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

References

[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.