Compactness, Contractibility and Fixed Point Properties of the Pareto Sets in Multi-Objective Programming

ABSTRACT

This paper presents the Pareto solutions in continuous multi-objective mathematical programming. We discuss the role of some assumptions on the objective functions and feasible domain, the relationship between them, and compactness, contractibility and fixed point properties of the Pareto sets. The authors have tried to remove the concavity assumptions on the objective functions which are usually used in multi-objective maximization problems. The results are based on constructing a retraction from the feasible domain onto the Pareto-optimal set.

This paper presents the Pareto solutions in continuous multi-objective mathematical programming. We discuss the role of some assumptions on the objective functions and feasible domain, the relationship between them, and compactness, contractibility and fixed point properties of the Pareto sets. The authors have tried to remove the concavity assumptions on the objective functions which are usually used in multi-objective maximization problems. The results are based on constructing a retraction from the feasible domain onto the Pareto-optimal set.

KEYWORDS

Multi-Objective Programming, Pareto-Optimal, Pareto-Front, Compact, Contractible, Fixed Point, Retraction

Multi-Objective Programming, Pareto-Optimal, Pareto-Front, Compact, Contractible, Fixed Point, Retraction

Cite this paper

nullZ. Slavov and C. Evans, "Compactness, Contractibility and Fixed Point Properties of the Pareto Sets in Multi-Objective Programming,"*Applied Mathematics*, Vol. 2 No. 5, 2011, pp. 556-561. doi: 10.4236/am.2011.25073.

nullZ. Slavov and C. Evans, "Compactness, Contractibility and Fixed Point Properties of the Pareto Sets in Multi-Objective Programming,"

References

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[2] Y. Collette and P. Siarry, “Multi-Objective Optimization,” Springer, Berlin, 2003.

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

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

[5] B. Peleg, “Topological Properties of the Efficient Point Set,” Proceedings of the American Mathematical Society, Vol. 35, No. 2, 1972, pp. 531-536. doi:10.1090/S0002-9939-1972-0303916-2

[6] Z. Slavov and C. Evans, “On the Structure of the Efficient Set,” Mathematics and Education in Mathematics, Vol. 33, 2004, pp. 247-250.

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

[8] R. Steuer, “Multiple Criteria Optimization: Theory, Com- putation and Application,” John Wiley and Sons, Hoboken, 1986.

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

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

[11] Z. Slavov, “Compactness of the Pareto Sets in Multi- Objective Optimization with Quasi-Concave Functions,” Mathematics and Education in Mathematics, Vol. 35, 2006, pp. 298-301.

[12] J. Benoist, “Contractibility of the Efficient Set in Strictly Quasi-Concave Vector Maximization,” Journal of Optimization Theory and Applications, Vol. 110, No. 2, 2001, pp. 325-336. doi:10.1023/A:1017527329601

[13] 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

[14] Z. Slavov, “Contractibility of Pareto Solutions Sets in Concave Vector Maximization,” Mathematics and Education in Mathematics, Vol. 36, 2007, pp. 299-304.

[15] 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.

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

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

[18] A. Cellina, “Fixed Point of Noncontinuous Mapping,” Atti della Accademia Nazionale dei Lincei Serie Ottava Rendiconti, Vol. 49, 1970, pp. 30-33.

[19] J. Franklin, “Methods of Mathematical Economics: Linear and Nonlinear Programming, Fixed Point Theorems,” Springer, Berlin, 1980.

[20] A. Mukherjea and K. Pothoven, “Real and Functional Analysis,” Plenum Press, New York, 1978.

[21] R. Sundaran, “A First Course in Optimization Theory,” Cambridge University Press, London, 1996.

[22] A. Wilansky, “Topology for Analysis,” Dover Publications, New York, 1998.

[1] J. Cohon, “Multi-Objective Programming and Planning,” Academic Press, Cambridge, 1978.

[2] Y. Collette and P. Siarry, “Multi-Objective Optimization,” Springer, Berlin, 2003.

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

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

[5] B. Peleg, “Topological Properties of the Efficient Point Set,” Proceedings of the American Mathematical Society, Vol. 35, No. 2, 1972, pp. 531-536. doi:10.1090/S0002-9939-1972-0303916-2

[6] Z. Slavov and C. Evans, “On the Structure of the Efficient Set,” Mathematics and Education in Mathematics, Vol. 33, 2004, pp. 247-250.

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

[8] R. Steuer, “Multiple Criteria Optimization: Theory, Com- putation and Application,” John Wiley and Sons, Hoboken, 1986.

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

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

[11] Z. Slavov, “Compactness of the Pareto Sets in Multi- Objective Optimization with Quasi-Concave Functions,” Mathematics and Education in Mathematics, Vol. 35, 2006, pp. 298-301.

[12] J. Benoist, “Contractibility of the Efficient Set in Strictly Quasi-Concave Vector Maximization,” Journal of Optimization Theory and Applications, Vol. 110, No. 2, 2001, pp. 325-336. doi:10.1023/A:1017527329601

[13] 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

[14] Z. Slavov, “Contractibility of Pareto Solutions Sets in Concave Vector Maximization,” Mathematics and Education in Mathematics, Vol. 36, 2007, pp. 299-304.

[15] 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.

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

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

[18] A. Cellina, “Fixed Point of Noncontinuous Mapping,” Atti della Accademia Nazionale dei Lincei Serie Ottava Rendiconti, Vol. 49, 1970, pp. 30-33.

[19] J. Franklin, “Methods of Mathematical Economics: Linear and Nonlinear Programming, Fixed Point Theorems,” Springer, Berlin, 1980.

[20] A. Mukherjea and K. Pothoven, “Real and Functional Analysis,” Plenum Press, New York, 1978.

[21] R. Sundaran, “A First Course in Optimization Theory,” Cambridge University Press, London, 1996.

[22] A. Wilansky, “Topology for Analysis,” Dover Publications, New York, 1998.