An Application of the Maximum Theorem in Multi-Criteria Optimization, Properties of Pareto-Retract Mappings, and the Structure of Pareto Sets

Affiliation(s)

Varna Free University, Varna, Bulgaria.

The George Washington University, Washington DC, USA.

Varna Free University, Varna, Bulgaria.

The George Washington University, Washington DC, USA.

ABSTRACT

In this paper we consider three problems in continuous multi-criteria optimization: An application of the Berge Maximum Theorem, properties of Pareto-retract mappings, and the structure of Pareto sets. The key goal of this work is to present the relationship between the three problems mentioned above. First, applying the Maximum Theorem we construct the Pareto-retract mappings from the feasible domain onto the Pareto-optimal solutions set if the feasible domain is compact. Next, using these mappings we analyze the structure of the Pareto sets. Some basic topological properties of the Pareto solutions sets in the general case and in the convex case are also discussed.

In this paper we consider three problems in continuous multi-criteria optimization: An application of the Berge Maximum Theorem, properties of Pareto-retract mappings, and the structure of Pareto sets. The key goal of this work is to present the relationship between the three problems mentioned above. First, applying the Maximum Theorem we construct the Pareto-retract mappings from the feasible domain onto the Pareto-optimal solutions set if the feasible domain is compact. Next, using these mappings we analyze the structure of the Pareto sets. Some basic topological properties of the Pareto solutions sets in the general case and in the convex case are also discussed.

KEYWORDS

Multi-Criteria Optimization; Maximum Theorem; Pareto-Retract Mapping; Pareto-Optimal; Pareto-Front

Multi-Criteria Optimization; Maximum Theorem; Pareto-Retract Mapping; Pareto-Optimal; Pareto-Front

Cite this paper

Z. Slavov and C. Evans, "An Application of the Maximum Theorem in Multi-Criteria Optimization, Properties of Pareto-Retract Mappings, and the Structure of Pareto Sets,"*Applied Mathematics*, Vol. 3 No. 10, 2012, pp. 1415-1422. doi: 10.4236/am.2012.330199.

Z. Slavov and C. Evans, "An Application of the Maximum Theorem in Multi-Criteria Optimization, Properties of Pareto-Retract Mappings, and the Structure of Pareto Sets,"

References

[1] C. Berge, “Topological Spaces, Including Treatment of Multi-Valued Function, Vector Spaces, and Convexity,” Oliver and Boyd, Edinburgh, 1963.

[2] R. Sundaran, “A First Course in Optimization Theory,” Cambridge University Press, Cambridge, 1996. doi:10.1017/CBO9780511804526

[3] D. Luc, “Theory of Vector Optimization,” Springer, Berlin, 1989.

[4] J. Jahn, “Vector Optimization: Theory, Applications, and Extensions,” Springer, Berlin, 2004.

[5] R. Steuer, “Multiple Criteria Optimization: Theory, Computation and Application,” John Wiley and Sons, New York, 1986.

[6] M. Ehrgott, “Multi-Criteria Optimization,” Springer, Berlin, 2005.

[7] A. Hatcher, “Algebraic Topology,” Cambridge University Press, Cambridge, 2002.

[8] J. Benoist, “The Structure of the Efficient Frontier of Finite-Dimensional Completely-Shaded Sets,” Journal of Mathematical Analysis and Application, Vol. 250, No. 1, 2000, pp. 98-117. doi:10.1006/jmaa.2000.6960

[9] N. Huy and N. Yen, “Contractibility of the Solution Sets in Strictly Quasi-Concave Vector Maximization on Noncompact Domains,” Journal of Optimization Theory and Applications, Vol. 124, No. 3, 2005, pp. 615-635. doi:10.1007/s10957-004-1177-9

[10] Z. Slavov, “On the Engineering Multi-Objective Maximization and Properties of the Pareto-Optimal Set,” International e-Journal of Engineering Mathematics: Theory and Application, Vol. 7, 2009, pp. 32-46.

[11] Z. Slavov, “On Pareto Sets in Multi-Criteria Optimization,” Mathematics and Education in Mathematics, Vol. 40, 2011, pp. 207-212.

[12] Z. Slavov and C. Evans, “Compactness, Contractibility and Fixed Point Properties of Pareto Sets in Multi-Objective Programming,” Applied Mathematics, Vol. 2, No. 5, 2011, pp. 556-561. doi:10.4236/am.2011.25073

[13] A. Wilansky, “Topology for Analysis,” Dover Publications, 1998.

[14] H. Benson and E. Sun, “New Closedness Results for Efficient Sets in Multiple Objective Mathematical Programming,” Journal of Mathematical Analysis and Application, Vol. 238, No. 1, 1999, pp. 277-296.

[15] G. Bitran and T. Magnanti, “The Structure of Admissible Points with Respect to Cone Dominance,” Journal of Optimization Theory and Application, Vol. 29, 1979, pp. 573-614.

[16] C. Malivert and N. Boissard, “Structure of Efficient Sets for Strictly Quasi-Convex Objectives,” Journal of Convex Analysis, Vol. 1, No. 2, 1994, pp. 143-150.

[17] J. Benoist, “Connectedness of the Efficient Set for Strictly Quasi-Concave Sets,” Journal of Optimization Theory and Application, Vol. 96, No. 3, 1998, pp. 627654. doi:10.1023/A:1022616612527

[18] A. Danilidis, N. Hajisavvas and S. Schaible, “Connectedness of the Efficient Set for Three-Objective Maximization Problems,” Journal of Optimization Theory and Application, Vol. 93, 1997, pp. 517-524.

[19] M. Hirschberger, “Connectedness of Efficient Points in Convex and Convex Transformable Vector Optimization,” Optimization, Vol. 54, No. 3, 2005, pp. 283-304. doi:10.1080/02331930500096270

[20] Y. Hu and E. Sun, “Connectedness of the Efficient Set in Strictly Quasi-Concave Vector Maximization,” Journal of Optimization Theory and Application, Vol. 78, No. 3, 1993, pp. 613-622. doi:10.1007/BF00939886

[21] D. Luc, “Connectedness of the Efficient Point Sets in Quasi-Concave Vector Maximization,” Journal of Mathematical Analysis and Application, Vol. 122, No. 2, 1987, pp. 346-354.

[22] P. Naccache, “Connectedness of the Set of Nondominated Outcomes in Multi-Criteria Optimization,” Journal of Optimization Theory and Application, Vol. 29, No. 3, 1978, pp. 459-466. doi:10.1007/BF00932907

[23] E. Sun, “On the Connectedness of the Efficient Set for Strictly Quasi-Concave Vector Maximization Problems,” Journal of Optimization Theory and Application, Vol. 89 1996, pp. 475-581. doi:10.1007/BF02192541

[24] A. Warburton, “Quasi-Concave Vector Maximization: Connectedness of the Sets of Pareto-Optimal and Weak Pareto-Optimal Alternatives,” Journal of Optimization Theory and Application, Vol. 40, No. 4, 1983, pp. 537-557. doi:10.1007/BF00933970

[25] J. Benoist, “Contractibility of the Efficient Set in Strictly Quasi-Concave Vector Maximization,” Journal of Optimization Theory and Applications, Vol. 110, 2001, pp. 325-336. doi:10.1023/A:1017527329601

[26] Z. Slavov, “The Fixed Point Property in Convex MultiObjective Optimization Problem,” Acta Universitatis Apulensis, Vol. 15, 2008, pp. 405-414.

[27] J. Borwein and A. Lewis, “Convex Analysis and Nonlinear Optimization: Theory and Examples,” Springer, Berlin, 2000.

[28] S. Boyd and L. Vandenberghe, “Convex Optimization,” Cambridge University Press, Cambridge, 2004.

[1] C. Berge, “Topological Spaces, Including Treatment of Multi-Valued Function, Vector Spaces, and Convexity,” Oliver and Boyd, Edinburgh, 1963.

[2] R. Sundaran, “A First Course in Optimization Theory,” Cambridge University Press, Cambridge, 1996. doi:10.1017/CBO9780511804526

[3] D. Luc, “Theory of Vector Optimization,” Springer, Berlin, 1989.

[4] J. Jahn, “Vector Optimization: Theory, Applications, and Extensions,” Springer, Berlin, 2004.

[5] R. Steuer, “Multiple Criteria Optimization: Theory, Computation and Application,” John Wiley and Sons, New York, 1986.

[6] M. Ehrgott, “Multi-Criteria Optimization,” Springer, Berlin, 2005.

[7] A. Hatcher, “Algebraic Topology,” Cambridge University Press, Cambridge, 2002.

[8] J. Benoist, “The Structure of the Efficient Frontier of Finite-Dimensional Completely-Shaded Sets,” Journal of Mathematical Analysis and Application, Vol. 250, No. 1, 2000, pp. 98-117. doi:10.1006/jmaa.2000.6960

[9] N. Huy and N. Yen, “Contractibility of the Solution Sets in Strictly Quasi-Concave Vector Maximization on Noncompact Domains,” Journal of Optimization Theory and Applications, Vol. 124, No. 3, 2005, pp. 615-635. doi:10.1007/s10957-004-1177-9

[10] Z. Slavov, “On the Engineering Multi-Objective Maximization and Properties of the Pareto-Optimal Set,” International e-Journal of Engineering Mathematics: Theory and Application, Vol. 7, 2009, pp. 32-46.

[11] Z. Slavov, “On Pareto Sets in Multi-Criteria Optimization,” Mathematics and Education in Mathematics, Vol. 40, 2011, pp. 207-212.

[12] Z. Slavov and C. Evans, “Compactness, Contractibility and Fixed Point Properties of Pareto Sets in Multi-Objective Programming,” Applied Mathematics, Vol. 2, No. 5, 2011, pp. 556-561. doi:10.4236/am.2011.25073

[13] A. Wilansky, “Topology for Analysis,” Dover Publications, 1998.

[14] H. Benson and E. Sun, “New Closedness Results for Efficient Sets in Multiple Objective Mathematical Programming,” Journal of Mathematical Analysis and Application, Vol. 238, No. 1, 1999, pp. 277-296.

[15] G. Bitran and T. Magnanti, “The Structure of Admissible Points with Respect to Cone Dominance,” Journal of Optimization Theory and Application, Vol. 29, 1979, pp. 573-614.

[16] C. Malivert and N. Boissard, “Structure of Efficient Sets for Strictly Quasi-Convex Objectives,” Journal of Convex Analysis, Vol. 1, No. 2, 1994, pp. 143-150.

[17] J. Benoist, “Connectedness of the Efficient Set for Strictly Quasi-Concave Sets,” Journal of Optimization Theory and Application, Vol. 96, No. 3, 1998, pp. 627654. doi:10.1023/A:1022616612527

[18] A. Danilidis, N. Hajisavvas and S. Schaible, “Connectedness of the Efficient Set for Three-Objective Maximization Problems,” Journal of Optimization Theory and Application, Vol. 93, 1997, pp. 517-524.

[19] M. Hirschberger, “Connectedness of Efficient Points in Convex and Convex Transformable Vector Optimization,” Optimization, Vol. 54, No. 3, 2005, pp. 283-304. doi:10.1080/02331930500096270

[20] Y. Hu and E. Sun, “Connectedness of the Efficient Set in Strictly Quasi-Concave Vector Maximization,” Journal of Optimization Theory and Application, Vol. 78, No. 3, 1993, pp. 613-622. doi:10.1007/BF00939886

[21] D. Luc, “Connectedness of the Efficient Point Sets in Quasi-Concave Vector Maximization,” Journal of Mathematical Analysis and Application, Vol. 122, No. 2, 1987, pp. 346-354.

[22] P. Naccache, “Connectedness of the Set of Nondominated Outcomes in Multi-Criteria Optimization,” Journal of Optimization Theory and Application, Vol. 29, No. 3, 1978, pp. 459-466. doi:10.1007/BF00932907

[23] E. Sun, “On the Connectedness of the Efficient Set for Strictly Quasi-Concave Vector Maximization Problems,” Journal of Optimization Theory and Application, Vol. 89 1996, pp. 475-581. doi:10.1007/BF02192541

[24] A. Warburton, “Quasi-Concave Vector Maximization: Connectedness of the Sets of Pareto-Optimal and Weak Pareto-Optimal Alternatives,” Journal of Optimization Theory and Application, Vol. 40, No. 4, 1983, pp. 537-557. doi:10.1007/BF00933970

[25] J. Benoist, “Contractibility of the Efficient Set in Strictly Quasi-Concave Vector Maximization,” Journal of Optimization Theory and Applications, Vol. 110, 2001, pp. 325-336. doi:10.1023/A:1017527329601

[26] Z. Slavov, “The Fixed Point Property in Convex MultiObjective Optimization Problem,” Acta Universitatis Apulensis, Vol. 15, 2008, pp. 405-414.

[27] J. Borwein and A. Lewis, “Convex Analysis and Nonlinear Optimization: Theory and Examples,” Springer, Berlin, 2000.

[28] S. Boyd and L. Vandenberghe, “Convex Optimization,” Cambridge University Press, Cambridge, 2004.