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An algorithm for the mixed-integer nonlinear bilevel programming problem

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Abstract

The bilevel programming problem (BLPP) is a two-person nonzero sum game in which play is sequential and cooperation is not permitted. In this paper, we examine a class of BLPPs where the leader controls a set of continuous and discrete variables and tries to minimize a convex nonlinear objective function. The follower's objective function is a convex quadratic in a continuous decision space. All constraints are assumed to be linear. A branch and bound algorithm is developed that finds global optima. The main purpose of this paper is to identify efficient branching rules, and to determine the computational burden of the numeric procedures. Extensive test results are reported. We close by showing that it is not readily possible to extend the algorithm to the more general case involving integer follower variables.

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Authors and Affiliations

  1. Systems Research Group, Lawrence Livermore National Laboratory, University of California, 94550, Livermore, CA, USA

    Thomas A. Edmunds

  2. Department of Mechanical Engineering, Operations Research Group, University of Texas, 78712, Austin, TX, USA

    Jonathan F. Bard

Authors
  1. Thomas A. Edmunds
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  2. Jonathan F. Bard
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Additional information

This work was supported by a grant from the Advanced Research Program of the Texas Higher Education Coordinating Board.

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Edmunds, T.A., Bard, J.F. An algorithm for the mixed-integer nonlinear bilevel programming problem. Ann Oper Res 34, 149–162 (1992). https://doi.org/10.1007/BF02098177

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