Links Between Linear Bilevel and Mixed 0–1 Programming Problems

We study links between the linear bilevel and linear mixed 0–1 programming problems. A new reformulation of the linear mixed 0–1 programming problem into a linear bilevel programming one, which does not require the introduction of a large finite constant, is presented. We show that solving a linear mixed 0–1 problem by a classical branch-and-bound algorithm is equivalent in a strong sense to solving its bilevel reformulation by a bilevel branch-and-bound algorithm. The mixed 0–1 algorithm is embedded in the bilevel algorithm through the aforementioned reformulation; i.e., when applied to any mixed 0–1 instance and its bilevel reformulation, they generate sequences of subproblems which are identical via the reformulation.

[1]  Charles E. Blair,et al.  Computational Difficulties of Bilevel Linear Programming , 1990, Oper. Res..

[2]  Charles Blair,et al.  The computational complexity of multi-level linear programs , 1992, Ann. Oper. Res..

[3]  José Fortuny-Amat,et al.  A Representation and Economic Interpretation of a Two-Level Programming Problem , 1981 .

[4]  Ue-Pyng Wen,et al.  Algorithms for solving the mixed integer two-level linear programming problem , 1990, Comput. Oper. Res..

[5]  Ue-Pyng Wen,et al.  The hybrid algorithm for solving the three-level linear programming problem , 1986, Comput. Oper. Res..

[6]  Warren P. Adams,et al.  A hierarchy of relaxation between the continuous and convex hull representations , 1990 .

[7]  Jonathan F. Bard,et al.  An investigation of the linear three level programming problem , 1984, IEEE Transactions on Systems, Man, and Cybernetics.

[8]  Wayne F. Bialas,et al.  On two-level optimization , 1982 .

[9]  G. Mitra,et al.  Reformulation of mathematical programming problems as linear complementarity problems and investigation of their solution methods , 1988 .

[10]  Omar Ben-Ayed,et al.  Bilevel linear programming , 1993, Comput. Oper. Res..

[11]  Egon Balas,et al.  A lift-and-project cutting plane algorithm for mixed 0–1 programs , 1993, Math. Program..

[12]  Harold P. Benson,et al.  On the structure and properties of a linear multilevel programming problem , 1989 .

[13]  Wilfred Candler,et al.  A linear two-level programming problem, , 1982, Comput. Oper. Res..

[14]  Pierre Hansen,et al.  New Branch-and-Bound Rules for Linear Bilevel Programming , 1989, SIAM J. Sci. Comput..

[15]  L. N. Vicente,et al.  Discrete linear bilevel programming problem , 1996 .

[16]  John A. Tomlin,et al.  Technical Note - An Improved Branch-and-Bound Method for Integer Programming , 1971, Oper. Res..

[17]  James E. Falk,et al.  A linear max—min problem , 1973, Math. Program..

[18]  J. Bard.  Some properties of the bilevel programming problem , 1991 .

[19]  Jonathan F. Bard,et al.  A Branch and Bound Algorithm for the Bilevel Programming Problem , 1990, SIAM J. Sci. Comput..

[20]  Paul H. Calamai,et al.  Bilevel and multilevel programming: A bibliography review , 1994, J. Glob. Optim..

[21]  Robert G. Jeroslow,et al.  The polynomial hierarchy and a simple model for competitive analysis , 1985, Math. Program..

[22]  Jonathan F. Bard,et al.  An explicit solution to the multi-level programming problem , 1982, Comput. Oper. Res..

[23]  Alexander Schrijver,et al.  Cones of Matrices and Set-Functions and 0-1 Optimization , 1991, SIAM J. Optim..

[24]  Joaquim Júdice,et al.  A sequential LCP method for bilevel linear programming , 1992, Ann. Oper. Res..