Bilevel Linear Problem Research Articles

On a stochastic bilevel programming problem

AbstractIn this article, a mixed integer bilevel problem having a probabilistic knapsack constraint in the first level is proposed. The problem formulation is mainly motivated by practical pricing and service provision problems as it can be interpreted as a model for the interaction between a service provider and customers. A discrete probability space is assumed which allows a reformulation of the problem as an equivalent deterministic bilevel problem. The problem is further transformed into a linear bilevel problem, which in turn yields a quadratic optimization problem, namely the global linear complementarity problem. Based on this quadratic problem, a procedure to compute upper bounds on the initial problem by using a Lagrangian relaxation and an iterative linear minmax scheme is proposed. Numerical experiments confirm that the scheme practically converges.© 2011 Wiley Periodicals, Inc. NETWORKS, 2012

Linear bilevel programming with interval coefficients

In this paper, we address linear bilevel programs when the coefficients of both objective functions are interval numbers. The focus is on the optimal value range problem which consists of computing the best and worst optimal objective function values and determining the settings of the interval coefficients which provide these values. We prove by examples that, in general, there is no precise way of systematizing the specific values of the interval coefficients that can be used to compute the best and worst possible optimal solutions. Taking into account the properties of linear bilevel problems, we prove that these two optimal solutions occur at extreme points of the polyhedron defined by the common constraints. Moreover, we develop two algorithms based on ranking extreme points that allow us to compute them as well as determining settings of the interval coefficients which provide the optimal value range.

Natural gas cash-out problem: Bilevel stochastic optimization approach

A stochastic formulation of the natural gas cash-out problem is given in a 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.

Numerical solution of a linear bilevel problem

The linear bilevel programming problem in the optimistic formulation is studied. It is reduced to an optimization problem with a nonconvex constraint in the form of a d.c. function (that is, the difference of two convex functions). For this problem, local and global search methods are developed. Numerical experiments performed for numerous specially generated problems, including large-scale ones, demonstrate the efficiency of the proposed approach.

Constraint programming approach to a bilevel scheduling problem

Bilevel optimization problems involve two decision makers who make their choices sequentially, either one according to its own objective function. Many problems arising in economy and management science can be modeled as bilevel optimization problems. Several special cases of bilevel problem have been studied in the literature, e.g., linear bilevel problems. However, up to now, very little is known about solution techniques of discrete bilevel problems. In this paper we show that constraint programming can be used to model and solve such problems. We demonstrate our first results on a simple bilevel scheduling problem.

On linear bilevel problems with multiple objectives at the lower level

Bilevel programming problems provide a framework to deal with decision processes involving two decision makers with a hierarchical structure. They are characterized by the existence of two optimization problems in which the constraint region of the upper level problem is implicitly determined by the lower level optimization problem. This paper focuses on bilevel problems for which the lower level problem is a linear multiobjective program and constraints at both levels define polyhedra. This bilevel problem is reformulated as an optimization problem over a nonconvex region given by a union of faces of the polyhedron defined by all constraints. This reformulation is obtained when dealing with efficient solutions as well as weakly efficient solutions for the lower level problem. Assuming that the upper level objective function is quasiconcave, then an extreme point exists which solves the problem. An exact and a metaheuristic algorithm are developed and their performance is analyzed and compared.

Linear bilevel programs with multiple objectives at the upper level

Bilevel programming has been proposed for dealing with decision processes involving two decision makers with a hierarchical structure. They are characterized by the existence of two optimization problems in which the constraint region of the upper level problem is implicitly determined by the lower level optimization problem. Focus of the paper is on general bilevel optimization problems with multiple objectives at the upper level of decision making. When all objective functions are linear and constraints at both levels define polyhedra, it is proved that the set of efficient solutions is non-empty. Taking into account the properties of the feasible region of the bilevel problem, some methods of computing efficient solutions are given based on both weighted sum scalarization and scalarization techniques. All the methods result in solving linear bilevel problems with a single objective function at each level.

A new approach for solving linear bilevel problems using genetic algorithms

Bilevel programming involves two optimization problems where the constraint region of the first level problem is implicitly determined by another optimization problem. This paper develops a genetic algorithm for the linear bilevel problem in which both objective functions are linear and the common constraint region is a polyhedron. Taking into account the existence of an extreme point of the polyhedron which solves the problem, the algorithm aims to combine classical extreme point enumeration techniques with genetic search methods by associating chromosomes with extreme points of the polyhedron. The numerical results show the efficiency of the proposed algorithm. In addition, this genetic algorithm can also be used for solving quasiconcave bilevel problems provided that the second level objective function is linear.

Resolution method for mixed integer bi-level linear problems based on decomposition technique

In this article, we propose a new algorithm for the resolution of mixed integer bi-level linear problem (MIBLP). The algorithm is based on the decomposition of the initial problem into the restricted master problem (RMP) and a series of problems named slave problems (SP). The proposed approach is based on Benders decomposition method where in each iteration a set of variables are fixed which are controlled by the upper level optimization problem. The RMP is a relaxation of the MIBLP and the SP represents a restriction of the MIBLP. The RMP interacts in each iteration with the current SP by the addition of cuts produced using Lagrangian information from the current SP. The lower and upper bound provided from the RMP and SP are updated in each iteration. The algorithm converges when the difference between the upper and lower bound is within a small difference ?. In the case of MIBLP Karush---Kuhn---Tucker (KKT) optimality conditions could not be used directly to the inner problem in order to transform the bi-level problem into a single level problem. The proposed decomposition technique, however, allows the use of KKT conditions and transforms the MIBLP into two single level problems. The algorithm, which is a new method for the resolution of MIBLP, is illustrated through a modified numerical example from the literature. Additional examples from the literature are presented to highlight the algorithm convergence properties.

Linear bilevel multi-follower programming with independent followers

This paper considers a particular case of linear bilevel programming problems with one leader and multiple followers. In this model, the followers are independent, meaning that the objective function and the set of constraints of each follower only include the leader's variables and his own variables. We prove that this problem can be reformulated into a linear bilevel problem with one leader and one follower by defining an adequate second level objective function and constraint region. In the second part of the paper we show that the results on the optimality of the linear bilevel problem with multiple independent followers presented in Shi et al. [The kth-best approach for linear bilevel multi-follower programming, J. Global Optim. 33, 563---578 (2005)] are based on a misconstruction of the inducible region.

An extended branch and bound algorithm for linear bilevel programming

For linear bilevel programming, the branch and bound algorithm is the most successful algorithm to deal with the complementary constraints arising from Kuhn–Tucker conditions. However, one principle challenge is that it could not well handle a linear bilevel programming problem when the constraint functions at the upper-level are of arbitrary linear form. This paper proposes an extended branch and bound algorithm to solve this problem. The results have demonstrated that the extended branch and bound algorithm can solve a wider class of linear bilevel problems can than current capabilities permit.

Linear bilevel problems: Genericity results and an efficient method for computing local minima

The paper is concerned with linear bilevel problems. These nonconvex problems are known to be NP-complete. So, no efficient method for solving the global bilevel problem can be expected. In this paper we give a genericity analysis of linear bilevel problems and present a new algorithm for computing efficiently local minimizers. The method is based on the given structural analysis and combines ideas of the Simplex method with projected gradient steps.

On a new class of bilevel programming problems and its use for reformulating mixed integer problems

We extend some known results about the Bilevel Linear Problem (BLP), a hierarchical two-stage optimization problem, showing how it can be used to reformulate any Mixed Integer (Linear) Problem; then, we introduce some new concepts, which might be useful to fasten almost all the known algorithms devised for BLP. As this kind of reformulation appears to be somewhat artificial, we define a natural generalization of BLP, the Bilevel Linear/Quadratic Problem (BL/QP), and show that most of the exact and/or approximate algorithms originally devised for the BLP, such as GSA or K-th Best, can be extended to this new class of Bilevel Programming Problems. For BL/QP, more ‘natural’ reformulations of MIPs are available, leading to the use of known (nonexact) algorithms for BLP as (heuristic) approaches to MIPs: we report some contrasting results obtained in the Network Design Problem case, showing that, although the direct application of our (Dual) GSA algorithm is not of any practical use, we obtain as a by-product a good theoretical characterization of the optimal solutions set of the NDP, along with a powerful scheme for constructing fast algorithms for the Minimum Cost Flow Problem with piecewise convex linear cost functions.

A note on the Pareto optimality of solutions to the linear bilevel programming problem

In this note it is shown that no general hypothesis on the cost functions can guarantee that a Pareto optimal solution to a linear bilevel programming problem exists. Indeed, whatever the leader's and follower's objectives, it is always possible to design constraints such that the resulting linear bilevel problem's solution is not Pareto optimal.