Cone-valued maps in optimization
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TL;DR: In this paper, it was shown that classical concepts for set-valued maps such as cone-convexity or monotonicity are not appropriate for characterizing conevalued maps.
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Abstract: Cone-valued maps are special set-valued maps where the image sets are cones. Such maps play an important role in optimization, for instance in optimality conditions or in the context of Bishop–Phelps cones. In vector optimization with variable ordering structures, they have recently attracted even more interest. We show that classical concepts for set-valued maps as cone-convexity or monotonicity are not appropriate for characterizing cone-valued maps. For instance, every convex or monotone cone-valued map is a constant map. Similar results hold for cone-convexity, sublinearity, upper semicontinuity or the local Lipschitz property. Therefore, we also propose new concepts for cone-valued maps.
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Citations
SET-Valued Analysis
Zdzisław Denkowski,Stanisław Migórski,Nikolas S. Papageorgiou +2 more
- 01 Jan 2003
TL;DR: “Multivalued Analysis” is the theory of set-valued maps (called multifonctions) and has important applications in many different areas and there is no doubt that a modern treatise on “Nonlinear functional analysis” can not afford the luxury of ignoring multivalued analysis.
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Numerical Procedures in Multiobjective Optimization with Variable Ordering Structures
TL;DR: This paper proposes numerical approaches for solving multiobjective optimization problems with a variable ordering structure using scalarization functionals and the Jahn–Graef–Younes method, which is adapted to allow the determination of all optimal elements with a reduced effort compared to a pairwise comparison.
Properly optimal elements in vector optimization with variable ordering structures
TL;DR: Proper optimality concepts in vector optimization with variable ordering structures are introduced for the first time and characterization results via scalarizations are given.
Variable Ordering Structures
Gabriele Eichfelder
- 01 Jan 2014
TL;DR: This chapter introduces variable ordering structures and examines their properties and focuses on special ordering maps where the images are Bishop-Phelps cones and discuss several applications.
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References
SET-Valued Analysis
Zdzisław Denkowski,Stanisław Migórski,Nikolas S. Papageorgiou +2 more
- 01 Jan 2003
TL;DR: “Multivalued Analysis” is the theory of set-valued maps (called multifonctions) and has important applications in many different areas and there is no doubt that a modern treatise on “Nonlinear functional analysis” can not afford the luxury of ignoring multivalued analysis.
1.5K
Nonconvex separation theorems and some applications in vector optimization
C. Gerth,Petra Weidner +1 more
TL;DR: In this paper, separation theorems for an arbitrary set and a not necessarily convex set in a linear topological space are proved and applied to vector optimization and scalarization results for weakly efficient points and properly efficient points are deduced.
386
•Book
Introduction to the Theory of Nonlinear Optimization
Johannes Jahn
- 01 Sep 1994
TL;DR: In this paper, the authors present an introductory text to optimization theory in normed spaces and cover all areas of nonlinear optimization, with particular emphasis on the application to problems in the calculus of variations, approximation and optimal control theory.
376
•Book
Variable Ordering Structures in Vector Optimization
Gabriele Eichfelder
- 01 May 2014
TL;DR: In vector optimization one assumes in general that a partial ordering is given by some nontrivial convex cone K in the considered space Y and a candidate element is called a minimal (or nondominated-like) element if it is not dominated by any other reference element w.r.t. the cone of the candidate element.
Optimal Elements in Vector Optimization with a Variable Ordering Structure
TL;DR: Two different optimality concepts for vector optimization problems with a variable ordering structure, called minimal elements and nondominated elements, are shown to be connected by duality properties.
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