Linear Bilevel Programming Problem Research Articles

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.

A probabilistic bi-level linear multi-objective programming problem to supply chain planning

Bi-level programming, a tool for modeling decentralized decisions, consists of the objective(s) of the leader at its first level and that is of the follower at the second level. Three level programming results when second level is itself a bi-level programming. By extending this idea it is possible to define multi-level programs with any number of levels. In most of the real life problems in mathematical programming, the parameters are considered as random variables. The branch of mathematical programming which deals with the theory and methods for the solution of conditional extremum problems under incomplete information about the random parameters is called “stochastic programming”. Supply chain planning problems are concerned with synchronizing and optimizing multiple activities involved in the enterprise, from the start of the process, such as procurement of the raw materials, through a series of process operations, to the end, such as distribution of the final product to customers. Enterprise-wide supply chain planning problems naturally exhibit a multi-level decision network structure, where for example, one level may correspond to a local plant control/scheduling/planning problem and another level to a corresponding plant-wide planning/network problem. Such a multi-level decision network structure can be mathematically represented by using “multi-level programming” principles. In this paper, we consider a “probabilistic bi-level linear multi-objective programming problem” and its application in enterprise-wide supply chain planning problem where (1) market demand, (2) production capacity of each plant and (3) resource available to all plants for each product are random variables and the constraints may consist of joint probability distributions or not. This probabilistic model is first converted into an equivalent deterministic model in each level, to which fuzzy programming technique is applied to solve the multi-objective nonlinear programming problem to obtain a compromise solution.

A penalty function method based on Kuhn–Tucker condition for solving linear bilevel programming

Using the Kuhn–Tucker optimality condition of the lower level problem, we transform the linear bilevel programming problem into a corresponding single level programming. The complementary and slackness condition of the lower level problem is appended to the upper level objective with a penalty. Then we decompose the linear bilevel programming into a series of linear programming problems and get the optimal solution of the linear bilevel programming using linear programming method.

A penalty function method for solving weak price control problem

The price control problem is an important linear bilevel programming problem. In this paper, we are concerned with a class of weak price control problems with non-unique lower level solutions. For such problems, we study the existence of solution via a penalty method. Then, we propose a simple algorithm for this problem, and present the convergence of the algorithm. The example illustrates that the method is feasible and efficient.

Disjunctive cuts for continuous linear bilevel programming

This work shows how disjunctive cuts can be generated for a bilevel linear programming problem (BLP) with continuous variables. First, a brief summary on disjunctive programming and bilevel programming is presented. Then duality theory is used to reformulate BLP as a disjunctive program and, from there, disjunctive programming results are applied to derive valid cuts. These cuts tighten the domain of the linear relaxation of BLP. An example is given to illustrate this idea, and a discussion follows on how these cuts may be incorporated in an algorithm for solving BLP.

A note on the definition of a linear bilevel programming solution

An alternative definition of the linear bilevel programming problem BLP has recently been proposed by Lu, Shi, and Zhang. This note shows that the proposed definition is a restriction of BLP. Indeed, the new definition is equivalent to transferring the first-level constraints involving second-level variables into the second level, resulting in a special case of BLP in which there are no first-level constraint involving second-level variables. Thus, contrary to what is stated by the authors who suggested the new definition, this does not allow to solve a wider class of problems, but rather relaxes the feasible region, allowing for infeasible points to be considered as feasible.

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 programming with upper level constraints depending on the lower level solution

Focus in the paper is on the definition of linear bilevel programming problems, the existence of optimal solutions and necessary as well as sufficient optimality conditions. In the papers [C. Shi, G. Zhang, J. Lu, An extended Kuhn–Tucker approach for linear bilevel programming, Appl. Math. Comput. 162 (2005) 51–63] and [C. Shi, G. Zhang, J. Lu, On the definition of linear bilevel programming solution, Appl. Math. Comput. 160 (2005) 169–176], the authors claim to suggest a refined definition of linear bilevel programming problems and related optimality conditions. Mainly their attempt reduces to shifting upper level constraints involving both the upper and the lower level variables into the lower level. We investigate such a shift in more details and show that it is not allowed in general. We show that an optimal solution of the bilevel program exists under the conditions in [C. Shi, G. Zhang, J. Lu, On the definition of linear bilevel programming solution, Appl. Math. Comput. 160 (2005) 169–176] if we add the assumption that the inducible region is not empty. The necessary optimality condition reduces to check optimality in one linear programming problem. Optimality of one feasible point for a certain number of linear programs implies optimality for the bilevel problem.

A Simplex Approach for Finding Local Solutions of a Linear Bilevel Program by Equilibrium Points

In this paper, a linear bilevel programming problem (LBP) is considered. Local optimality conditions are derived. They are based on the notion of equilibrium point of an exact penalization for LBP. It is described how an equilibrium point can be obtained with the simplex method. It is shown that the information in the simplex tableaux can be used to get necessary and sufficient local optimality conditions for LBP. Based on these conditions, a simplex type algorithm is proposed, which attains a local solution of LBP by moving in equilibrium points. A numerical example illustrates how the algorithm works. Some computational results are reported.

A Study of Local Solutions in Linear Bilevel Programming

In this paper, a linear bilevel programming problem (LBP) is considered. Local optimality in LBP is studied via two related problems (P) and P(M). Problem (P) is a one-level model obtained by replacing the innermost problem of LBP by its KKT conditions. Problem P(M) is a penalization of the complementarity constraints of (P) with a penalty parameter M. Characterizations of a (strict) local solution of LBP are derived. In particular, the concept of equilibrium point of P(M) is used to characterize the local optima of (P) and LBP.

Weak linear bilevel programming problems: existence of solutions via a penalty method

We are concerned with a class of weak linear bilevel programs with nonunique lower level solutions. For such problems, we give via an exact penalty method an existence theorem of solutions. Then, we propose an algorithm.

An extended Kth-best approach for linear bilevel programming

Kth-best approach is one of the three popular and workable approaches for linear bilevel programming. However, it could not well deal with a linear bilevel programming problem when the constraint functions at the upper-level are of arbitrary linear form. Based on our previous work, this paper proposes theoretical properties of linear bilevel programming and presents an extended Kth-best approach for linear bilevel programming which can effectively solve above problem.

An extended Kuhn–Tucker approach for linear bilevel programming

Kuhn–Tucker approach has been applied with remarkable success in linear bilevel programming (BLP). However, it still has some extent unsatisfactory and incomplete. One principle challenges is that it could not well handle a linear BLP problem when the constraint functions at the upper-level are of arbitrary linear form. This paper describes theoretical foundation of Kuhn–Tucker approach and proposes an extended Kuhn–Tucker approach to deal with the problem. The results have demonstrated that the extended Kuhn–Tucker approach can solve a wider class of linear BLP problems can than current capabilities permit.

A note on ‘bilevel linear fractional programming problem’

In this note, the bilevel linear fractional programming problem is considered. We examine a previous paper published by Thirwani and Arora [Cah. CERO 35 (1993) 135]. We show that several proofs in that paper are incorrect and provide alternative proofs.

On the definition of linear bilevel programming solution

Linear bilevel programming theory has been studied for many years by a number of researchers from different aspects, yet it still remains to some extent unsatisfactory and incomplete. The main challenge is how to solve a linear bilevel programming problem when the upper-level's constraint functions are of arbitrary linear form. This paper proposes a definition for linear bilevel programming solution. The performance comparisons have demonstrated that the new model can solve a wider class of problems than current capabilities permit.

A note on a modified simplex approach for solving bilevel linear programming problems

We analyze the article “A modified simplex approach for solving bilevel linear programming problems” (EJOR, 67, 126–135). We point out some problems in its theoretical analysis. Moreover, the algorithm proposed may not find a global solution as it is claimed. We give some examples in order to illustrate these remarks.

Deriving the Properties of Linear Bilevel Programming via a Penalty Function Approach

For the linear bilevel programming problem, we propose an assumption weaker than existing assumptions, while achieving similar results via a penalty function approach. The results include: equivalence between (i) existence of a solution to the problem, (ii) existence of an exact penalty function approach for solving the problem, and (iii) achievement of the optimal value of the equivalent form of the problem at some vertex of a certain polyhedral convex set. We prove that the assumption is both necessary and sufficient for the linear bilevel programming problem to admit an exact penalty function formulation, provided that the equivalent form of the problem has a feasible solution. A method is given for computing the minimal penalty function parameter value. This method can be executed by solving a set of linear programming problems. Lagrangian duality is also presented.

A note on efficient solutions for the linear bilevel programming problem

The solutions of linear bilevel programming problems frequently are non-Pareto-optimal. The potential increase in payoffs generated by Pareto improvements makes it worthwhile to consider methods with which to move the solution to the efficient frontier. Bargaining models offer one class of solutions, which, contrary to the original non-cooperative, sequential decision-making situation, however, assume cooperation. We make an attempt to maintain the original power structure by introducing the asymmetric Nash bargaining solution.

A Multilevel Programming Approach to Robust Optimal Design of Energy Supply Systems.

A multilevel programming approach is proposed to optimal design of energy supply systems in consideration of their robustness in economic and energy saving characteristics against the energy demand uncertainty. Equipment capacities and utility contract demands are determined so as to minimize the annual total cost for average energy demands, to minimize the difference between the annual operational costs for the worst and average energy demands, to maximize the difference between the annual operational costs for average and the best energy demands, and to satisfy all the possible energy demands. This optimization problem is formulated as a three-level linear programming problem, and its solution is obtained by repeatedly solving a linear and a set of linear and bilevel linear programming problems which give, respectively, lower and upper bounds for the original objective function. Through a numerical case study on a cogeneration system, the effectiveness of this approach is ascertained, and the influence of consideration of the robustness on the cogeneration unit sizing is clarified.

An algorithm for the integer linear fractional bilevel programming problem

The bilevel programming problem is an example of a two stage, non co-operative game in which the first player can influence but not control the actions of the second. In this paper we examine the case when the objective functions are linear fractional and present an algorithm for solving the integer case. We begin by solving the leader’s objective function and finding its integral optimal solution. Using the ranking in integer linear fractional programming problem, we are able to solve the given bilevel programming problem