# Exact Solution Methodologies for Linear and (Mixed) Integer Bilevel Programming

• Georgios K. D. Saharidis
• Antonio J. Conejo
• George Kozanidis
Chapter
Part of the Studies in Computational Intelligence book series (SCI, volume 482)

## Abstract

Bilevel programming is a special branch of mathematical programming that deals with optimization problems which involve two decision makers who make their decisions hierarchically. The problem’s decision variables are partitioned into two sets, with the first decision maker (referred to as the leader) controlling the first of these sets and attempting to solve an optimization problem which includes in its constraint set a second optimization problem solved by the second decision maker (referred to as the follower), who controls the second set of decision variables. The leader goes first and selects the values of the decision variables that he controls.With the leader’s decisions known, the follower solves a typical optimization problem in his self-controlled decision variables. The overall problem exhibits a highly combinatorial nature, due to the fact that the leader, anticipating the follower’s reaction, must choose the values of his decision variables in such a way that after the problem controlled by the follower is solved, his own objective function will be optimized. Bilevel optimization models exhibit wide applicability in various interdisciplinary research areas, such as biology, economics, engineering, physics, etc. In this work, we review the exact solution algorithms that have been developed both for the case of linear bilevel programming (both the leader’s and the follower’s problems are linear and continuous), as well as for the case of mixed integer bilevel programming (discrete decision variables are included in at least one of these two problems). We also document numerous applications of bilevel programming models from various different contexts. Although several reviews dealing with bilevel programming have previously appeared in the related literature, the significant contribution of the present work lies in that a) it is meant to be complete and up to date, b) it puts together various related works that have been revised/corrected in follow-up works, and reports in sequence the works that have provided these corrections, c) it identifies the special conditions and requirements needed for the application of each solution algorithm, and d) it points out the limitations of each associated methodology. The present collection of exact solution methodologies for bilevel optimization models can be proven extremely useful, since generic solution methodologies that solve such problems to global or local optimality do not exist.

## Keywords

Network Design Problem Karush Kuhn Tucker Bilevel Programming Lower Level Problem Bilevel Optimization
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Al-Khayyal, F.A.: An implicit enumeration procedure for the general linear complementarity problem. Mathematical Programming Studies 31, 1–20 (1987)
2. 2.
Amouzegar, M.A., Moshirvaziri, K.: Determining optimal pollution control policies: An application of bilevel programming. European Journal of Operational Research 119, 100–120 (1999)
3. 3.
Anandalingam, G., Apprey, V.: Multi-level programming and conflict resolution. European Journal of Operational Research 51, 233–247 (1991)
4. 4.
Anandalingam, G., White, D.J.: A solution method for the linear static stackelberg problem using penalty functions. IEEE Transactions on Automatic Control 35(10), 1170–1173 (1990)
5. 5.
Audet, C., Hansen, P., Jaumard, B., Savard, G.: Links between linear bilevel and mixed 0-1 programming problems. Journal of Optimization Theory and Applications 93(2), 273–300 (1997)
6. 6.
Audet, C., Haddad, J., Savard, G.: A note on the definition of a linear bilevel programming solution. Applied Mathematics and Computation 181, 351–355 (2006)
7. 7.
Audet, C., Savard, G., Zghal, W.: New branch and cut algorithm for bilevel linear programming. Journal of Optimization Theory and Applications 134(2), 353–370 (2007)
8. 8.
Bard, J.F.: An efficient point algorithm for a linear two-stage optimization problem. Operations Research 31(4), 670–684 (1983)
9. 9.
Bard, J.F.: An investigation of the linear three level programming problem. IEEE Transactions on Systems, Man, and Cybernetics 14(5), 711–717 (1984)
10. 10.
Bard, J.F.: Practical bilevel optimization. Kluwer Academic Publishers, Dordrecht (1998)
11. 11.
Bard, J.F., Falk, J.E.: An explicit solution to the multilevel programming problem. Computers and Operations Research 9(1), 77–100 (1982)
12. 12.
Bard, J.F., Moore, J.T.: A branch and bound algorithm for the bilevel programming problem. SIAM Journal on Scientific and Statistical Computing 11, 281–292 (1990)
13. 13.
Bard, J.F., Moore, J.T.: An algorithm for the discrete bilevel programming problem. Naval Research Logistics 39(3), 419–435 (1992)
14. 14.
Baringo, L., Conejo, A.J.: Wind power investment within a market environment. Applied Energy 88(9), 3239–3247 (2011a)
15. 15.
Baringo, L., Conejo, A.J.: Transmission and wind power investment. IEEE Transactions on Power Systems (2011b) (accepted)Google Scholar
16. 16.
BenAyed, O.: Bilevel linear programming. Computers and Operations Research 20(5), 485–501 (1993)
17. 17.
BenAyed, O., Blair, C.E.: Computational difficulties of bilevel linear programming. Operations Research 38(3), 556–560 (1990)
18. 18.
BenAyed, O., Boyce, D.E., Blair, C.E.: A general bilevel linear programming formulation of the network design problem. Transportation Research - Part B 22B(4), 311–318 (1988)
19. 19.
BenAyed, O., Blair, C.E., Boyce, D.E., LeBlanc, L.J.: Construction of a real-world bilevel linear programming model of the highway network design problem. Annals of Operations Research 34, 219–254 (1992)
20. 20.
Bialas, W.F., Karwan, M.H.: Multilevel linear programming. Research Report No. 78-1, Operation Research Program, Department of Industrial Engineering, State University of New York at Buffalo (1978)Google Scholar
21. 21.
Bialas, W.F., Karwan, M.H.: On two-level optimization. IEEE Transactions on Automatic Control 27(1), 211–214 (1982)
22. 22.
Bialas, W.F., Karwan, M.H.: Two-level linear programming. Management Science 30(8), 1004–1020 (1984)
23. 23.
Bracken, J., McGill, J.T.: Defense applications of mathematical programs with optimization problems in the constraints. Operations Research 22(5), 1086–1096 (1974)
24. 24.
Brown, G., Carlyle, M., Salmerón, J., Wood, K.: Defending critical infrastructure. Interfaces 36, 530–544 (2006)
25. 25.
Burton, R.M., Obel, B.: The multilevel approach to organizational issues of the firm - A critical review. OMEGA The International Journal of Management Science 5(4), 395–414 (1977)
26. 26.
Campelo, M., Scheimberg, S.: A note on a modified simplex approach for solving bilevel linear programming problems. European Journal of Operations Research 126(2), 454–458 (2000)
27. 27.
Campelo, M., Dantas, S., Scheimberg, S.: A note on a penalty function approach for solving bilevel linear programs. Journal of Global Optimization 16, 245–255 (2000)
28. 28.
Candler, W.: A linear bilevel programming algorithm: A comment. Computers and Operations Research 15(3), 297–298 (1988)
29. 29.
Candler, W., Norton, R.: Multi-level programming and development policy. World Bank, Bank Staff Working Paper No. 258 (1977)Google Scholar
30. 30.
Candler, W., Townsley, R.: A linear two-level programming problem. Computers and Operations Research 9, 59–76 (1982)
31. 31.
Cao, D., Chen, M.: Capacitated plant selection in a decentralized manufacturing environment: A bilevel optimization approach. European Journal of Operational Research 169, 97–110 (2006)
32. 32.
Cassidy, R.G., Kirby, M.J.L., Raike, W.M.: Efficient distribution of resources through three levels of government. Management Science 17(8), 462–473 (1971)
33. 33.
Chinchuluun, A., Pardalos, P.M., Huang, H.-X.: Multilevel (hierarchical) optimization: Complexity issues, optimality conditions, algorithms. In: Advances in Applied Mathematics and Global Optimization -Advances in Mechanics and Mathematics, vol. 17, ch. 6, pp. 197–221 (2009)Google Scholar
34. 34.
Clark, P.A., Westerberg, A.W.: Optimization for design problems having more than one objective. Computers and Chemical Engineering 7(4), 259–278 (1983)
35. 35.
Clark, P.A., Westerberg, A.W.: Bilevel programming for steady state chemical process design - I. Fundamentals and algorithms. Computers and Chemical Engineering 14(1), 87–97 (1990)
36. 36.
Colson, B., Marcotte, P., Savard, G.: Bilevel programming: A survey. 4OR 3, 87–107 (2005)
37. 37.
Colson, B., Marcotte, P., Savard, G.: An overview of bilevel optimization. Annals of Operations Research 153, 235–256 (2007)
38. 38.
Constantin, I., Florian, M.: Optimizing frequencies in a transit network: A non-linear bilevel programming approach. International Transactions in Operational Research 2(2), 149–164 (1995)
39. 39.
Cote, J.-P., Marcotte, P., Savard, G.: A bilevel modelling approach to pricing and fare optimisation in the airline industry. Journal of Revenue and Pricing Management 2(1), 23–36 (2003)
40. 40.
Dempe, S.: A simple algorithm for the linear bilevel programming problem. Optimization 18(3), 373–385 (1987)
41. 41.
Dempe, S.: Foundations of bilevel programming. Kluwer Academic Publishers, Dordrecht (2002)
42. 42.
Dempe, S.: Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization 52(3), 333–359 (2003)
43. 43.
Dempe, S.: Bilevel programming. In: Audet, C., Hansen, P., Savard, G. (eds.) Essays and Surveys in Global Optimization, ch. 6, pp. 165–194 (2005)Google Scholar
44. 44.
DeNegre, S.T., Ralphs, T.K.: A branch and cut algorithm for integer bilevel linear programs. Operations research and cyber-infrastructure, Operations Research/Computer Science Interfaces Series 47(2), Part 1, 65–78 (2009)Google Scholar
45. 45.
Erkut, E., Gzara, F.: Solving the hazmat transport network design problem. Computers and Operations Research 35, 2234–2247 (2008)
46. 46.
Falk, J.E., Soland, R.M.: An algorithm for separable nonconvex programming problems. Management Science 15, 550–569 (1969)
47. 47.
Fortuny-Amat, J., McCarl, B.: A representation and economic interpretation of a two-level programming problem. Journal of the Operational Research Society 32(9), 783–792 (1981)
48. 48.
Garcés, L.P., Conejo, A.J., García-Bertrand, R., Romero, R.: A bi-level approach to transmission expansion planning within a market environment. IEEE Transactions on Power Systems 24(3), 1513–1522 (2009)
49. 49.
Glackin, J., Ecker, J.G., Kupferschmid, M.: Solving bilevel linear programs using multiple objective linear programming. Journal of Optimization Theory and Applications 140(2), 197–212 (2009)
50. 50.
Golias, M.M., Saharidis, G.K.D., Boile, M., Theofanis, S.: Scheduling of inbound trucks at a cross-docking facility: Bi-objective vs bi-level modeling approaches. International Journal of Information Systems and Supply Chain Management 5(1) (2012) (in press)Google Scholar
51. 51.
Hansen, P., Jaumard, B., Savard, G.: New branch and bound rules for linear bilevel programming. SIAM Journal on Scientific and Statistical Computing 13(5), 1194–1217 (1992)
52. 52.
Haurie, A., Savard, G., White, D.J.: A note on: An efficient point algorithm for a linear two-stage optimization problem. Operations Research 38(3), 553–555 (1990)
53. 53.
Hobbs, B.F., Nelson, S.K.: A nonlinear bilevel model for analysis of electric utility demand-side planning issues. Annals of Operations Research 34, 255–274 (1992)
54. 54.
Júdice, J.J., Faustino, A.M.: The solution of the linear bilevel programming problem by using the linear complementarity problem. Investigavio Operacional 8, 77–95 (1988)Google Scholar
55. 55.
Júdice, J.J., Faustino, A.M.: A sequential LCP method for bilevel linear programming. Annals of Operations Research 34(1), 89–106 (1992)
56. 56.
Kazempour, J., Conejo, A.J., Ruiz, C.: Strategic generation investment using a complementarity approach. IEEE Transactions on Power Systems 26(2), 940–948 (2011)
57. 57.
Koppe, M., Queyranne, M., Ryan, C.T.: Parametric integer programming algorithm for bilevel mixed integer programs. Journal of Optimization Theory and Applications 146(1), 137–150 (2010)
58. 58.
Labbe, M., Marcotte, P., Savard, G.: A bilevel model of taxation and its application to optimal highway pricing. Management Science 44(12), 1608–1622 (1998)
59. 59.
LeBlanc, L.J., Boyce, D.E.: A bilevel programming algorithm for exact solution of the network design problem with user-optimal flows. Transportation Research - Part B 20B(3), 259–265 (1986)
60. 60.
Liu, Y.H., Spencer, T.H.: Solving a bilevel linear program when the inner decision maker controls few variables. European Journal of Operational Research 81(3), 644–651 (1995)
61. 61.
Loridan, P., Morgan, J.: Weak via strong stackelberg problem: New results. Journal of Global Optimization 8, 263–287 (1996)
62. 62.
Maher, M.J., Zhang, X., Vliet, D.V.: A bi-level programming approach for trip matrix estimation and traffic control problems with stochastic user equilibrium link flows. Transportation Research - Part B 35, 23–40 (2001)
63. 63.
Marinakis, Y., Migdalas, A., Pardalos, P.M.: A new bilevel formulation for the vehicle routing problem and a solution method using a genetic algorithm. Journal of Global Optimization 38, 555–580 (2007)
64. 64.
Migdalas, A.: Bilevel programming in traffic planning: Models, methods and challenge. Journal of Global Optimization 7, 381–405 (1995)
65. 65.
Mitsos, A., Bollas, G.M., Barton, P.I.: Bilevel optimization formulation for parameter estimation in liquid-liquid phase equilibrium problems. Chemical Engineering Science 64, 548–559 (2009)
66. 66.
Moore, J.T., Bard, J.F.: The mixed integer linear bilevel programming problem. Operations Research 38(5), 911–921 (1990)
67. 67.
Motto, A.L., Arroyo, J.M., Galiana, F.D.: A mixed-integer LP procedure for the analysis of electric grid security under disruptive threat. IEEE Transactions on Power Systems 20(3), 1357–1365 (2005)
68. 68.
Nicholls, M.G.: Aluminum production modeling - A nonlinear bilevel programming approach. Operations Research 43(2), 208–218 (1995)
69. 69.
Onal, H.: A modified simplex approach for solving bilevel linear programming problems. European Journal of Operations Research 67(1), 126–135 (1993)
70. 70.
Onal, H., Darmawan, D.H., Johnson, S.H.: A multilevel analysis of agricultural credit distribution in East Java, Indonesia. Computers and Operations Research 22(2), 227–236 (1995)
71. 71.
Pandzic, H., Conejo, A.J., Kuzle, I., Caro, E.: Yearly maintenance scheduling of transmission lines within a market environment. IEEE Transactions on Power Systems (2011) (in press)Google Scholar
72. 72.
Ruiz, C., Conejo, A.J.: Pool strategy of a producer with endogenous formation of locational marginal prices. IEEE Transactions on Power Systems 24(4), 1855–1866 (2009)
73. 73.
Ryu, J.-H., Dua, V., Pistikopoulos, E.N.: A bilevel programming framework for enterprise-wide process networks under uncertainty. Computers and Chemical Engineering 28, 1121–1129 (2004)
74. 74.
Saharidis, G.K.D., Ierapetritou, M.G.: Resolution method for mixed integer bilevel linear problems based on decomposition technique. Journal of Global Optimization 44(1), 29–51 (2009)
75. 75.
Saharidis, G.K.D., Golias, M.M., Boile, M., Theofanis, S., Ierapetritou, M.G.: The berth scheduling problem with customer differentiation: A new methodological approach based on hierarchical optimization. International Journal of Advanced Manufacturing Technology 46, 377–393 (2010)
76. 76.
Saharidis, G.K.D., Androulakis, I.P., Ierapetritou, M.G.: Model building using bi-level optimization. Journal of Global Optimization 49, 49–67 (2011)
77. 77.
Salmerón, J., Wood, K., Baldick, R.: Analysis of electric grid security under terrorist threat. IEEE Transactions on Power Systems 19(2), 905–912 (2004)
78. 78.
Salmerón, J., Wood, K., Baldick, R.: Worst-case interdiction analysis of large-scale electric power grids. IEEE Transactions on Power Systems 24, 96–104 (2009)
79. 79.
Shi, C., Zhang, G., Lu, J.: On the definition of linear bilevel programming solution. Applied Mathematics and Computation 160(1), 169–176 (2005a)
80. 80.
Shi, C., Lu, J., Zhang, G.: An extended Kuhn-Tucker approach for linear bilevel programming. Applied Mathematics and Computation 162, 51–63 (2005b)
81. 81.
Shi, C., Lu, J., Zhang, G.: An extended Kth best approach for linear bilevel programming. Applied Mathematics and Computation 164, 843–855 (2005c)
82. 82.
Shi, C., Lu, J., Zhang, G., Zhou, H.: An extended branch and bound algorithm for linear bilevel programming. Applied Mathematics and Computation 180, 529–537 (2006)
83. 83.
Suh, S., Kim, T.J.: Solving nonlinear bilevel programming models of the equilibrium network design problem: A comparative review. Annals of Operations Research 34, 203–218 (1992)
84. 84.
Tuy, H., Migdalas, A., Varbrand, P.: A global optimization approach for the linear two-level program. Journal of Global Optimization 3, 1–23 (1993)
85. 85.
Unlu, G.: A linear bilevel programming algorithm based on bicriteria programming. Computers and Operations Research 14(2), 173–179 (1987)
86. 86.
Vicente, L.N., Calamai, P.H.: Bilevel and multilevel programming: A bibliography review. Journal of Global Optimization 5(3), 291–306 (1994)
87. 87.
Wen, U.P., Hsu, S.T.: A note on a linear bilevel programming algorithm based on bicriteria programming. Computers and Operations Research 16(1), 79–83 (1989)
88. 88.
Wen, U.P., Hsu, S.T.: Linear bi-level programming problems - A review. Journal of the Operational Research Society 42(2), 125–133 (1991)
89. 89.
Wen, U.P., Yang, Y.H.: Algorithms for solving the mixed integer two-level linear programming problem. Computers and Operations Research 17(2), 133–142 (1990)
90. 90.
White, D.J.: Solving bi-level linear programmes. Journal of Mathematical Analysis and Applications 200(1), 254–258 (1996)
91. 91.
White, D.J., Anandalingam, G.: A penalty function approach for solving bilevel linear programs. Journal of Global Optimization 3(4), 397–419 (1993)
92. 92.
Yin, Y.: Multiobjective bilevel optimization for transportation planning and management problems. Journal of Advanced Transportation 36(1), 93–105 (2002)

## Authors and Affiliations

• Georgios K. D. Saharidis
• 1
• 2
• Antonio J. Conejo
• 3
• George Kozanidis
• 4
1. 1.Department of Mechanical EngineeringUniversity of ThessalyVolosGreece
2. 2.Greece Kathikas Institute of Research and TechnologyPaphosCyprus
3. 3.Department of Electrical EngineeringUniv. Castilla - La ManchaCuencaSpain
4. 4.Systems Optimization Laboratory Department of Mechanical EngineeringUniversity of ThessalyVolosGreece