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On the notion of proper efficiency in vector optimization. (English) Zbl 0827.90123
This is a survey paper concerned with proper efficiency in vector optimization. The paper considers thirteen types of notions of proper efficiency, including superefficiency and the Geoffrion’s definition, for the following vector optimization problem: \[ \text{maximize}\quad f(x),\qquad\text{subject to}\quad x\in A, \tag{P} \] where \(X\) is a topological vector space and \(Y\) is an ordered topological vector space with a closed, convex, and pointed cone \(K\), which induces a partial ordering to \(Y\), \(A\) is a subset of \(X\), and \(f: X\to Y\).
The authors classify them into four groups. The first group consists of the scalar maximum solutions (Pos) and Hurwicz’s properly efficient solutions (Hu) for problem (P). Pos\(\subset\)Hu; Hu\(\subset\)Pos if \(Y\) is a locally convex space and \(K\) has a compact base. The second group is made up of Hartley’s properly efficient solutions (Ha), Benson’s properly efficient solutions (Be), Borwein’s globally properly efficient solutions (GBo), Henig’s globally properly efficient solutions (GHe), and the superefficient solutions (SE), \(\text{SE}\subset\text{HA}\subset \text{BE} \subset \text{GBo}\); \(\text{Pos} \subset \text{GHe} \subset \text{Be}\); \(\text{SE} \subset {GHe}\) if \(K\) has a base; and \(\text{GBo} \subset \text{SE}\) if \(K\) has a weakly compact base. The third group is formed by Borwein’s properly efficient solutions (Bo) and Henig’s locally properly efficient solutions (LHe). \(\text{GHe} \subset \text{LHe}\); \(\text{Be} \subset \text{Bo}\); \(\text{LHe} \subset \text{Bo}\) if \(Y\) is a normed space; \(\text{Bo} \subset \text{LHe}\) if \(Y\) is a normed space and \(K\) has a compact base. The others are Kuhn-Tucker’s properly efficient solutions \((\text{KT}_\infty)\), Klinger’s properly efficient solutions \((\text{K}_\infty)\), and Borwein’s locally properly efficient solutions (LBo). The sets \(\text{KT}_\infty\) and \(\text{K}_\infty\) are defined for \(A= \{x\in X: g(x)\in D\}\) and Fréchet differentiability of \(f\) and \(g\). \(\text{KT}_\infty\subset \text{K}_\infty\); \(\text{LBo}\subset K_\infty\); \(\text{Bo}\subset \text{LBo}\); \(\text{GBo}\subset \text{LBo}\), etc.

MSC:
90C29 Multi-objective and goal programming
90C48 Programming in abstract spaces
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