Projective geometry and the outer approximation algorithm for multiobjective linear programming

A key problem in multiobjective linear programming is to find the set of all efficient extreme points in objective space. In this paper we introduce oriented projective geometry as an efficient and effective framework for solving this problem. The key advantage of oriented projective geometry is that we can work with an "optimally simple" but unbounded efficiency-equivalent polyhedron, yet apply to it the familiar theory and algorithms that are traditionally restricted to bounded polytopes. We apply these techniques to Benson's outer approximation algorithm, using oriented projective geometry to remove an exponentially large complexity from the algorithm and thereby remove a significant burden from the running time.

[1]  Heinz Isermann,et al.  The Enumeration of the Set of All Efficient Solutions for a Linear Multiple Objective Program , 1977 .

[2]  Heinz Isermann,et al.  Technical Note - Proper Efficiency and the Linear Vector Maximum Problem , 1974, Oper. Res..

[3]  G. W. Evans,et al.  An Overview of Techniques for Solving Multiobjective Mathematical Programs , 1984 .

[4]  Daniel P. Giesy,et al.  Multicriteria Optimization Methods for Design of Aircraft Control Systems , 1988 .

[5]  Jorge Stolfi,et al.  Oriented projective geometry , 1987, SCG '87.

[6]  Jared L. Cohon,et al.  Multiobjective programming and planning , 2004 .

[7]  Tomás Pajdla,et al.  Oriented Matching Constraints , 2001, BMVC.

[8]  Ralph E. Steuer.  Sausage Blending Using Multiple Objective Linear Programming , 1984 .

[9]  J. P. Dauer,et al.  Constructing the set of efficient objective values in multiple objective linear programs , 1990 .

[10]  P. Yu,et al.  The set of all nondominated solutions in linear cases and a multicriteria simplex method , 1975 .

[11]  G. M.,et al.  Projective Geometry , 1938, Nature.

[12]  R. Horst,et al.  On finding new vertices and redundant constraints in cutting plane algorithms for global optimization , 1988 .

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

[14]  Anthony Przybylski,et al.  A Recursive Algorithm for Finding All Nondominated Extreme Points in the Outcome Set of a Multiobjective Integer Programme , 2010, INFORMS J. Comput..

[15]  Milan Zeleny,et al.  Multiple Criteria Decision Making (MCDM) , 2004 .

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

[17]  Erjiang Sun,et al.  A weight set decomposition algorithm for finding all efficient extreme points in the outcome set of a multiple objective linear program , 2002, Eur. J. Oper. Res..

[18]  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..

[19]  R. J. Gallagher,et al.  A representation of an efficiency equivalent polyhedron for the objective set of a multiple objective linear program , 1995 .

[20]  P. Armand,et al.  Determination of the efficient set in multiobjective linear programming , 1991 .

[21]  N. Schulz.  Welfare Economics and the Vector Maximum Problem , 1988 .

[22]  Harold P. Benson,et al.  Hybrid Approach for Solving Multiple-Objective Linear Programs in Outcome Space , 1998 .

[23]  H. P. Benson,et al.  Outcome Space Partition of the Weight Set in Multiobjective Linear Programming , 2000 .

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

[26]  Pierre Hansen,et al.  On-line and off-line vertex enumeration by adjacency lists , 1991, Oper. Res. Lett..

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

[28]  Ralph E. Steuer,et al.  A regression study of the number of efficient extreme points in multiple objective linear programming , 2005, Eur. J. Oper. Res..

[29]  W. Stadler.  Multicriteria Optimization in Engineering and in the Sciences , 1988 .

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

[31]  J. Ecker,et al.  Generating all maximal efficient faces for multiple objective linear programs , 1980 .

[32]  Ute Rosenbaum,et al.  Projective Geometry: From Foundations to Applications , 1998 .

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

[34]  H. Raiffa,et al.  3. The Double Description Method , 1953 .

[35]  Olivier D. Faugeras,et al.  Oriented Projective Geometry for Computer Vision , 1996, ECCV.

[36]  T. Gal.  A general method for determining the set of all efficient solutions to a linear vectormaximum problem , 1977 .

[37]  Bernd Gärtner,et al.  Understanding and using linear programming , 2007, Universitext.