Existence Results for Minimal Points
In this chapter we establish several existence results for minimal points with respect to transitive relations; then we apply them in topological vector spaces for quasiorders generated by convex cones. We continue with the presentation of several types o
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Existence Results for Minimal Points
In this chapter we establish several existence results for minimal points with respect to transitive relations; then we apply them in topological vector spaces for quasiorders generated by convex cones. We continue with the presentation of several types of convex cones and compactness notions with respect to cones. We end the chapter with existence results for vector and set optimization problems. The presentation in Sects. 9.1, 9.2, 9.3 and 9.4 follows that in [214] or [539], where one can find the proofs for the results which are only stated.
9.1 Preliminary Notions and Results Concerning Transitive Relations In the sequel Y is a nonempty set and t Y Y , that is, t is a relation on Y . If ; ¤ A Y and t Y Y , the restriction of t to A is denoted by t A ; i.e., t A WD t \ .A A/. With the relation t on Y we associate the following relations: tR WD t [ Y ;
tN WD t n t 1 D t n .t \ t 1 /;
tNR D .tN /R :
Hazen and Morin [240] call tN the asymmetric part of t. Some properties of these relations are given in the following proposition; the first three properties mentioned in (ii) are stated by Dolecki and Malivert in [144]. Proposition 9.1.1. Let t be a transitive relation on Y and ; ¤ A Y . (i) tR is reflexive and transitive; tN \ Y D ;; (ii) t ı tN tN , tN ı t tN , tN ı tN tN , .tR /N D tN , tR ı tN D tN ı tR D tN ; .tN /N D tN ;
© Springer-Verlag Berlin Heidelberg 2015 A.A. Khan et al., Set-valued Optimization, Vector Optimization, DOI 10.1007/978-3-642-54265-7__9
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9 Existence Results for Minimal Points
(iii) tNR is reflexive, antisymmetric, and transitive; .tN /N D .tNR /N D tN ; (iv) t A R D .tR /A , t A N D .tN /A . A A A Taking into the above account proposition, we denote by tR , tN , and tNR the A A A relations t R , t N , and t NR , respectively. The preceding proposition shows that with every transitive relation one can associate a partial order. It is useful to know whether they determine the same maximal and minimal points. As noted in Sect. 2.1, Max.AI t/ D Min.A; t 1 / for ; ¤ A Y ; so it is sufficient to study only the problems related to minimal points.
Corollary 9.1.2. Let t be a transitive relation on Y and ; ¤ A Y ; then Min.AI t/ D Min.AI tR / D Min.AI tN / D Min.AI tNR /: The above corollary shows that the problem of existence (for example) of minimal points w.r.t. to a transitive relation t reduces, theoretically, to the same problem for the partial order tNR . Another way to reduce this problem to one for a partial order is given by the following known result. Proposition 9.1.3. Let t be a transitive relation on Y and take WD tR \ .tR /1 . Then is an equivalence relation, and tO D f.x; O y/ O j .x; y/ 2 tg is a partial order on YO WD Y =, where xO is the class of x 2 Y with respect to . Moreover, if x 2 Y , x 2 Min.Y I t/ if and only if xO 2 Min.YO I tO/. In the sequel we shall also use the notation AC t .x/ and At .x/ for the upper and lower sections of A Y with respect to t and x 2 Y . S
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