Abstract
In real linear spaces, partial orderings are usually generated by ordering cones. In many situations, however, such an ordering cone is too small with respect to the whole space. Therefore, in this paper, we extend the concept of ordering cones to a more general concept. For this purpose, we define a parameterized binary relation, based on a convex cone and a binary function. We investigate some geometrical and topological properties of this relation in detail.
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Jahn, J.: Vector Optimization: Theory, Applications, and Extensions. Springer, Berlin (2011)
Weidner, P.: Charakterisierung von Mengen effizienter Elemente in linearen Räumen auf der Grundlage allgemeiner Bezugsmengen. Dissertation (A), Universität Halle-Wittenberg (1985)
Schiel, R.: Eine parametrisierte binäre Relation in der Vektoroptimierung. Diploma thesis, Universität Erlangen-Nürnberg (2010)
Sommer, C.: Eine neue Ordnungsrelation in der Vektoroptimierung. Dissertation, Universität Erlangen-Nürnberg. http://www.mso.math.fau.de/applied-mathematics-2/staff-members/sommer-christian/dr-christian-sommer.html (2012). Accessed 17 July 2012
Yu, P.-L.: Multiple-Criteria Decision Making. Plenum Press, New York (1985)
Eppler, K.: Optimale Steuerung für elliptische RWA. Technische Universität Dresden, Manuskript (2010)
Alt, H.: Lineare Funktionalanalysis. Springer, Berlin (2006)
Bergstresser, K., Charnes, A., Yu, P.-L.: Generalization of domination structures and nondominated solutions in multicriteria decision making. J. Optim. Theory Appl. 18, 3–13 (1976)
Yu, P.-L.: Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives. J. Optim. Theory Appl. 14, 319–377 (1974)
Eichfelder, G.: Cone-valued maps in optimization. Appl. Anal. 91, 1831–1846 (2012)
Pruckner, M.: Kegelwertige Abbildungen in der Optimierung. Diploma thesis, Universität Erlangen-Nürnberg (2011)
Eichfelder, G., Truong, X.D.H.: Optimality conditions for vector optimization problems with variable ordering structures. Optimization 62, 597–627 (2013)
Eichfelder, G.: Optimal elements in vector optimization with a variable ordering structure. J. Optim. Theory Appl. 151, 217–240 (2011)
Engau, A.: Variable preference modeling with ideal-symmetric convex cones. J. Global Optim. 42, 295–311 (2008)
Jahn, J.: Bishop–Phelps cones in optimization. Int. J. Optim. Theory Methods Appl. 1, 123–139 (2009)
Krasnosel’skii, M.A.: Positive Solutions of Operator Equations. Noordhoff, Groningen (1964)
Petschke, M.: On a theorem of Arrow, Barankin, and Blackwell. SIAM J. Control Optim. 28, 395–401 (1990)
Truong, X.D.H., Jahn, J.: Properties of Bishop-Phelps cones. Preprint, Universität Erlangen-Nürnberg, No. 343 (2010)
Sommer, C.: Geometric properties of a parameterized order relation in vector optimization. Optimization (2014, to appear)
Sommer, C.: Minimality concepts using a new parameterized binary relation in vector optimization. Appl. Math. Sci. 7, 2841–2861 (2013)
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Communicated by Johannes Jahn.
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Sommer, C. Geometrical and Topological Properties of a Parameterized Binary Relation in Vector Optimization. J Optim Theory Appl 163, 815–840 (2014). https://doi.org/10.1007/s10957-014-0529-3
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DOI: https://doi.org/10.1007/s10957-014-0529-3