Generating equidistant representations in biobjective programming

In recent years, emphasis has been placed on generating quality representations of the nondominated set of multiobjective optimization problems. This paper presents two methods for generating discrete representations with equidistant points for biobjective problems with solution sets determined by convex, polyhedral cones. The Constraint Controlled-Spacing method is based on the epsilon-constraint method with an additional constraint to control the spacing of generated points. The Bilevel Controlled-Spacing method has a bilevel structure with the lower-level generating the nondominated points and the upper-level controlling the spacing, and is extended to multiobjective problems. Both methods are proven to produce (weakly) nondominated points and are demonstrated on a variety of test problems.

[1]  Yacov Y. Haimes,et al.  Multiobjective Decision Making: Theory and Methodology , 1983 .

[2]  G. R. Bitran,et al.  The structure of admissible points with respect to cone dominance , 1979 .

[3]  Gabriele Eichfelder.  A Constraint Method in Nonlinear Multi-Objective Optimization , 2009 .

[4]  Matthias Ehrgott,et al.  Multiple criteria decision analysis: state of the art surveys , 2005 .

[5]  H. P. Benson,et al.  A face search heuristic algorithm for optimizing over the efficient set , 1993 .

[6]  Siegfried Helbig,et al.  On a constructive approximation of the efficient outcomes in bicriterion vector optimization , 1994, J. Glob. Optim..

[7]  Michael Masin,et al.  Diversity Maximization Approach for Multiobjective Optimization , 2008, Oper. Res..

[8]  Alejandro Crema,et al.  A method for finding well-dispersed subsets of non-dominated vectors for multiple objective mixed integer linear programs , 2007, Eur. J. Oper. Res..

[9]  Ralph E. Steuer,et al.  An interactive weighted Tchebycheff procedure for multiple objective programming , 1983, Math. Program..

[10]  Brian J. Hunt.  Multiobjective Programming with Convex Cones: Methodology and Applications , 2004 .

[11]  S. Scholtes,et al.  Nondifferentiable and two-level mathematical programming , 1997 .

[12]  Yaochu Jin,et al.  Knowledge incorporation in evolutionary computation , 2005 .

[13]  Xavier Gandibleux,et al.  Multiobjective Programming and Goal Programming: Theoretical Results and Practical Applications , 2009 .

[14]  Bernd Schandl.  Norm-Based Evaluation and Approximation in Multicriteria Programming , 1999 .

[15]  Stacey L. Faulkenberg,et al.  On the quality of discrete representations in multiple objective programming , 2010 .

[16]  Serpil Sayin,et al.  A Procedure to Find Discrete Representations of the Efficient Set with Specified Coverage Errors , 2003, Oper. Res..

[17]  M. Wiecek,et al.  Inclusion of Preferences in the Design of Evolutionary Optimization Algorithms: An Empirical Study , 2008 .

[18]  Matthias Ehrgott,et al.  Saddle Points and Pareto Points in Multiple Objective Programming , 2005, J. Glob. Optim..

[19]  S Scholtes,et al.  Nondifferentiable and two-level mathematical programming - Shimizu,K, Ishizuka,Y, Bard,JF , 1997 .

[20]  Serpil Sayin,et al.  Algorithm robust for the bicriteria discrete optimization problem , 2006, Ann. Oper. Res..

[21]  Mokhtar S. Bazaraa,et al.  Nonlinear Programming: Theory and Algorithms , 1993 .

[22]  Harold P. Benson,et al.  A bisection-extreme point search algorithm for optimizing over the efficient set in the linear dependence case , 1993, J. Glob. Optim..

[23]  P. Yu.  Multiple-Criteria Decision Making: "Concepts, Techniques, And Extensions" , 2012 .

[24]  Alexander V. Lotov,et al.  Interactive Decision Maps: Approximation and Visualization of Pareto Frontier , 2004 .

[25]  Serpil Sayin,et al.  Measuring the quality of discrete representations of efficient sets in multiple objective mathematical programming , 2000, Math. Program..

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

[27]  H. P. Benson,et al.  A finite, nonadjacent extreme-point search algorithm for optimization over the efficient set , 1992 .

[28]  Matthew L. Tenhuisen,et al.  On the Structure of the Non-dominated Set for Bicriteria Programmes , 1996 .

[29]  Gabriele Eichfelder,et al.  An Adaptive Scalarization Method in Multiobjective Optimization , 2008, SIAM J. Optim..

[30]  Brian J. Hunt,et al.  MCDM with Relative Importance of Criteria: Application to Configuration Design of Vehicles , 2009 .

[31]  Horst W. Hamacher,et al.  Finding representative systems for discrete bicriterion optimization problems , 2007, Oper. Res. Lett..

[32]  Margaret M. Wiecek,et al.  Modeling relative importance of design criteria with a modified pareto preference , 2007 .

[33]  Johannes Jahn,et al.  Multiobjective Search Algorithm with Subdivision Technique , 2006, Comput. Optim. Appl..

[34]  H. P. Benson,et al.  An algorithm for optimizing over the weakly-efficient set , 1986 .

[35]  Alexander Engau,et al.  Domination and decomposition in multiobjective programming , 2007 .

[36]  V. Noghin.  Relative importance of criteria: a quantitative approach , 1997 .

[37]  Sven Leyffer,et al.  A Complementarity Constraint Formulation of Convex Multiobjective Optimization Problems , 2009, INFORMS J. Comput..

[38]  Murat Köksalan,et al.  Generating a Representative Subset of the Nondominated Frontier in Multiple Criteria Decision Making , 2009, Oper. Res..

[39]  J. Jahn.  Introduction to the Theory of Nonlinear Optimization , 1994 .

[40]  Kalyanmoy Deb,et al.  Integrating User Preferences into Evolutionary Multi-Objective Optimization , 2005 .

[41]  H. P. Benson,et al.  Towards finding global representations of the efficient set in multiple objective mathematical programming , 1997 .

[42]  Serpil Sayin,et al.  The Multiobjective Discrete Optimization Problem: A Weighted Min-Max Two-Stage Optimization Approach and a Bicriteria Algorithm , 2005, Manag. Sci..

[43]  Masahiro Tanaka,et al.  GA-based decision support system for multicriteria optimization , 1995, 1995 IEEE International Conference on Systems, Man and Cybernetics. Intelligent Systems for the 21st Century.

[44]  S. Ruzika,et al.  Approximation Methods in Multiobjective Programming , 2005 .

[45]  S. Yu. Churkina,et al.  Search for Efficient Solutions of Multi-Criterion Problems by Target-Level Method , 2002 .

[46]  R. Armann,et al.  Solving multiobjective programming problems by discrete representation , 1989 .

[47]  Jörg Fliege,et al.  Gap-free computation of Pareto-points by quadratic scalarizations , 2004, Math. Methods Oper. Res..

[48]  P. Serafini,et al.  Scalarizing vector optimization problems , 1984 .

[49]  Johannes Jahn,et al.  Reference point approximation method for the solution of bicriterial nonlinear optimization problems , 1992 .

[50]  Harold P. Benson,et al.  Outcome-based algorithm for optimizing over the efficient set of a bicriteria linear programming problem , 1996 .

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

[52]  M. Wiecek.  Advances in Cone-Based Preference Modeling for Decision Making with Multiple Criteria , 2007 .