A branch-and-cut algorithm for mixed integer bilevel linear optimization problems and its implementation

In this paper, we describe a comprehensive algorithmic framework for solving mixed integer bilevel linear optimization problems (MIBLPs) using a generalized branch-and-cut approach. The framework presented merges features from existing algorithms (for both traditional mixed integer linear optimization and MIBLPs) with new techniques to produce a flexible and robust framework capable of solving a wide range of bilevel optimization problems. The framework has been fully implemented in the open-source solver MibS . The paper describes the algorithmic options offered by MibS and presents computational results evaluating the effectiveness of the various options for the solution of a number of classes of bilevel optimization problems from the literature.

[1]  Giovanni Rinaldi,et al.  A Branch-and-Cut Algorithm for the Resolution of Large-Scale Symmetric Traveling Salesman Problems , 1991, SIAM Rev..

[2]  Matteo Fischetti,et al.  A New General-Purpose Algorithm for Mixed-Integer Bilevel Linear Programs , 2017, Oper. Res..

[3]  Daniel Bienstock,et al.  The N-k Problem in Power Grids: New Models, Formulations, and Numerical Experiments , 2009, SIAM J. Optim..

[4]  D. Bienstock,et al.  The Nk Problem in Power Grids : New Models , Formulations and Numerical Experiments 1 , 2008 .

[5]  Claire S. Adjiman,et al.  Branch-and-Sandwich: a deterministic global optimization algorithm for optimistic bilevel programming problems. Part II: Convergence analysis and numerical results , 2014, Journal of Global Optimization.

[6]  Eitan Israeli.  System Interdiction and Defense. , 1999 .

[7]  Laurence A. Wolsey,et al.  Cutting planes in integer and mixed integer programming , 2002, Discret. Appl. Math..

[8]  Massimiliano Caramia,et al.  Enhanced exact algorithms for discrete bilevel linear problems , 2015, Optim. Lett..

[9]  Jerome Bracken,et al.  Mathematical Programs with Optimization Problems in the Constraints , 1973, Oper. Res..

[10]  Bo Zeng.  Solving Bilevel Mixed Integer Program by Reformulations and Decomposition June , 2014 , 2014 .

[11]  Gerhard J. Woeginger,et al.  Bilevel Knapsack with Interdiction Constraints , 2016, INFORMS J. Comput..

[12]  J. P. Secrétan,et al.  Der Saccus endolymphaticus bei Entzündungsprozessen , 1944 .

[13]  Ted K. Ralphs,et al.  The Symphony Callable Library for Mixed Integer Programming , 2005 .

[14]  Tobias Achterberg,et al.  Constraint integer programming , 2007 .

[15]  Ailsa H. Land,et al.  An Automatic Method of Solving Discrete Programming Problems , 1960 .

[16]  Berç Rustem,et al.  Parametric global optimisation for bilevel programming , 2007, J. Glob. Optim..

[17]  A. Land,et al.  An Automatic Method for Solving Discrete Programming Problems , 1960, 50 Years of Integer Programming.

[18]  Matthew J. Saltzman,et al.  Computational Experience with a Software Framework for Parallel Integer Programming , 2009, INFORMS J. Comput..

[19]  Ted K. Ralphs.  The SYMPHONY Callable Library for Mixed-Integer Linear Programming , 2005 .

[20]  Fengqi You,et al.  Design and optimization of shale gas energy systems: Overview, research challenges, and future directions , 2017, Comput. Chem. Eng..

[21]  T. Ralphs.  Duality and Warm Starting in Integer Programming , 2006 .

[22]  Pan Xu,et al.  An exact algorithm for the bilevel mixed integer linear programming problem under three simplifying assumptions , 2014, Comput. Oper. Res..

[23]  M. Padberg,et al.  Addendum: Optimization of a 532-city symmetric traveling salesman problem by branch and cut , 1990 .

[24]  Alexander Mitsos,et al.  Global solution of nonlinear mixed-integer bilevel programs , 2010, J. Glob. Optim..

[25]  A. M. Geoffrion.  Proper efficiency and the theory of vector maximization , 1968 .

[26]  Pan Xu,et al.  The Watermelon Algorithm for The Bilevel Integer Linear Programming Problem , 2017, SIAM J. Optim..

[27]  Jonathan F. Bard,et al.  Practical Bilevel Optimization , 1998 .

[28]  H. Stackelberg,et al.  Marktform und Gleichgewicht , 1935 .

[29]  Ying Zhang,et al.  The competitive facility location problem under disruption risks , 2016 .

[30]  Richard D. Wollmer,et al.  Removing Arcs from a Network , 1964 .

[31]  Matteo Fischetti,et al.  Intersection Cuts for Bilevel Optimization , 2016, IPCO.

[32]  Jonathan Cole Smith,et al.  A mixed-integer bilevel programming approach for a competitive prioritized set covering problem , 2016, Discret. Optim..

[33]  U. Wen,et al.  A simple Tabu Search method to solve the mixed-integer linear bilevel programming problem , 1996 .

[34]  Claire S. Adjiman,et al.  Branch-and-Sandwich: a deterministic global optimization algorithm for optimistic bilevel programming problems. Part I: Theoretical development , 2014, Journal of Global Optimization.

[35]  J. Bard,et al.  An algorithm for the discrete bilevel programming problem , 1992 .

[36]  Martin W. P. Savelsbergh,et al.  An Updated Mixed Integer Programming Library: MIPLIB 3.0 , 1998 .

[37]  M. Saltzman,et al.  Alps: A Framework for Implementing Parallel Tree Search Algorithms , 2005 .

[38]  T. Ralphs,et al.  Interdiction and discrete bilevel linear programming , 2011 .

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

[40]  Jonathan Cole Smith,et al.  A Value-Function-Based Exact Approach for the Bilevel Mixed-Integer Programming Problem , 2017, Oper. Res..

[41]  J. Salmeron,et al.  Analysis of electric grid security under terrorist threat , 2004, IEEE Transactions on Power Systems.

[42]  Jonathan F. Bard,et al.  The Mixed Integer Linear Bilevel Programming Problem , 1990, Oper. Res..

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

[44]  E. Balas,et al.  Canonical Cuts on the Unit Hypercube , 1972 .

[45]  M. Saltzman,et al.  ALPS : A Framework for Implementing Parallel Search Algorithms , 2004 .

[46]  Matthew J. Saltzman,et al.  A Library Hierarchy for Implementing Scalable Parallel Search Algorithms , 2004, The Journal of Supercomputing.

[47]  Thorsten Koch,et al.  Branching rules revisited , 2005, Oper. Res. Lett..

[48]  Jonathan F. Bard,et al.  Practical Bilevel Optimization: Algorithms and Applications , 1998 .

[49]  Jacqueline Morgan,et al.  Weak via strong Stackelberg problem: New results , 1996, J. Glob. Optim..

[50]  Matteo Fischetti,et al.  On the use of intersection cuts for bilevel optimization , 2018, Math. Program..

[51]  Egon Balas,et al.  Intersection Cuts - A New Type of Cutting Planes for Integer Programming , 1971, Oper. Res..