

In this article, we consider a Bayesian bi-level variable selection problem in high-dimensional regressions. In many practical situations, it is natural to assign group membership to each predictor. Examples include that genetic variants can be grouped at the gene level and a covariate from different tasks naturally forms a group. Thus, it is of interest to select important groups as well as important members from those groups. The existing Markov chain Monte Carlo methods are often computationally intensive and not scalable to large datasets. To address this problem, we consider variational inference for bi-level variable selection. In contrast to the commonly used mean-field approximation, we propose a hierarchical factorization to approximate the posterior distribution, by using the structure of bi-level variable selection. Moreover, we develop a computationally efficient and fully parallelizable algorithm based on this variational approximation. We further extend the developed method to model datasets from multitask learning. The comprehensive numerical results from both simulation studies and real data analysis demonstrate the advantages of BIVAS for variable selection, parameter estimation, and computational efficiency over existing methods. The method is implemented in R package “bivas” available at https://github.com/mxcai/bivas. Supplementary materials for this article are available online.

ε-Constraint method for bi-objective competitive facility location problem with uncertain demand scenario

We consider a model of two parties’ competition organized as a Stackelberg game. The parties open their facilities intending to maximize profit from serving the customers that behave following a binary rule. The set of customers is unknown to the party which opens its facilities first and is called the Leader. Instead, a finite list of possible scenarios specifying this set is provided to the Leader. One of the scenarios is to be realized in the future before the second party, called the Follower, would make their own decision. The scenarios are supplied with known probabilities of realization, and the Leader aims to maximize both the probability to get a profit not less than a specific value, called a guaranteed profit, and the value of a guaranteed profit itself. We formulate the Leader’s problem as a bi-objective bi-level mathematical program. To approximate the set of efficient solutions of this problem, we develop an ε -constraint method where a branch-and-bound algorithm solves a sequence of bi-level problems with a single objective. Based on the properties of feasible solutions of a bi-level program and mathematical programming techniques, we developed three upper bound procedures for the branch-and-bound method mentioned. In numerical experiments, we compare these procedures with each other. Besides that, we discuss relations of the model under investigation and the stochastic competitive location model with uncertain profit values.

A Julia package for bilevel optimization problems

Mathematical optimization is the discipline dealing with the determination of the best (or almost best) decision with respect to a specific cost function and to a set of constraints on the decision. Bilevel optimization is a class of mathematical optimization problems with the optimality conditions of a lower-level problem embedded in the constraints. BilevelOptimization.jl is a toolbox built on top of the JuMP.jl ecosystem for mathematical optimization (Dunning, Huchette, & Lubin, 2017). Bilevel optimization is used to tackle various problems in areas such as power systems, security applications, network design or market equilibria. See Dempe (2018) for an overview of applications and recent formulations and theoretical progress. The computation of an optimal solution to a bilevel problem is in general hard. Even with all the constraints and the objectives at the two levels being linear, the resulting problem is non-convex and NP-hard, with a possibly disjoint feasible set. Optimization practitioners often rely on problem-specific properties and modeling techniques or heuristics. The goal of this package is to offer a both flexible model of a general class of bilevel problems and a solver which is compatible with the JuMP workflow.

Bi‐level operational planning of microgrids with considering demand response technology and contingency analysis

Considering financial, ecological and also reliability challenges in planning and operation of microgrids is an indispensable issue. On the other hand, an appropriate design of microgrids in planning stages has a major impact on the daily operation of the system. According to these matters, in this study, a hierarchical decision-making framework has been presented for both planning and operation of microgrids with wide ranges of means and concepts. The proposed model has been formulated as a bi-level optimisation problem in which every level optimises its own objectives independently, however, in interaction with each other. The upper level (leader) problem, which is related to the planning of microgrids, minimises the utility's demand, investment, and emission costs, while the lower level (follower) problem minimises the operation and maintenance costs through the implementation of an energy management system. The mentioned model is a non-linear bi-level problem, which is transformed into a linear single-level problem through Karush–Kuhn–Tucker conditions. Moreover, the contingency based energy management, demand response programme and uncertain nature of renewable resources have been taken into account. Finally, the proposed method has been applied to a typical microgrid and its results are compared with the weighted-sum multi-objective approach to depict its efficiency.

When is it beneficial to provide freelance suppliers with choice? A hierarchical approach for peer-to-peer logistics platforms

This paper proposes and evaluates a new hierarchical approach to peer-to-peer logistics platforms, recasting the platform’s role as one providing personalized menus of requests to freelance suppliers. A bilevel optimization formulation explicitly models the two stage decision process: first, the platform determines which set of requests to recommend to which suppliers, and second, suppliers have a choice to select which request (if any) to fulfill. By harnessing the problem’s structure, the computationally expensive mixed-integer linear bilevel problem is transformed into an equivalent single-level problem that is computationally superior. A computational study based on ride-sharing quantifies the value of providing suppliers with choices. When a platform’s knowledge of suppliers’ selections is imperfect, our hierarchical approach outperforms existing recommendation methods, namely, centralized, many-to-many stable matching, and decentralized approaches. We show that a platform’s lack of knowledge over suppliers’ selections can be compensated by providing choices in environments with either inflexible suppliers or when suppliers’ utilities have higher variance than the platform’s utilities. In these environments, providing choices and recommending alternatives to more than one supplier can be beneficial to not only the platform, but also freelance suppliers and demand requests.

A bilevel model for maintenance scheduling of power units including wind farms

This paper proposes a comprehensive bilevel model for the maintenance scheduling of power units including wind farms in a restructured environment. The maintenance-scheduling problem of generating companies (GENCOs) is constrained by technical and security constraints set by the independent system operator (ISO), who has the responsibility of operating the power system. To consider the impact of these constraints on the scheduling problem, we propose a bilevel approach with a strategic GENCO at the upper level and an ISO at the lower level. In the upper level, the scheduling problem considering the participation of wind farms is presented. The lower level, also called the ISO level, maximizes the social welfare with network security constraints. This bilevel problem is modeled as a mathematical program with equilibrium constraints (MPEC), which is then transformed into a mixed-integer linear problem through a linearization procedure. The resultant model is tested on the IEEE 24-Bus Reliability Test System considering various case studies to illustrate the effectiveness of the proposed model. The results show that the proposed model can properly schedule the outages under different strict conditions preserving system’s security.

A novel evolutionary algorithmic solution approach for bilevel reliability-redundancy allocation problem

Reliability-redundancy allocation problem (RRAP) is an optimization problem with objectives such as maximization of the system reliability and minimizations of cost, weight etc. It considers component reliability and the number of redundant components as decision variables. RRAP was mostly solved as a single level optimization problem. However, this problem has numerous variables and constraints. In this paper, we report a novel mixed-integer non-linear bilevel formulation for the RRAP of the series-parallel systems considering reliability, cost and weight as the main objectives. Then we show a novel way of solving the newly formulated bilevel RRAP using a latest bilevel optimization algorithm called BLEAQ (bilevel evolutionary algorithm based on quadratic approximations). Here, optimization is needed at two separate levels, where one problem is encircled within another problem. The inner problem is known as lower-level problem and the external problem is called upper-level problem. We demonstrate the working of the model considering a real-life example. For a thorough comparison, we also solve the problem using two existing bilevel evolutionary approaches to show the efficacy of our proposed solution approach.

A parametric programming approach to bilevel optimisation with lower-level variables in the upper level

This paper examines linearly constrained bilevel programming problems in which the upper-level objective function depends on both the lower-level primal and dual optimal solutions. We parametrize the lower-level solutions and thereby the upper-level objective function by the upper-level variables and argue that it may be non-convex and even discontinuous. However, when the upper-level objective is affine in the lower-level primal optimal solution, the parametric function is piece-wise linear. We show how this property facilitates the application of parametric programming and demonstrate how the approach allows for decomposition of a separable lower-level problem. When the upper-level objective is bilinear in the lower-level primal and dual optimal solutions, we also provide an exact linearisation method that reduces the bilevel problem to a single-level mixed-integer linear programme (MILP). We assess the performance of the parametric programming approach on two case studies of strategic investment in electricity markets and benchmark against state-of-the-art MILP and non-linear solution methods for bilevel optimisation problems. Preliminary results indicate substantial computational advantages over several standard solvers, especially when the lower-level problem separates into a large number of subproblems. Furthermore, we show that the parametric programming approach succeeds in solving problems to global optimality for which standard methods can fail.