Scalarization and Separation by Translation Invariant Functions

Tammer, Christiane; Weidner, Petra

doi:10.1007/978-3-030-44723-6

Cited by 22 publications

(20 citation statements)

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“…In this Section we consider two examples of scalarization methods that are based respectively on the Gerstewitz and the oriented distance functions. Similar approaches to scalarization are widely used in vector optimization (see, e.g., [20] or [29]). Both the approaches are placed within the theoretical framework outlined in Sect.…”

Section: Some Special Casesmentioning

confidence: 99%

“…The function φ e,A has many applications in the context of nonlinear analysis and it was used as a scalarizing function to obtain optimality conditions in vector optimization problems (see [20] and [29] and the references therein). Scalarizations through the Gerstewitz function have also been applied to consider robustness in set-valued frameworks (see e.g.…”

Section: Formulation Of Problem Rc − S ' − Vpmentioning

confidence: 99%

“…6, two applications using the Gerstewitz and the oriented distance approach to scalarization are considered in our framework. Such functions are widely used in scalarizations of vector optimization problems (see, e.g., [29] or [20] and the references therein). Moreover, in the special case of multiobjective optimization, several well known scalarization methods, such as Pascoletti-Serafini, -constraint or Chebyshev scalarization, can be reduced to the use of the Gerstewitz scalarizing function (see e.g.…”

Section: Introductionmentioning

confidence: 99%

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Scalarization and robustness in uncertain vector optimization problems: a non componentwise approach

2022

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The robust optimization approach can be used to tackle uncertain vector problems by considering worst case scenarios. In this context, notions of robust efficient solutions which are coherent with a set-valued minimization process have been introduced in literature in order to avoid unduly pessimistic attitudes (see e.g. Ehrgott et al. in Eur. J. Oper. Res. 239(1), 17–31, 2014). We address the question whether scalarization and robustification can be commuted in a non componentwise framework. We prove that the commutation of the two approaches is ensured under appropriate assumptions. To this purpose, we identify a class of scalarization processes that ensure necessary and sufficient robust optimality conditions through the direct scalarization of the uncertain vector optimization problem, without explicitly passing through the set-valued formulation of the problem.

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Section: Some Special Casesmentioning

confidence: 99%

Section: Formulation Of Problem Rc − S ' − Vpmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Scalarization and robustness in uncertain vector optimization problems: a non componentwise approach

2022

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“…One idea could be to involve further generalized convexity concepts, and another idea could be to consider nonlinear scalarization techniques (for instance based on the so-called Gerstewitz function, see [35,Sec. 5.2] and [42]). In this context, Arrow-Barankin-Blackwell type theorems for our considered Henig-type proper efficiency concepts are of interest (see, e.g., [22]).…”

Section: Remark 55 Bymentioning

confidence: 99%

Vector Optimization w.r.t. Relatively Solid Convex Cones in Real Linear Spaces

Günther

Khazayel

Tammer

2021

J Optim Theory Appl

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In vector optimization, it is of increasing interest to study problems where the image space (a real linear space) is preordered by a not necessarily solid (and not necessarily pointed) convex cone. It is well-known that there are many examples where the ordering cone of the image space has an empty (topological/algebraic) interior, for instance in optimal control, approximation theory, duality theory. Our aim is to consider Pareto-type solution concepts for such vector optimization problems based on the intrinsic core notion (a well-known generalized interiority notion). We propose a new Henig-type proper efficiency concept based on generalized dilating cones which are relatively solid (i.e., their intrinsic cores are nonempty). Using linear functionals from the dual cone of the ordering cone, we are able to characterize the sets of (weakly, properly) efficient solutions under certain generalized convexity assumptions. Toward this end, we employ separation theorems that are working in the considered setting.

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“…The following properties will be useful and are collected from [15]. Lemma 2.1 (see Tammer and Weidner [15,Lemma 2.3.24]). Let X be a vector space over R, A ⊆ X and K ∈ X \{0}.…”

Section: Preliminariesmentioning

confidence: 99%

A new view on risk measures associated with acceptance sets

Marohn¹,

Tammer

2021

J. Appl. Numer. Optim.

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In this paper, we study the properties of certain risk measures associated with acceptance sets. These sets describe the regulatory preconditions that have to be fulfilled by financial institutions to pass a given acceptance test. If the financial position of an institution is not acceptable, the decision maker has to raise new capital and invest it into a basket of so called eligible assets to change the current position such that the resulting one corresponds with an element of the acceptance set. Risk measures have been widely studied in the literature. The risk measure that is considered here determines the minimal costs of making a financial position acceptable. In the literature, monetary risk measures are often defined as translation invariant functions and, thus, there is an equivalent formulation as the Gerstewitz-Functional, which is an useful tool for separation and scalarization in multiobjective optimization in the non-convex case. In our paper, we study properties of the sublevel sets, strict sublevel sets, and level lines of a risk measure defined on a linear space. Furthermore, we discuss the finiteness of the risk measure and relax the closedness assumptions.

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Scalarization and Separation by Translation Invariant Functions

Cited by 22 publications

References 0 publications

Scalarization and robustness in uncertain vector optimization problems: a non componentwise approach

Scalarization and robustness in uncertain vector optimization problems: a non componentwise approach

Vector Optimization w.r.t. Relatively Solid Convex Cones in Real Linear Spaces

A new view on risk measures associated with acceptance sets

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