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

Show more

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.