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Scalarization of vector optimization problems and properties of the positive cone in normed vector lattices
Abstract
A useful technique in identifying efficient (maximal) points of a partially ordered set A is to appropriately choose a scalar-valued (i.e. real-valued) function f so that any solution of the real-valued optimization problem max$\{$f(a): a $\in$ A $\}$ is an efficient point of A. This dissertation considers the setting where the set A is a subset of a partially ordered topological vector space ${\cal Y}$ and where f is a continuous linear functional on ${\cal Y}$ which is strictly positive on the ordering cone. If a$\sb0\in$ A and if there exists some strictly positive f $\in {\cal Y}$* such that f(a$\sb0$ = max$\{$f(a): a $\in$ A$\}$, then the point a$\sb0$ is called a positive proper efficient point of A. A geometric characterization of positive proper efficient points is given; and it is shown that, under appropriate conditions, the set of positive proper efficient points is dense in the set of all efficient points. This density result is shown to be applicable in the normed vector lattices C (a,b), c, c$\sb0$, $\ell\sp{\rm p}$ and L$\sp{\rm p}$ for 1 $\leq$ p $\leq \infty$; and it is shown that similar density results given by previous authors are not applicable in these spaces. This dissertation also develops a scalarization and corresponding density result when the set A is a subset of a dual space. In this setting, the scalarizing linear functional is restricted to be a positive element from the predual. An investigation of the topological properties of the positive cones in partially ordered normed spaces shows that results which assume the cone possesses a weak-compact or even a bounded base exclude applications in most infinite dimensional normed vector lattices. Such assumptions on the base are common in the literature.
Subject Area
Mathematics|Operations research
Recommended Citation
Gallagher, Richard Joseph, "Scalarization of vector optimization problems and properties of the positive cone in normed vector lattices" (1988). ETD collection for University of Nebraska-Lincoln. AAI8824928.
https://digitalcommons.unl.edu/dissertations/AAI8824928