Bilevel Linear Problem Research Articles

An algorithm to find stable solutions in linear–linear bilevel problems

An algorithm to find stable solutions in linear–linear bilevel problems

On a Computationally Ill-Behaved Bilevel Problem with a Continuous and Nonconvex Lower Level

It is well known that bilevel optimization problems are hard to solve both in theory and practice. In this paper, we highlight a further computational difficulty when it comes to solving bilevel problems with continuous but nonconvex lower levels. Even if the lower-level problem is solved to varepsilon -feasibility regarding its nonlinear constraints for an arbitrarily small but positive varepsilon , the obtained bilevel solution as well as its objective value may be arbitrarily far away from the actual bilevel solution and its actual objective value. This result even holds for bilevel problems for which the nonconvex lower level is uniquely solvable, for which the strict complementarity condition holds, for which the feasible set is convex, and for which Slater’s constraint qualification is satisfied for all feasible upper-level decisions. Since the consideration of varepsilon -feasibility cannot be avoided when solving nonconvex problems to global optimality, our result shows that computational bilevel optimization with continuous and nonconvex lower levels needs to be done with great care. Finally, we illustrate that the nonlinearities in the lower level are the key reason for the observed bad behavior by showing that linear bilevel problems behave much better at least on the level of feasible solutions.

Learning Active Constraints to Efficiently Solve Linear Bilevel Problems: Application to the Generator Strategic Bidding Problem

Bilevel programming can be used to formulate many problems in the field of power systems, such as strategic bidding. However, common reformulations of bilevel problems to mixed-integer linear programs make solving such problems hard, which impedes their implementation in real-life. In this paper, we significantly improve solution speed and tractability by introducing decision trees to learn the active constraints of the lower-level problem, while avoiding to introduce binaries and big-M constants. The application of machine learning reduces the online solving time, by moving the selection of active constraints to an offline process, and becomes particularly beneficial when the same problem has to be solved multiple times. We apply our approach to the strategic bidding of generators in electricity markets, where generators solve the same problem many times for varying load demand or renewable production. Three methods are developed and applied to the problem of a strategic generator, with a DCOPF in the lower-level. These methods are heuristic and as so, do not provide guarantees of optimality or solution quality. Yet, we show that for networks of varying sizes, the computational burden is significantly reduced, while we also manage to find solutions for strategic bidding problems that were previously intractable.

Using neural networks to solve linear bilevel problems with unknown lower level

Bilevel problems are used to model the interaction between two decision makers in which the lower-level problem, the so-called follower’s problem, appears as a constraint in the upper-level problem of the so-called leader. One issue in many practical situations is that the follower’s problem is not explicitly known by the leader. For such bilevel problems with unknown lower-level model we propose the use of neural networks to learn the follower’s optimal response for given decisions of the leader based on available historical data of pairs of leader and follower decisions. Integrating the resulting neural network in a single-level reformulation of the bilevel problem leads to a challenging model with a black-box constraint. We exploit Lipschitz optimization techniques from the literature to solve this reformulation and illustrate the applicability of the proposed method with some preliminary case studies using academic and linear bilevel instances.

Why there is no need to use a big-M in linear bilevel optimization: a computational study of two ready-to-use approaches

Linear bilevel optimization problems have gained increasing attention both in theory as well as in practical applications of Operations Research (OR) during the last years and decades. The latter is mainly due to the ability of this class of problems to model hierarchical decision processes. However, this ability makes bilevel problems also very hard to solve. Since no general-purpose solvers are available, a “best-practice” has developed in the applied OR community, in which not all people want to develop tailored algorithms but “just use” bilevel optimization as a modeling tool for practice. This best-practice is the big-M reformulation of the Karush–Kuhn–Tucker (KKT) conditions of the lower-level problem—an approach that has been shown to be highly problematic by Pineda and Morales (2019). Choosing invalid values for M yields solutions that may be arbitrarily bad. Checking the validity of the big-Ms is however shown to be as hard as solving the original bilevel problem in Kleinert et al. (2019). Nevertheless, due to its appealing simplicity, especially w.r.t. the required implementation effort, this ready-to-use approach still is the most popular method. Until now, there has been a lack of approaches that are competitive both in terms of implementation effort and computational cost. In this note we demonstrate that there is indeed another competitive ready-to-use approach: If the SOS-1 technique is applied to the KKT complementarity conditions, adding the simple additional root-node inequality developed by Kleinert et al. (2020) leads to a competitive performance—without having all the possible theoretical disadvantages of the big-M approach.

A new exact method for linear bilevel problems with multiple objective functions at the lower level

In this paper we consider linear bilevel programming problems with multiple objective functions at the lower level. We propose a general-purpose exact method to compute the optimistic optimal solution, which is based on the search of efficient extreme solutions of an associated multiobjective linear problem with many objective functions. We also explore a heuristic procedure relying on the same principles. Although this procedure cannot ensure the global optimal solution but just a local optimum, it has shown to be quite effective in problems where the global optimum is difficult to obtain within a reasonable timeframe. A computational study is presented to evaluate the performance of the exact method and the heuristic procedure, comparing them with an exact and an approximate method proposed by other authors, using randomly generated instances. Our approach reveals interesting results in problems with few upper-level variables.

On linear bilevel optimization problems with complementarity-constrained lower levels

We consider a novel class of linear bilevel optimization models with a lower level that is a linear program with complementarity constraints (LPCC). We present different single-level reformulations depending on whether the linear complementarity problem (LCP) as part of the lower-level constraint set depends on the upper-level decisions or not as well as on whether the LCP matrix is positive definite or positive semidefinite. Moreover, we illustrate the connection to linear trilevel models that can be reduced to bilevel problems with LPCC lower levels having positive (semi)definite matrices. Finally, we provide two generic and illustrative bilevel models from the fields of transportation and energy to show the practical relevance of the newly introduced class of bilevel problems and show related theoretical results.

A reformulation strategy for mixed-integer linear bi-level programming problems

Bi-level programming has been used widely to model interactions between hierarchical decision-making problems, and their solution is challenging, especially when the lower-level problem contains discrete decisions. The solution of such mixed-integer linear bi-level problems typically need decomposition, approximation or heuristic-based strategies which either require high computational effort or cannot guarantee a global optimal solution. To overcome these issues, this paper proposes a two-step reformulation strategy in which the first part consists of reformulating the inner mixed-integer problem into a nonlinear one, while in the second step the well-known Karush-Kuhn-Tucker conditions for the nonlinear problem are formulated. This results in a mixed-integer nonlinear problem that can be solved with a global optimiser. The computational and numerical benefits of the proposed reformulation strategy are demonstrated by solving five examples from the literature.

Outer approximation for global optimization of mixed-integer quadratic bilevel problems

Bilevel optimization problems have received a lot of attention in the last years and decades. Besides numerous theoretical developments there also evolved novel solution algorithms for mixed-integer linear bilevel problems and the most recent algorithms use branch-and-cut techniques from mixed-integer programming that are especially tailored for the bilevel context. In this paper, we consider MIQP-QP bilevel problems, i.e., models with a mixed-integer convex-quadratic upper level and a continuous convex-quadratic lower level. This setting allows for a strong-duality-based transformation of the lower level which yields, in general, an equivalent nonconvex single-level reformulation of the original bilevel problem. Under reasonable assumptions, we can derive both a multi- and a single-tree outer-approximation-based cutting-plane algorithm. We show finite termination and correctness of both methods and present extensive numerical results that illustrate the applicability of the approaches. It turns out that the proposed methods are capable of solving bilevel instances with several thousand variables and constraints and significantly outperform classical solution approaches.

Presolving linear bilevel optimization problems

• There is almost no literature on presolving linear bilevel problems up to now. • We carry over standard presolve techniques from single-level optimization to bilevel problems. • We show that this needs to be done with great caution. • Our numerical study shows that presolve can be very beneficial for bilevel problems. Linear bilevel optimization problems are known to be strongly NP-hard and the computational techniques to solve these problems are often motivated by techniques from single-level mixed-integer optimization. Thus, during the last years and decades many branch-and-bound methods, cutting planes, or heuristics have been proposed. On the other hand, there is almost no literature on presolving linear bilevel problems although presolve is a very important ingredient in state-of-the-art mixed-integer optimization solvers. In this paper, we carry over standard presolve techniques from single-level optimization to bilevel problems and show that this needs to be done with great caution since a naive application of well-known techniques does often not lead to correctly presolved bilevel models. Our numerical study shows that presolve can also be very beneficial for bilevel problems but also highlights that these methods have a more heterogeneous effect on the solution process compared to what is known from single-level optimization. As a side result, our numerical experiments reveal that there is an urgent need for better and more heterogeneous test instance libraries to further propel the field of computational bilevel optimization.

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.

Solving Linear Bilevel Problems Using Big-Ms: Not All That Glitters Is Gold

The most common procedure to solve a linear bilevel problem in the PES community is, by far, to transform it into an equivalent single-level problem by replacing the lower level with its KKT optimality conditions. Then, the complementarity conditions are reformulated using binary variables and large enough constants (big-Ms) to cast the single-level problem as a mixed-integer linear program that can be solved using optimization software. In most cases, such large constants are tuned by trial and error. We show, through a counterexample, that this widely used trial-and-error approach may lead to highly suboptimal solutions. Then, further research is required to properly select big-M values to solve linear bilevel problems.

A Hierarchical Approach to Enabling Supplier Choice In On-Demand Platforms

In peer-to-peer logistics platforms, capacity cannot be set. Instead, capacity must be enticed from decentralized suppliers (who provide access to their resources). Current peer-to-peer platforms for crowdsourced delivery and ride-sharing do not operationally prioritize suppliers' underutilized resources when making assignments. This paper is interested in evaluating a new hierarchical approach, recasting the platform's role as one providing personalized menus of requests for suppliers to choose from. Supplier choice can increase participation (capacity) and resource utilization. However, coordinating decentralized resources using multiple, simultaneous recommendations is complicated because platform performance is the result of interdependent suppliers' selection outcomes, in which some requests may not be selected and other requests are selected by more than one supplier. Therefore, the objective of this research is to determine when providing suppliers with choices is beneficial to the platform and to quantify this benefit for different environments. A novel 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 having a choice to select which request (if any) they would like to serve. 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. We find that 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 dynamic reformulation heuristic for Generalized Interdiction Problems

We consider a subfamily of mixed-integer linear bilevel problems that we call Generalized Interdiction Problems. This class of problems includes, among others, the widely-studied interdiction problems, i.e., zero-sum Stackelberg games where two players (called the leader and the follower) share a set of items, and the leader can interdict the usage of certain items by the follower. Problems of this type can be modeled as Mixed-Integer Nonlinear Programming problems, whose exact solution can be very hard. In this paper we propose a new heuristic scheme based on a single-level and compact mixed-integer linear programming reformulation of the problem obtained by relaxing the integrality of the follower variables. A distinguished feature of our method is that general-purpose mixed-integer cutting planes for the follower problem are exploited, on the fly, to dynamically improve the reformulation. The resulting heuristic algorithm proved very effective on a large number of test instances, often providing an (almost) optimal solution within very short computing times.

Operational scheduling of microgrids via parametric programming

We present a parametric programming based approach for energy management in microgrids. An operational planning problem for a grid-connected microgrid with energy sources including solar photovoltaic, wind turbine and battery energy storage system, in addition to a household load demand, is captured as a parametric mixed-integer linear programming problem (p-MILP) through parameterizations of the uncertain coordinates of wind and solar energy resources. Thus, the energy management problem - typically nonlinear - is transformed into a linear bi-level optimization problem, where choice of the parameterization scheme is made at the upper level while system operation decisions are made at the lower level. The p-MILP formulation leads to significant improvements in uncertainty handling, solution quality and computational ease; by removing dependency of the solution on meteorological forecasts and avoiding the multiple computational cycles of the traditional online optimization techniques. The problem is solved offline on a flexible time-scale basis, allowing online implementation to be achievable on real-time system state updates. The proposed parametric programming approach extends the state-of-the-art in microgrid energy management methods and the results from various case studies are used to demonstrate the feasibility of our method.

A Stochastic Approach to Solving Bilevel Natural Gas Cash-Out Problems

We study a special bilevel programming problem that arises in transactions between a Natural Gas Shipping Company and a Pipeline Operator. Because of the business relationships between these two actors, the timing, and objectives of their decision-making process are different. In order to model that, bilevel programming was traditionally used. Apart from the theoretical studies of the problem to facilitate its solution a linear reformulation is required, as well as heuristic approaches, and branch-and-bound techniques may be applied. We present a linear programming reformulation of the latest version of the model, which is easier and faster to solve numerically. This reformulation makes it easier to theoretically analyze the problem, allowing us to draw some conclusions about the nature of the solution.Since elements of uncertainty are definitely present in the bilevel natural gas cash-out problem, its stochastic formulation is developed in the form of a bilevel multi-stage stochastic programming model with recourse. After reducing the original formulation to a bilevel linear problem, a stochastic scenario tree is defined by its node events, and time series forecasting is used to produce stochastic values for data of natural gas price and demand. Numerical experiments were run to compare the stochastic solution with the perfect information solution and the expected value solutions.

Enhanced exact algorithms for discrete bilevel linear problems

We address a particular class of bilevel linear programming problems in which all the variables are discrete. The main computational complexities are analyzed and two enhanced exact algorithms are proposed. The rationale behind these two algorithms is described and a modified version is presented for both. A common test bed is used to assess their computational efficiency along with a comparison with an existing benchmark algorithm.

On the choice of the penalty parameter for discrete-continuous linear bi-level problems reformulation

In this paper we focus on linear bi-level problems in which the variables controlled by the leader are discrete. It is known that such problems are equivalent to continuous linear bi-level problems in which the integrality requirements are relaxed and the leader’s objective function is modified including a concave penalty function weighted by a parameter µ. The equivalence holds for a sufficiently large value of µ. A valid lower bound for µ is known in the literature. In the following, we provide an improvement of this lower bound and experiment the new lower bound on a set of test problems.

Benders Decomposition for Discrete–Continuous Linear Bilevel Problems with application to traffic network design

We propose a new fast solution method for linear Bilevel Problems with binary leader and continuous follower variables under the partial cooperation assumption. We reformulate the Bilevel Problem into a single-level problem by using the Karush–Kuhn–Tucker conditions. This non-linear model can be linearized because of the special structure achieved by the binary leader decision variables and subsequently solved by a Benders Decomposition Algorithm to global optimality. We illustrate the capability of the approach on the Discrete Network Design Problem which adds arcs to an existing road network at the leader stage and anticipates the traffic equilibrium for the follower stage. Because of the non-linear objective functions of this problem, we use a linearization method for increasing, convex and non-linear functions based on continuous variables. Numerical tests show that this algorithm can solve even large instances of Bilevel Problems.

An algorithm based on particle swarm optimization for multiobjective bilevel linear problems

This paper presents an improved multiple objective particle swarm optimization (MOPSO) algorithm to solve bilevel linear programming problems with multiple objective functions at the upper level. The algorithm aims to produce a good approximation of the entire Pareto front of the problem. We have previously designed a MOPSO algorithm for the same class of problems, in which several techniques for the global best selection were tested, including a new one. The algorithm revealed a good convergence towards the Pareto front but the diversity of the solutions was a drawback. The algorithm we propose herein uses a hybrid strategy for the global best selection and an adaptive mutation operator. The incorporation of these mechanisms led to an improved algorithm, which also showed better overall performance than considering alternative options usually employed in MOPSO algorithms. The algorithm and computational results are presented.