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Enhanced exact algorithms for discrete bilevel linear problems

OPTIMIZATION LETTERS 2015年第7期9卷 1447-1468页

作者： Caramia, Massimiliano Mari, Renato Univ Roma Tor Vergata Dipartimento Ingn Impresa I-00133 Rome Italy

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.

关键词： bilevel linear programming Discrete bilevel linear problems Exact methods Cutting planes Branch and cut

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The Improvement of Optimality Test over Possible Reaction Set in bilevel linear Optimization with Ambiguous Objective Function of the Follower

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JOURNAL OF ADVANCED COMPUTATIONAL INTELLIGENCE AND INTELLIGENT INFORMATICS 2015年第5期19卷 645-654页

作者： Sariddichainunta, Puchit Inuiguchi, Masahiro Osaka Univ Grad Sch Engn Sci 1-3 Kanemachiyama Toyonaka Osaka 5608531 Japan

Verifying a rational response is the most crucial step in searching for an optimal solution in bilevel linear programming. Such verification is even difficult in a model with ambiguous objective function of the follower who reacts rationally to a leader's decision. In our model, we assume that the ambiguous coefficient vector of follower lies in a convex polytope and we formulate bilevel linear programming with the ambiguous objective function of the follower as a special three-level programming problem. We use the k-th best method that sequentially enumerates a solution and examine whether it is the best of all possible reactions. The optimality test process over possible reactions in lower-level problems usually encounters degenerate bases that become obstacles to verifying the optimality of an enumerated solution efficiently. To accelerate optimality verification, we propose search strategies and the evaluation of basic possible reactions adjacent to a degenerate basic solution. We introduce these methods in both local and global optimality testing, confirming the effectiveness of our proposed methods in numerical experiments.

关键词： bilevel linear programming degenerate solution maximin optimization

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

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APPLIED MATHEMATICS AND COMPUTATION 2014年 247卷 547-561页

作者： Alves, Maria Joao Costa, Joao Paulo Univ Coimbra INESC Coimbra Fac Econ P-3004512 Coimbra Portugal

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. (C) 2014 Elsevier Inc. All rights reserved.

关键词： Multiobjective optimization bilevel linear programming Particle swarm optimization

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A method for solving bilevel linear programming problems

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JOURNAL OF INFORMATION & OPTIMIZATION SCIENCES 2008年第2期29卷 335-358页

作者： Pieume, C. O. Fotso, L. P. Siarry, P. Univ Yaounde I Fac Sci Dept Math POB 812 Yaounde Cameroon Univ Yaounde I Fac Sci Dept Comp Sci Yaounde Cameroon Univ Paris XII Val Marne Fac Sci & Technol F-94010 Creteil France

This paper presents an approach for solving bilevel linear programming problems (BLPP). It is based on the result that an optimal solution to the BLPP is reachable at an extreme point of the underlying region. Consequently, we develop a pivot technique to find the global optimal solution on an expanded tableau that represents the data of the BLPP. The pivot technique allows to rank in increasing order the outer level objective function value until a value is reached with a corresponding extreme point feasible for the BLPP. This solution is then the required global solution. Numerical examples are provided. Solutions obtained through our algorithm to some problems available in the literature show that these problems were until now wrongly solved.

关键词： Optimization bilevel linear programming feasible solution

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Farm planning in nitrate sensitive agricultural areas

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INTERNATIONAL JOURNAL OF ENVIRONMENT AND SUSTAINABLE DEVELOPMENT 2012年第2期11卷 105-117页

作者： Moulogianni, Christina Bournaris, Thomas Manos, Basil Nastis, Stefanos Aristotle Univ Thessaloniki Sch Agr Dept Agr Econ Thessaloniki 54124 Greece

One of the main aims of the Rural Development Plan under the EU Common Agricultural Policy is the protection of nitrate sensitive areas through agri-environmental schemes. This paper presents a mathematical programming model for farm planning in agricultural areas that are sensitive to nitrates. A bilevel linear programming (BLP) model is developed, that can achieve the optimal farm production plan assuming two conflicting goals: the maximisation of farm gross margin and the minimisation of fertilisers' use. The first goal is pursued by farmers, and comprises the first level of BLP. The second goal is pursued by society, through the government, and comprises the second level of BLP. The model is applied to an agricultural area in Northern Greece, which belongs to the nitrate sensitive areas scheme of the Greek Rural Development Plan 2007-2013. The model is further used to simulate the impacts of the measure under two scenarios proposed for farms located in nitrate sensitive areas. The result shows that the model can achieve the two goals set by increasing gross margin and reducing fertilisers use.

关键词： farm planning bilevel linear programming BLP nitrate sensitive areas NSAs agri-environmental schemes AESs

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bilevel linear programming model of charging for effluent based on price control

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Journal of Harbin Institute of Technology(New Series) 2007年第2期14卷 292-296页

作者：李煜华李磊胡运权邵海宏 College of Economics and Management Harbin University of Science and Technology Harbin 150040 China School of Management Harbin Institute of Technology Harbin 150001 China School of Management Harbin Institute of Technology

For the optimum price problem of charging for effluent, this paper analyzes the optimal Pigovian Tax and the serious information asymmetry problem existing in the application process of optimal Pigovian Tax, which is predominant in theory. Then the bilevel system optimizing decision-making theory is applied to give bilevel linear programming decision-making model of charging for effluent, in which the government (environmental protection agency) acts as the upper level decision-making unit and the polluting enterprises act as the lower level decision-making unit. To some extent, the model avoids the serious information asymmetry between the government and the polluting enterprises on charging for effluent.

关键词： charging for effluent Pigovian Tax price control bilevel linear programming information asymmetry

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New branch-and-cut algorithm for bilevel linear programming

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2007年第2期134卷 353-370页

作者： Audet, C. Savard, G. Zghal, W. Ecole Hautes Etud Commerciales Gerad Montreal PQ Canada Ecole Polytech Montreal PQ H3C 3A7 Canada

linear mixed 0-1 integer programming problems may be reformulated as equivalent continuous bilevel linear programming (BLP) problems. We exploit these equivalences to transpose the concept of mixed 0-1 Gomory cuts to BLP. The first phase of our new algorithm generates Gomory-like cuts. The second phase consists of a branch-and-bound procedure to ensure finite termination with a global optimal solution. Different features of the algorithm, in particular, the cut selection and branching criteria are studied in details. We propose also a set of algorithmic tests and procedures to improve the method. Finally, we illustrate the performance through numerical experiments. Our algorithm outperforms pure branch-and-bound when tested on a series of randomly generated problems.

关键词： bilevel linear programming Gomory cuts linear mixed 0-1 integer programming branch-and-cut algorithms

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An adaptive genetic algorithm for solving bilevel linear programming problem

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Applied Mathematics and Mechanics(English Edition) 2007年第12期28卷 1605-1612页

作者：王广民王先甲万仲平贾世会 School of Management China University of Geosciences Institute of Systems Engineering Wuhan University School of Mathematics and Statistics Wuhan University School of Science Wuhan University of Science and Technology

bilevel linear programming, which consists of the objective functions of the upper level and lower level, is a useful tool for modeling decentralized decision problems. Various methods are proposed for solving this problem. Of all the algorithms, the ge- netic algorithm is an alternative to conventional approaches to find the solution of the bilevel linear programming. In this paper, we describe an adaptive genetic algorithm for solving the bilevel linear programming problem to overcome the difficulty of determining the probabilities of crossover and mutation. In addition, some techniques are adopted not only to deal with the difficulty that most of the chromosomes maybe infeasible in solving constrained optimization problem with genetic algorithm but also to improve the efficiency of the algorithm. The performance of this proposed algorithm is illustrated by the examples from references.

关键词： bilevel linear programming genetic algorithm fitness value adaptive operator probabilities crossover and mutation

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Solving bilevel linear Programs Using Multiple Objective linear programming

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2009年第2期140卷 197-212页

作者： Glackin, J. Ecker, J. G. Kupferschmid, M. US Mil Acad Dept Math Sci West Point NY 10996 USA Rensselaer Polytech Inst Dept Math Sci Troy NY 12180 USA Rensselaer Polytech Inst Acad & Res Comp Troy NY 12180 USA

We present an algorithm for solving bilevel linear programs that uses simplex pivots on an expanded tableau. The algorithm uses the relationship between multiple objective linear programs and bilevel linear programs along with results for minimizing a linear objective over the efficient set for a multiple objective problem. Results in multiple objective programming needed are presented. We report computational experience demonstrating that this approach is more effective than a standard branch-and-bound algorithm when the number of leader variables is small.

关键词： bilevel linear programming Multiple objective linear programming Efficient set Rational reaction set Simplex algorithm pivot

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DC programming and DCA for globally solving the value-at-risk

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COMPUTATIONAL MANAGEMENT SCIENCE 2009年第4期6卷 477-501页

作者： Tao Pham Dinh Nguyen Canh Nam Hoai An Le Thi Natl Inst Appl Sci Rouen Lab Modelling Optimizat & Operat Res BP 08Pl Emile Blondel F-76131 Mont St Aignan France Paul Verlaine Univ Metz Lab Theoret & Appl Comp Sci LITA UFR MIM EA 3097 F-57045 Metz France

The value-at-risk is an important risk measure that has been used extensively in recent years in portfolio selection and in risk analysis. This problem, with its known bilevel linear program, is reformulated as a polyhedral DC program with the help of exact penalty techniques in DC programming and solved by DCA. To check globality of computed solutions, a global method combining the local algorithm DCA with a well adapted branch- and- bound algorithm is investigated. An illustrative example and numerical simulations are reported, which show the robustness, the globality and the efficiency of DCA.

关键词： Value-at-risk bilevel linear programming DC programming DCA Branch- and -bound

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