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A CHARACTERIZATION OF PROPER MINIMAL POINTS AS SOLUTIONS OF SUBLINEAR OPTIMIZATION PROBLEMS
Abstract
We consider the problem of finding the set of proper minimal (proper efficient, proper Pareto optimal) points of a given subset of an infinite dimensional locally convex space with partial order induced by a convex cone. Different definitions for a proper minimal point of a set, not necessarily equivalent, have been proposed following Geoffrion's definition, in 1968, in Euclidean space. Borwein proposed a definition, in 1977, based on the concept of tangent cones, Benson, in 1978, provided another definition, and in 1980 Borwein introduced another notion for proper efficiency. This thesis provides the relationship between these notions of properness. With convexity assumptions, the density property of proper minimal points in the set of minimal points, i.e., each minimal point can be approximated by a proper minimal point, has been discussed by Hartely in Euclidean spaces in 1978. Borwein, in 1980, proved a density result, in the weak sense, in normed spaces. This thesis provides a density in reflexive normed spaces in the strong sense. The main results of the thesis are to provide a characterization for proper minimal points of a subset of a locally convex space as solutions for scalar optimization problems. It should be noted that, with convexity assumptions, such a problem has been solved on the basis of classical results of separation of two disjoint convex sets by a hyperplane (continuous linear functional). This thesis provides separation results for two cones, one of them is not necessarily convex, having only the vertex in common. These separation results are extensions of earlier results by Henig (in Euclidean spaces) and Jahn (in normed spaces). These separation results are the basis for the characterization results. This thesis also provides necessary and sufficient conditions for proper minimal solutions of constrained vector problems as solutions for scalar Lagrangian function, with special type of multipliers.
Subject Area
Mathematics
Recommended Citation
SALEH, OSSAMA A, "A CHARACTERIZATION OF PROPER MINIMAL POINTS AS SOLUTIONS OF SUBLINEAR OPTIMIZATION PROBLEMS" (1987). ETD collection for University of Nebraska-Lincoln. AAI8722420.
https://digitalcommons.unl.edu/dissertations/AAI8722420