A new complexity result on multiobjective linear integer programming using short rational generating functions

This paper presents a new complexity result for solving multiobjective integer programming problems. We prove that encoding the entire set of nondominated solutions of the problem in a short sum of rational functions is polynomially doable, when the dimension of the decision space is fixed. This result extends a previous result presented in De Loera et al. (INFORMS J. Comput. 21(1):39–48, 2009) in that there the number of the objective functions is assumed to be fixed whereas ours allows this number to vary.

[1]  Xavier Gandibleux,et al.  Multiple Criteria Optimization: State of the Art Annotated Bibliographic Surveys , 2013 .

[2]  P. Pardalos,et al.  Pareto optimality, game theory and equilibria , 2008 .

[3]  Jean B. Lasserre.  Erratum to "Generating functions and duality for integer programs": [Discrete Optimization 1 (2) (2004) 167-187] , 2005, Discret. Optim..

[4]  M. Conceição Fonseca,et al.  A stochastic bi-objective location model for strategic reverse logistics , 2010 .

[5]  Daniel Scholz.  The multicriteria big cube small cube method , 2010 .

[6]  M. Brion.  Points entiers dans les polyèdres convexes , 1988 .

[7]  Alexander I. Barvinok,et al.  A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[8]  R. S. Laundy,et al.  Multiple Criteria Optimisation: Theory, Computation and Application , 1989 .

[9]  Mariano Luque,et al.  Portfolio selection in the Spanish stock market by interactive multiobjective programming , 2011 .

[10]  Justo Puerto,et al.  Multiobjective solution of the uncapacitated plant location problem , 2003, Eur. J. Oper. Res..

[11]  Anne Setämaa-Kärkkäinen,et al.  Heuristic for a new multiobjective scheduling problem , 2007, Optim. Lett..

[12]  Jesús A. De Loera,et al.  Short rational functions for toric algebra and applications , 2004, J. Symb. Comput..

[13]  A. Barvinok,et al.  Short rational generating functions for lattice point problems , 2002, math/0211146.

[14]  S. Zionts.  A Survey of Multiple Criteria Integer Programming Methods , 1979 .

[15]  Ralph E. Steuer.  Multiple criteria optimization , 1986 .

[16]  Justo Puerto,et al.  Some algebraic methods for solving multiobjective polynomial integer programs , 2009, J. Symb. Comput..

[17]  Jesús A. De Loera,et al.  Pareto Optima of Multicriteria Integer Linear Programs , 2009, INFORMS J. Comput..

[18]  Jean B. Lasserre,et al.  Generating functions and duality for integer programs , 2004, Discret. Optim..

[19]  Justo Puerto,et al.  Partial Gröbner Bases for Multiobjective Integer Linear Optimization , 2007, SIAM J. Discret. Math..

[20]  Panos M. Pardalos,et al.  A survey of recent developments in multiobjective optimization , 2007, Ann. Oper. Res..

[21]  Jesús A. De Loera,et al.  A computational study of integer programming algorithms based on Barvinok's rational functions , 2005, Discret. Optim..