Duality in vector optimization. I: Abstract duality scheme. (English) Zbl 0556.49010

The paper was conceived by the author in three parts. The first part presents an abstract duality scheme. The second one discusses dual problems in vectorial quasiconvex programming. The last part is devoted to fractional programming. In the following a summary of the first part is presented: Let Y be a topological linear space. For a set \(A\subset Y\) the notions \(Sup^ s\), \(Inf^ w\), \(Max^ s\) and \(Min^ w\) are defined. Let E be a nonempty set and \((\Lambda_*,\Lambda^*)\) an ordered interval from Y and let \(\{P,Q_ y\); \(y\in (\Lambda_*,\Lambda^*)\}\) be a system of subsets belonging to E so that: For all \(y',y''\in (\Lambda_*,\Lambda^*):\) \(y'\leq y''\Rightarrow Q_{y''}\subset Q_{y'}\). We define \(Q=\cup_{y\in (\Lambda_*,\Lambda^*)}Q_ y\), \(P_ 0=P\cap Q\). For a set \(E^*\in \exp E\) we define the sets \(P^*=\{F\in E^*|\) \(P\subset F\}\) and \(Q^*_ y=\{F\in E^*|\) \(Q_{y'}\cap F=\emptyset\), \(\forall y'\geq y\}\). We also define \(Q^*=\cap_{y\in (\Lambda_*,\Lambda^*)}Q_ y\), \(P^*_ 0=P^*\cap Q^*\), \(\mu (a)=\{y\in (\Lambda_*,\Lambda^*)|\) \(a\in Q_ y\}\), \(\nu (F)=\{y\in (\Lambda_*,\Lambda^*)|\) \(F\in Q^*_ y\}\) and consider the following pairs of optimization problems: \[ \text{''find}\quad Sup^{s(w)}\cup_{a\in P_ 0}\mu (a)=S_ p^{s(w)}\text{''\quad and\quad ''find}\quad Inf^{s(w)}\cup_{F\in P_ 0^*}\nu (F)=I_ d^{s(w)}\text{'',} \]
\[ \text{ ''find}\quad Max^{s(w)}\cup_{a\in P_ 0}\mu (a)=M_ p^{s(w)}\text{''\quad and\quad ''find}\quad Min^{s(w)}\cup_{F\in P^*_ 0}\nu (F)=M_ d^{s(w)}\text{''.} \] The author establishes duality principles for the two pairs of dual problems.
Reviewer: S.Mititelu


49N15 Duality theory (optimization)
46A40 Ordered topological linear spaces, vector lattices
90C48 Programming in abstract spaces
90C30 Nonlinear programming
49J27 Existence theories for problems in abstract spaces
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