##
**Bilevel and multilevel programming: A bibliography review.**
*(English)*
Zbl 0822.90127

Summary: This paper contains a bibliography of all references central to bilevel and multilevel programming that the authors know of. It should be regarded as a dynamic and permanent contribution since all the new and appropriate references that are brought to our attention will be periodically added to this bibliography. Readers are invited to suggest such additions, as well as corrections or modifications, and to obtain a copy of the LaTeX and BibTeX files that constitute this manuscript, using the guidelines contained in this paper.

To classify some of the references in this bibliography a short overview of past and current research in bilevel and multilevel programming is included. For those who are interested in but unfamiliar with the references in this area, we hope that this bibliography facilities and encourages their research.

To classify some of the references in this bibliography a short overview of past and current research in bilevel and multilevel programming is included. For those who are interested in but unfamiliar with the references in this area, we hope that this bibliography facilities and encourages their research.

### MSC:

90C30 | Nonlinear programming |

90-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to operations research and mathematical programming |

91A65 | Hierarchical games (including Stackelberg games) |

93A13 | Hierarchical systems |

### Keywords:

static Stackelberg problems; hierarchical optimization; minimax problems; bilevel programming; bibliography; multilevel programming### Software:

Algorithm 728
PDFBibTeX
XMLCite

\textit{L. N. Vicente} and \textit{P. H. Calamai}, J. Glob. Optim. 5, No. 3, 291--306 (1994; Zbl 0822.90127)

Full Text:
DOI

### References:

[1] | E. Aiyoshi and K. Shimizu. Hierarchical decentralized systems and its new solution by a barrier method.IEEE Transactions on Systems, Man, and Cybernetics, 11:444-449, 1981. · Zbl 0472.93004 |

[2] | E. Aiyoshi and K. Shimizu. A solution method for the static constrained Stackelberg problem via penalty method.IEEE Transactions on Automatic Control, 29:1111-1114, 1984. · Zbl 0553.90104 |

[3] | F. Al-Khayyal, R. Horst, and P. Pardalos. Global optimization of concave functions subject to quadratic constraints: an application in nonlinear bilevel programming.Annals of Operations Research, 34:125-147, 1992. · Zbl 0751.90066 |

[4] | G. Anandalingam. An analysis of information and incentives in bi-level programming. InIEEE 1985 Proceedings of the International Conference on Cybernetics and Society, pages 925-929, 1985. |

[5] | G. Anandalingam. A mathematical programming model of decentralized multi-level systems.Journal of the Operational Research Society, 39:1021-1033, 1988. · Zbl 0657.90061 |

[6] | G. Anandalingam and V. Aprey. Multi-level programming and conflict resolution.European Journal of Operational Research, 51:233-247, 1991. · Zbl 0743.90127 |

[7] | G. Anandalingam and T. Friesz. Hierarchical optimization: an introduction.Annals of Operations Research, 34:1-11, 1992. · Zbl 0751.90067 |

[8] | G. Anandalingam, R. Mathieu, L. Pittard, and N. Sinha. Artificial intelligence based approaches for solving hierarchical optimization problems. In R. Sharda, B. Golden, E. Wasil, O. Balci and W. Stewart, editor,Impacts of Recent Computer Advances on Operations Research, pages 289-301. Elsevier Science Publishing Co., Inc., 1983. |

[9] | G. Anandalingam and D. White. A solution method for the linear static Stackelberg problem using penalty functions.IEEE Transactions on Automatic Control, 35:1170-1173, 1990. · Zbl 0721.90098 |

[10] | J. Bard. A grid search algorithm for the linear bilevel programming problem. InProceedings of the 14th Annual Meeting of the American Institute for Decision Science, pages 256-258, 1982. |

[11] | J. Bard. An algorithm for solving the general bilevel programming problem.Mathematics of Operations Research, 8:260-272, 1983. · Zbl 0516.90061 |

[12] | J. Bard. Coordination of a multidivisional organization through two levels of management.OMEGA, 11:457-468, 1983. |

[13] | J. Bard. An efficient point algorithm for a linear two-stage optimization problem.Operations Research, 31:670-684, 1983. · Zbl 0525.90086 |

[14] | J. Bard. An investigation of the linear three level programming problem.IEEE Transactions on Systems, Man, and Cybernetics, 14:711-717, 1984. · Zbl 0552.90081 |

[15] | J. Bard. Optimality conditions for the bilevel programming problem.Naval Research Logistics Quarterly, 31:13-26, 1984. · Zbl 0537.90087 |

[16] | J. Bard. Geometric and algorithm developments for a hierarchical planning problem.European Journal of Operational Research, 19:372-383, 1985. · Zbl 0574.90045 |

[17] | J. Bard. Convex two-level optimization.Mathematical Programming, 40:15-27, 1988. · Zbl 0655.90060 |

[18] | J. Bard. Some properties of the bilevel programming problem.Journal of Optimization Theory and Applications, 68:371-378, 1991. Technical Note. · Zbl 0696.90086 |

[19] | J. Bard and J. Falk. An explicit solution to the multi-level programming problem.Computers and Operations Research, 9:77-100, 1982. |

[20] | J. Bard and J. Moore. A branch and bound algorithm for the bilevel programming problem.SIAM Journal on Scientific and Statistical Computing, 11:281-292, 1990. · Zbl 0702.65060 |

[21] | J. Bard and J. Moore. An algorithm for the discrete bilevel programming problem.Naval Research Logistics, 39:419-35, 1992. · Zbl 0751.90111 |

[22] | O. Ben-Ayed.Bilevel linear programming: analysis and application to the network design problem. PhD thesis, University of Illinois at Urbana-Champaign, 1988. |

[23] | O. Ben-Ayed. A bilevel linear programming model applied to the Tunisian inter-regional network design problem.Revue Tunisienne d’Économie et de Gestion, 5:235-279, 1990. |

[24] | O. Ben-Ayed. Bilevel linear programming.Computers and Operations Research, 20:485-501, 1993. · Zbl 0783.90068 |

[25] | O. Ben-Ayed and C. Blair. Computational difficulties of bilevel linear programming.Operations Research, 38:556-560, 1990. · Zbl 0708.90052 |

[26] | O. Ben-Ayed, C. Blair, D. Boyce, and L. LeBlanc. Construction of a real-world bilevel linear programming model of the highway design problem.Annals of Operations Research, 34:219-254, 1992. · Zbl 0729.91035 |

[27] | O. Ben-Ayed, D. Boyce, and C. Blair. A general bilevel linear programming formulation of the network design problem.Transportation Research, 22 B:311-318, 1988. |

[28] | H. Benson. On the structure and properties of a linear multilevel programming problem.Journal of Optimization Theory and Applications, 60:353-373, 1989. · Zbl 0632.90073 |

[29] | Z. Bi.Numerical methods for bilevel programming problems. PhD thesis, Department of Systems Design Engineering, University of Waterloo, 1992. |

[30] | Z. Bi and P. Calamai. Optimality conditions for a class of bilevel programming problems. Technical Report #191-O-191291, Department of Systems Design Engineering, University of Waterloo, 1991. |

[31] | Z. Bi, P. Calamai, and A. Conn. An exact penalty function approach for the linear bilevel programming problem. Technical Report #167-O-310789, Department of Systems Design Engineering, University of Waterloo, 1989. |

[32] | Z. Bi, P. Calamai, and A. Conn. An exact penalty function approach for the nonlinear bilevel programming problem. Technical Report #180-O-170591, Department of Systems Design Engineering, University of Waterloo, 1991. |

[33] | W. Bialas and M. Karwan. Multilevel linear programming. Technical Report 78-1, Operations Research Program, State University of New York at Buffalo, 1978. |

[34] | W. Bialas and M. Karwan. On two-level optimization.IEEE Transactions on Automatic Control, 27:211-214, 1982. · Zbl 0487.90005 |

[35] | W. Bialas and M. Karwan. Two-level linear programming.Management Science, 30:1004-1020, 1984. · Zbl 0559.90053 |

[36] | W. Bialas, M. Karwan, and J. Shaw. A parametric complementary pivot approach for two-level linear programming. Technical Report 80-2, Operations Research Program, State University of New York at Buffalo, 1980. |

[37] | J. Bisschop, W. Candler, J. Duloy, and G. O’Mara. The indus basin model: a special application of two-level linear programming.Mathematical Programming Study, 20:30-38, 1982. · Zbl 0488.90076 |

[38] | C. Blair. The computational complexity of multi-level linear programs.Annals of Operations Research, 34:13-19, 1992. · Zbl 0751.90046 |

[39] | J. Bracken, J. Falk, and J. McGill. Equivalence of two mathematical programs with optimization problems in the constraints.Operations Research, 22:1102-1104, 1974. · Zbl 0294.90071 |

[40] | J. Bracken and J. McGill. Mathematical programs with optimization problems in the constraints.Operations Research, 21:37-44, 1973. · Zbl 0263.90029 |

[41] | J. Bracken and J. McGill. Defense applications of mathematical programs with optimization problems in the constraints.Operations Research, 22:1086-1096, 1974. · Zbl 0292.90057 |

[42] | J. Bracken and J. McGill. A method for solving mathematical programs with nonlinear programs in the constraints.Operations Research, 22:1097-1101, 1974. · Zbl 0294.90070 |

[43] | J. Bracken and J. McGill. Production and marketing decisions with multiple objectives in a competitive environment.Journal of Optimization Theory and Applications, 24:449-458, 1978. · Zbl 0351.90001 |

[44] | P. Calamai and L. Vicente. Generating linear and linear-quadratic bilevel programming problems.SIAM Journal on Scientific and Statistical Computing, 14:770-782, 1993. · Zbl 0802.65079 |

[45] | P. Calamai and L. Vicente. Algorithm 728: Fortran subroutines for generating quadratic bilevel programming problems.ACM Transactions on Mathematical Software, 20:120-123, 1994. · Zbl 0888.65069 |

[46] | P. Calamai and L. Vicente. Generating quadratic bilevel programming problems.ACM Transactions on Mathematical Software, 20:103-119, 1994. · Zbl 0888.65068 |

[47] | W. Candler. A linear bilevel programming algorithm: A comment.Computers and Operations Research, 15:297-298, 1988. · Zbl 0641.90050 |

[48] | W. Candler, J. Fortuny-Amat, and B. McCarl. The potential role of multilevel programming in agricultural economics.American Journal of Agricultural Economics, 63:521-531, 1981. |

[49] | W. Candler and R. Norton. Multilevel programming. Technical Report 20, World Bank Development Research Center, Washington D.C., 1977. |

[50] | W. Candler and R. Norton. Multilevel programming and development policy. Technical Report 258, World Bank Staff, Washington D.C., 1977. |

[51] | W. Candler and R. Townsley. A linear two-level programming problem.Computers and Operations Research, 9:59-76, 1982. |

[52] | R. Cassidy, M. Kirby, and W. Raike. Efficient distribution of resources through three levels of government.Management Science, 17:462-473, 1971. |

[53] | M. Cellis, J. Dennis, and R. Tapia. A trust region strategy for nonlinear equality constrained optimization. InNumerical Optimization 1984, Proceedings 20, pages 71-82. SIAM, Philadelphia, 1985. |

[54] | Y. Chen.Bilevel programming problems: analysis, algorithms and applications. PhD thesis, Université de Montréal, 1993. |

[55] | Y. Chen and M. Florian. The nonlinear bilevel programming problem: a general formulation and optimality conditions. Technical Report CRT-794, Centre de Recherche sur les Transports, 1991. |

[56] | Y. Chen and M. Florian. On the geometry structure of linear bilevel programs: a dual approach. Technical Report CRT-867, Centre de Recherche sur les Transports, 1992. |

[57] | Y. Chen and M. Florian. The nonlinear bilevel programming problem: formulations, regularity and optimality conditions. Technical Report CRT-794, Centre de Recherche sur les Transports, 1993. · Zbl 0817.90101 |

[58] | Y. Chen, M. Florian, and S. Wu. A descent dual approach for linear bilevel programs. Technical Report CRT-866, Centre de Recherche sur les Transports, 1992. |

[59] | P. Clarke and A. Westerberg. A note on the optimality conditions for the bilevel programming problem.Naval Research Logistics, 35:413-418, 1988. · Zbl 0653.90045 |

[60] | S. Dempe. A simple algorithm for the linear bilevel programming problem.Optimization, 18:373-385, 1987. · Zbl 0634.90075 |

[61] | S. Dempe. On one optimality condition for bilevel optimization.Vestnik Leningrad Gos. University, pages 10-14, 1989. Serija I, in Russian, translation Vestnik Leningrad University, Math., 22:11-16, 1989. · Zbl 0695.90087 |

[62] | S. Dempe. A necessary and a sufficient optimality condition for bilevel programming problems.Optimization, 25:341-354, 1992. · Zbl 0817.90104 |

[63] | S. Dempe. Optimality conditions for bilevel programming problems. In P. Kall, editor,System modelling and optimization, pages 17-24. Springer-Verlag, 1992. · Zbl 0786.90064 |

[64] | A. deSilva.Sensitivity formulas for nonlinear factorable programming and their application to the solution of an implicitly defined optimization model of US crude oil production. PhD thesis, George Washington University, 1978. |

[65] | A. deSilva and G. McCormick. Implicitly defined optimization problems.Annals of Operations Research, 34:107-124, 1992. · Zbl 0756.90080 |

[66] | Y. Dirickx and L. Jennegren.Systems analysis by multi-level methods: with applications to economics and management. John Wiley, New York, 1979. |

[67] | T. Edmunds.Algorithms for nonlinear bilevel mathematical programs. PhD thesis, Department of Mechanical Engineering, University of Texas at Austin, 1988. |

[68] | T. Edmunds and J. Bard. Algorithms for nonlinear bilevel mathematical programming.IEEE Transactions on Systems, Man, and Cybernetics, 21:83-89, 1991. |

[69] | T. Edmunds and J. Bard. An algorithm for the mixed-integer nonlinear bilevel programming problem.Annals of Operations Research, 34:149-162, 1992. · Zbl 0751.90054 |

[70] | M. Florian and Y. Chen. A bilevel programming approach to estimating O-D matrix by traffic counts. Technical Report CRT-750, Centre de Recherche sur les Transports, 1991. |

[71] | M. Florian and Y. Chen. A coordinate descent method for bilevel O-D matrix estimation problems. Technical Report CRT-807, Centre de Recherche sur les Transports, 1993. |

[72] | J. Fortuny-Amat and B. McCarl. A representation and economic interpretation of a two-level programming problem.Journal of the Operational Research Society, 32:783-792, 1981. · Zbl 0459.90067 |

[73] | T. Friesz, C. Suwansirikul, and R. Tobin. Equilibrium decomposition optimization: a heuristic for the continuous equilibrium network design problem.Transportation Science, 21:254-263, 1987. · Zbl 0638.90097 |

[74] | T. Friesz, R. Tobin, H. Cho, and N. Mehta. Sensitivity analysis based heuristic algorithms for mathematical programs with variational inequality constraints.Mathematical Programming, 48:265-284, 1990. · Zbl 0723.90070 |

[75] | G. Gallo and A. Ülkücü. Bilinear programming: an exact algorithm.Mathematical Programming, 12:173-194, 1977. · Zbl 0363.90086 |

[76] | P. Hansen, B. Jaumard, and G. Savard. New branch-and-bound rules for linear bilevel programming.SIAM Journal on Scientific and Statistical Computing, 13:1194-1217, 1992. · Zbl 0760.65063 |

[77] | P. Harker and J.-S. Pang. Existence of optimal solutions to mathematical programs with equilibrium constraints.Operations Research Letters, 7:61-64, 1988. · Zbl 0648.90065 |

[78] | A. Haurie, R. Loulou, and G. Savard. A two-level systems analysis model of power cogeneration under asymmetric pricing. Technical Report G-89-34, Groupe d’Études et de Recherche en Analyse des Décisions, 1989. · Zbl 0800.90657 |

[79] | A. Haurie, G. Savard, and D. White. A note on: an efficient point algorithm for a linear two-stage optimization problem.Operations Research, 38:553-555, 1990. · Zbl 0708.90051 |

[80] | B. Hobbs and S. Nelson. A nonlinear bilevel model for analysis of electric utility demand-side planning issues.Annals of Operations Research, 34:255-274, 1992. · Zbl 0729.91036 |

[81] | Y. Ishizuka. Optimality conditions for quasi-differentiable programs with applications to two-level optimization.SIAM Journal on Control and Optimization, 26:1388-1398, 1988. · Zbl 0661.90081 |

[82] | Y. Ishizuka and E. Aiyoshi. Double penalty method for bilevel optimization problems.Annals of Operations Research, 34:73-88, 1992. · Zbl 0756.90083 |

[83] | R. Jan and M. Chern. Multi-level nonlinear integer programming. 1990. (Preprint from the Department of Computer and Information Science, National Chiao Tung University). |

[84] | R. Jeroslow. The polynomial hierarchy and a simple model for competitive analysis.Mathematical Programming, 32:146-164, 1985. · Zbl 0588.90053 |

[85] | J. Júdice and A. Faustino. The solution of the linear bilevel programming problem by using the linear complementarity problem.Investigação Operacional, 8:77-95, 1988. · Zbl 0647.90095 |

[86] | J. Júdice and A. Faustino. A sequential LCP method for bilevel linear programming.Annals of Operations Research, 34:89-106, 1992. · Zbl 0749.90049 |

[87] | J. Júdice and A. Faustino. The linear-quadratic bilevel programming problem.INFOR, 32:87-98, 1994. · Zbl 0805.90098 |

[88] | T. Kim and S. Suh. Toward developing a national transportation planning model: a bilevel programming approach for Korea.Annals of Regional Science, 22:65-80, 1988. |

[89] | M. Kocvara and J. Outrata. A nondifferentiable approach to the solution of optimum design problems with variational inequalities. In P. Kall, ed.,System Modelling and Optimization, Lecture Notes in Control and Information Sciences 180, pages 364-373. Springer-Verlag, Berlin, 1992. · Zbl 0789.49027 |

[90] | M. Kocvara and J. Outrata. A numerical solution of two selected shape optimization problems. Technical Report DFG (German Scientific Foundation) Research Report 464, University of Bayreuth, 1993. |

[91] | C. Kolstad. A review of the literature on bi-level mathematical programming. Technical Report LA-10284-MS, US-32, Los Alamos National Laboratory, 1985. |

[92] | C. Kolstad and L. Lasdon. Derivative evaluation and computational experience with large bilevel mathematical programs.Journal of Optimization Theory and Applications, 65:485-499, 1990. · Zbl 0676.90101 |

[93] | M. Labbé, P. Marcotte, and G. Savard. A bilevel model of taxation and its application to optimal highway policy. 1993. Preprint. · Zbl 0989.90014 |

[94] | L. Leblanc and D. Boyce. A bilevel programming algorithm for exact solution of the network design problem with user-optimal flows.Transportation Research, 20B:259-265, 1986. |

[95] | P. Loridan and J. Morgan. Approximate solutions for two-level optimization problems. In K. Hoffman, J. Hiriart-Urruty, C. Lamerachal and J. Zowe, editor,Trends in Mathematical Optimization, volume 84 of International Series of Numerical Mathematics, pages 181-196. Birkhäuser Verlag, Basel, 1988. · Zbl 0662.90090 |

[96] | P. Loridan and J. Morgan. A theoretical approximation scheme for Stackelberg problems.Journal of Optimization Theory and Applications, 61:95-l10, 1989. · Zbl 0642.90107 |

[97] | P. Loridan and J. Morgan. ?-Regularized two-level optimization problems: approximation and existence results. InOptimization -Fifth French-German Conference, Lecture Notes in Mathematics 1405, pages 99-113. Springer-Verlag, Berlin, 1989. |

[98] | P. Loridan and J. Morgan. New results on approximate solutions in two-level optimization.Optimization, 20:819-836, 1989. · Zbl 0684.90089 |

[99] | P. Loridan and J. Morgan.Quasi convex lower level problem and applications in two level optimization, volume 345 of Lecture Notes in Economics and Mathematical Systems, pages 325-341. Springer-Verlag, Berlin, 1990. · Zbl 0726.90111 |

[100] | P. Luh, T.-S. Chang, and T. Ning. Three-level Stackelberg decision problems.IEEE Transactions on Automatic Control, 29:280-282, 1984. · Zbl 0529.90104 |

[101] | Z.-Q. Luo, J.-S. Pang, and S. Wu. Exact penalty functions for mathematical programs and bilevel programs with analytic constraints. 1993. Preprint from the Department of Electrical and Computer Engineering, McMaster University. |

[102] | P. Marcotte. Network optimization with continuous control parameters.Transportation Science, 17:181-197, 1983. |

[103] | P. Marcotte. Network design problem with congestion effects: a case of bilevel programming.Mathematical Programming, 34:142-162, 1986. · Zbl 0604.90053 |

[104] | P. Marcotte. A note on bilevel programming algorithm by LeBlanc and Boyce.Transportation Research, 22 B:233-237, 1988. |

[105] | P. Marcotte and G. Marquis. Efficient implementation of heuristics for the continuous network design problem.Annals of Operations Research, 34:163-176, 1992. · Zbl 0756.90037 |

[106] | P. Marcotte and G. Savard. A note on the pareto optimality of solutions to the linear bilevel programming problem.Computers and Operations Research, 18:355-359, 1991. · Zbl 0717.90045 |

[107] | P. Marcotte and G. Savard. Novel approaches to the discrimination problem.ZOR ? Methods and Models of Operations Research, 36:517-545, 1992. · Zbl 0762.90090 |

[108] | P. Marcotte and D. Zhu. Exact and inexact penalty methods for the generalized bilevel programming problem. Technical Report CRT-920, Centre de Recherche sur les Transports, 1992. · Zbl 0855.90120 |

[109] | R. Mathieu, L. Pittard, and G. Anandalingam. Genetic algorithm based approach to bi-level linear programming.RAIRO: Recherche Operationelle, (forthcoming). · Zbl 0857.90083 |

[110] | M. Mesanovic, D. Macko, and Y. Takahara.Theory of hierarchical, multilevel systems. Academic Press, New York and London, 1970. · Zbl 0206.14501 |

[111] | T. Miller, T. Friesz, and R. Tobin. Heuristic algorithms for delivered price spatially competitive network facility location problems.Annals of Operations Research, 34:177-202, 1992. · Zbl 0751.90045 |

[112] | J. Moore.Extensions to the multilevel linear programming problem. PhD thesis, Department of Mechanical Engineering, University of Texas, Austin, 1988. |

[113] | J. Moore and J. Bard. The mixed integer linear bilevel programming problem.Operations Research, 38:911-921, 1990. · Zbl 0723.90090 |

[114] | S. Narula and A. Nwosu. A dynamic programming solution for the hierarchical linear programming problem. Technical Report 37-82, Department of Operations Research and Statistics, Rensselaer Polytechnic Institute, 1982. |

[115] | S. Narula and A. Nwosu. Two-level hierarchical programming problems. In P. Hansen, editor,Essays and surveys on multiple criteria decision making, pages 290-299. Springer-Verlag, Berlin, 1983. |

[116] | S. Narula and A. Nwosu. An algorithm to solve a two-level resource control pre-emptive hierarchical programming problem. In P. Serafini, editor,Mathematics of multiple-objective programming. Springer-Verlag, Berlin, 1985. |

[117] | P. Neittaanmäki and A. Stachurski. Solving some optimal control problems using the barrier penalty function method. In H.-J. Sebastian and K. Tammer, editors,Proceedings of the 14th 1F1P Conference on System Modelling and Optimization, Leipzig 1989, pages 358-367. Springer, 1990. |

[118] | A. Nwosu.Pre-emptive hierarchical programming problem: a decentralized decision model. PhD thesis, Department of Operations Research and Statistics, Rensselaer Polytechnic Institute, 1983. |

[119] | H. Önal. Computational experience with a mixed solution method for bilevel linear/quadratic programs. 1992. (Preprint from the University of Illinois at Urbana-Champaign). |

[120] | H. Önal. A modified simplex approach for solving bilevel linear programming problems.European Journal of Operational Research, 67:126-135, 1993. · Zbl 0806.90086 |

[121] | J. Outrata. On the numerical solution of a class of Stackelberg problems.ZOR-Methods and Models of Operations Research, 34:255-277, 1990. · Zbl 0714.90077 |

[122] | J. Outrata. Necessary optimality conditions for Stackelberg problems.Journal of Optimization Theory and Applications, 76:305-320, 1993. · Zbl 0802.49007 |

[123] | J. Outrata. On optimization problems with variational inequality constraints.SIAM Journal on Optimization, (forthcoming). · Zbl 0826.90114 |

[124] | J. Outrata and J. Zowe. A numerical approach to optimization problems with variational inequality constraints. Technical Report DFG (German Scientific Foundation) Research Report 463, University of Bayreuth, 1993. · Zbl 0835.90093 |

[125] | G. Papavassilopoulos. Algorithms for static Stackelberg games with linear costs and polyhedral constraints. InProceedings of the 21st IEEE Conference on Decisions and Control, pages 647-652, 1982. |

[126] | F. Parraga.Hierarchical programming and applications to economic policy. PhD thesis, Systems and Industrial Engineering Department, University of Arizona, 1981. |

[127] | G. Savard.Contributions à la programmation mathématique deux niveaux. PhD thesis, École Polytechnique, Université de Montréal, 1989. |

[128] | G. Savard and J. Gauvin. The steepest descent direction for the nonlinear bilevel programming problem. Technical Report G-90-37, Groupe d’Études et de Recherche en Analyse des Décisions, 1990. · Zbl 0816.90122 |

[129] | G. Schenk. A multilevel programming model for determining regional effluent charges. Master’s thesis, Department of Industrial Engineering, State University of New York at Buffalo, 1980. |

[130] | R. Segall. Bi-level geometric programming: a new optimization model. 1989. (Preprint from the Department of Mathematics, University of Lowell Olsen Hall). |

[131] | J. Shaw. A parametric complementary pivot approach to multilevel programming. Master’s thesis, Department of Industrial Engineering, State University of New York at Buffalo, 1980. |

[132] | H. Sherali. A multiple leader Stackelberg model and analysis.Operations Research, 32:390-404, 1984. · Zbl 0581.90008 |

[133] | K. Shimizu.Two-level decision problems and their new solution methods by a penalty method, volume 2 ofControl science and technology for the progress of society, pages 1303-1308. IFAC, 1982. |

[134] | K. Shimizu and E. Aiyoshi. A new computational method for Stackelberg and min-max problems by use of a penalty method.IEEE Transactions on Automatic Control, 26:460-466, 1981. · Zbl 0472.93004 |

[135] | K. Shimizu and E. Aiyoshi. Optimality conditions and algorithms for parameter design problems with two-level structure.IEEE Transactions on Automatic Control, 30:986-993, 1985. · Zbl 0573.90074 |

[136] | M. Simaan. Stackelberg optimization of two-level systems.IEEE Transactions on Systems, Man, and Cybernetics, 7:554-557, 1977. · Zbl 0381.49005 |

[137] | H. Stackelberg.The theory of the market economy. Oxford University Press, 1952. |

[138] | S. Suh and T. Kim. Solving nonlinear bilevel programming models of the equilibrium network design problem: a comparative review.Annals of Operations Research, 34:203-218, 1992. · Zbl 0751.90081 |

[139] | C. Suwansirikul, T. Friesz, and R. Tobin. Equilibrium decomposed optimization: a heuristic for the continuous equilibrium network design problem.Transportation Science, 21:254-263, 1987. · Zbl 0638.90097 |

[140] | T. Tanino and T. Ogawa. An algorithm for solving two-level convex optimization problems.International Journal of Systems Science, 15:163-174, 1984. · Zbl 0542.90075 |

[141] | R. Tobin and T. Friesz. Spatial competition facility location models: definition, formulation and solution approach.Annals of Operations Research, 6:49-74, 1986. |

[142] | H. Tuy, A. Migdalas, and P. Värbrand. A global optimization approach for the linear two-level program.Journal of Global Optimization, 3:1-23, 1993. · Zbl 0771.90108 |

[143] | G. Ünlü. A linear bilevel programming algorithm based on bicriteria programming.Computers and Operations Research, 14:173-179, 1987. · Zbl 0626.90086 |

[144] | L. Vicente. Bilevel programming. Master’s thesis, Department of Mathematics, University of Coimbra, 1992. Written in Portuguese. |

[145] | L. Vicente and P. Calamai. Geometry and local optimality conditions for bilevel programs with quadratic strictly convex lower levels. Technical Report #198-O-150294, Department of Systems Design Engineering, University of Waterloo, 1994. · Zbl 0847.90129 |

[146] | L. Vicente, G. Savard, and J. Júdice. Descent approaches for quadratic bilevel programming.Journal of Optimization Theory and Applications, (forthcoming). · Zbl 0819.90076 |

[147] | U. Wen.Mathematical methods for multilevel linear programming. PhD thesis, Department of Industrial Engineering, State University of New York at Buffalo, 1981. |

[148] | U. Wen. The ?Kth-Best? algorithm for multilevel programming. 1981. (Preprint from the Department of Operations Research, State University of New York at Buffalo). |

[149] | U. Wen. A solution procedure for the resource control problem in two-level hierarchical decision processes.Journal of Chinese Institute of Engineers, 6:91-97, 1983. |

[150] | U. Wen and W. Bialas. The hybrid algorithm for solving the three-level linear programming problem.Computers and Operations Research, 13:367-377, 1986. · Zbl 0643.90058 |

[151] | U. Wen and S. Hsu. A note on a linear bilevel programming algorithm based on bicriteria programming.Computers and Operations Research, 16:79-83, 1989. · Zbl 0659.90080 |

[152] | U. Wen and S. Hsu. Efficient solutions for the linear bilevel programming problem.European Journal of Operational Research, 62:354-362, 1991. · Zbl 0765.90083 |

[153] | U. Wen and S. Hsu. Linear bi-level programming problems ? a review.Journal of the Operational Research Society, 42:125-133, 1991. · Zbl 0722.90046 |

[154] | U. Wen and Y. Yang. Algorithms for solving the mixed integer two-level linear programming problem.Computers and Operations Research, 17:133-142, 1990. · Zbl 0683.90055 |

[155] | D. White and G. Anandalingam. A penalty function approach for solving bi-level linear programs.Journal of Global Optimization, 3:397-419, 1993. · Zbl 0791.90047 |

[156] | S. Wu, P. Marcotte, and Y. Chen. A cutting plane method for linear bilevel programs. 1993. (Preprint from the Centre de Research sur les Transports). · Zbl 0915.90205 |

[157] | J. Ye and D. Zhu. Optimality conditions for bilevel programming problems. Technical Report DMS-618-IR, Department of Mathematics and Statistics, University of Victoria, 1993. 1992, Revised 1993. · Zbl 0820.65032 |

[158] | J. Ye, D. Zhu, and Q. Zhu. Generalized bilevel programming problems. Technical Report DMS-646-IR, Department of Mathematics and Statistics, University of Victoria, 1993. · Zbl 0820.65032 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.