arXiv subjects

Petra Weidner

Publications and source records attributed to Petra Weidner.

7 recordsLinked to original sources

arXiv2018

Formulas for translative functions

In this report, we consider extended real-valued functions on some real vector space. Gerstewitz functionals are used to construct all translative functions. We derive formulas for translative functions which are lower semicontinuous, continuous or have sublevel sets that are given by linear inequalities. Each extended real-valued function is shown to be the restriction of some translative function to a hyperspace. Continuity of an arbitrary extended real-valued function is characterized by its epigraph. Moreover, we study the directional closedness of sets as a base for the presented results.

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arXiv2018

A Unified Approach to Extended Real-Valued Functions

Extended real-valued functions are often used in optimization theory, but in different ways for infimum problems and for supremum problems. We present an approach to extended real-valued functions that works for all types of problems and into which results of convex analysis can be embedded. Our approach preserves continuity and the Chebyshev norm when extending a functional to the entire space. The basic idea also works for other image spaces. Moreover, we illustrate that extended real-valued functions have to be handled in another way than real-valued functions and characterize semicontinuity, convexity, linearity and related properties of such functions.

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arXiv2017

Scalarization in vector optimization by functions with uniform sublevel sets

In this paper, vector optimization is considered in the framework of decision making and optimization in general spaces. Interdependencies between domination structures in decision making and domination sets in vector optimization are given. We prove some basic properties of efficient and of weakly efficient points in vector optimization. Sufficient conditions for solutions to vector optimization problems are shown using minimal solutions of functionals. We focus on the scalarization by functions with uniform sublevel sets, which also delivers necessary conditions for efficiency and weak efficiency. The functions with uniform sublevel sets may be, e.g., continuous or even Lipschitz continuous, convex, strictly quasiconcave or sublinear. They can coincide with an order unit norm on a subset of the space.

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arXiv2017

Functions with uniform sublevel sets and scalarization in linear spaces

Functions with uniform sublevel sets can represent orders, preference relations or other binary relations and thus turn out to be a tool for scalarization that can be used in multicriteria optimization, decision theory, mathematical finance, production theory and operator theory. Sets which are not necessarily convex can be separated by functions with uniform sublevel sets. This report focuses on properties of real-valued and extended-real-valued functions with uniform sublevel sets which are defined on a linear space without assuming topological properties. The functions may be convex or sublinear. They can coincide with a Minkowski functional or with an order unit norm on a subset of the space. The considered functionals are applied to the scalarization of vector optimization problems. These vector optimization problems refer to arbitrary domination sets. The consideration of such sets is motivated by their relationship to domination structures in decision making.

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arXiv2017

Proper efficiency and cone efficiency

In this report, two general concepts for proper efficiency in vector optimization are studied. Properly efficient elements can be defined as minimizers of functionals with certain monotonicity properties or as weakly efficient elements with respect to sets that contain the domination set. Interdependencies between both concepts are proved in topological vector spaces by means of Gerstewitz functionals. The investigation includes proper efficiency notions introduced by Henig and by Nehse and Iwanow. In contrary to Henig's notion, proper efficiency by Nehse and Iwanow is defined as efficiency with respect to certain convex sets which are not necessarily cones. For the finite-dimensional case, we turn to Geoffrion's proper efficiency as a special case of Henig's proper efficiency. It is characterized as efficiency with regard to subclasses of the set of polyhedral cones. Conditions for the existence of Geoffrion's properly efficient points are proved. For closed feasible point sets, Geoffrion's properly efficient point set is empty or coincides with that of Nehse and Iwanow. Properly efficient elements by Nehse and Iwanow are the minimizers of continuous convex functionals with certain monotonicity properties. Henig's proper efficiency can be described by means of minimizers of continuous sublinear functionals with certain monotonicity properties.

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arXiv2017

Minimizers of Gerstewitz functionals

Scalarization in vector optimization is essentially based on the minimization of Gerstewitz functionals. In this paper, the minimizer sets of Gerstewitz functionals are investigated. Conditions are given under which such a set is nonempty and compact. Interdependencies between solutions of problems with different parameters or with different feasible point sets are shown. Consequences for the parameter control in scalarization methods are derived. It is pointed out that the minimization of Gerstewitz functionals is equivalent to an optimization problem which generalizes the scalarization by Pascoletti and Serafini.

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arXiv2016

Functions with uniform level sets

Functions with uniform level sets can represent orders, preference relations or other binary relations and thus turn out to be a tool for scalarization that can be used, e.g., in multicriteria optimization, decision theory, mathematical finance, production theory and operator theory. Sets which are not necessarily convex can be separated by functions with uniform level sets. This has a deep impact on functional analysis, where many proofs require separation theorems. This report focuses on properties of real-valued and extended-real-valued functions with uniform level sets which are defined on a topological vector space. This includes the extension of aspects and results given in an earlier paper by Gerth (now Tammer) and Weidner. The functions may be, e.g., continuous, convex, strictly quasiconcave or sublinear. They can coincide with a Minkowski functional or with an order unit norm on a subset of the space. As a side result, we show that the core of a closed pointed convex cone is its interior in an appropriate norm topology.

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