The vector linear program solver Bensolve - notes on theoretical background

Bensolve is an open source implementation of Benson’s algorithm and its dual variant. Both algorithms compute primal and dual solutions of vector linear programs (VLP), which include the subclass of multiple objective linear programs (MOLP). The recent version of Bensolve can treat arbitrary vector linear programs whose upper image does not contain lines. This article surveys the theoretical background of the implementation. In particular, the role of VLP duality for the implementation is pointed out. Some numerical examples are provided. In contrast to the existing literature we consider a less restrictive class of vector linear programs.

[1]  László Csirmaz,et al.  Using multiobjective optimization to map the entropy region , 2013, Computational Optimization and Applications.

[2]  Andreas Löhne,et al.  Geometric Duality in Multiple Objective Linear Programming , 2008, SIAM J. Optim..

[3]  Komei Fukuda,et al.  Double Description Method Revisited , 1995, Combinatorics and Computer Science.

[4]  Yi-Hsin Liu,et al.  Solving multiple objective linear programs in objective space , 1990 .

[5]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[6]  H. P. Benson,et al.  Further Analysis of an Outcome Set-Based Algorithm for Multiple-Objective Linear Programming , 1998 .

[7]  J. Dauer.  Analysis of the objective space in multiple objective linear programming , 1987 .

[8]  M. E. Dyer,et al.  The Complexity of Vertex Enumeration Methods , 1983, Math. Oper. Res..

[9]  Andreas H. Hamel,et al.  Benson type algorithms for linear vector optimization and applications , 2013, Journal of Global Optimization.

[10]  Andreas Löhne,et al.  Equivalence between polyhedral projection, multiple objective linear programming and vector linear programming , 2015, Math. Methods Oper. Res..

[11]  Lizhen Shao,et al.  A dual variant of Benson’s “outer approximation algorithm” for multiple objective linear programming , 2012, J. Glob. Optim..

[12]  Harold P. Benson,et al.  An Outer Approximation Algorithm for Generating All Efficient Extreme Points in the Outcome Set of a Multiple Objective Linear Programming Problem , 1998, J. Glob. Optim..

[13]  H. P. Williams.  THEORY OF LINEAR AND INTEGER PROGRAMMING (Wiley-Interscience Series in Discrete Mathematics and Optimization) , 1989 .

[14]  Lizhen Shao,et al.  Approximately solving multiobjective linear programmes in objective space and an application in radiotherapy treatment planning , 2008, Math. Methods Oper. Res..

[15]  Andreas Löhne,et al.  Solution concepts in vector optimization: a fresh look at an old story , 2011 .

[16]  Andreas Löhne,et al.  Vector Optimization with Infimum and Supremum , 2011, Vector Optimization.

[17]  Ralph E. Steuer,et al.  A revised simplex method for linear multiple objective programs , 1973, Math. Program..

[18]  Petra Mutzel,et al.  Output-Sensitive Algorithms for Enumerating the Extreme Nondominated Points of Multiobjective Combinatorial Optimization Problems , 2015, ESA.