A new penalization tool in scalar and vector optimizations

Apetrii, Marius; Durea, Marius; Strugariu, Radu

doi:10.1016/j.na.2014.04.022

Cited by 10 publications

(2 citation statements)

References 25 publications

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“…In [6], Ye extended the exact penalty principle to vectorial problems (Clarke-Ye type penalisation), see also Apetrii, Durea and Strugariu [7] where the directional minimal time function is used as penalisation function and the references therein for some extension and more details. Furthermore, a Clarke-Ye type approach to penalisation for vector optimisation problems is derived by Fukuda, Drummond and Raupp in [8], where a vector external penalty function ν : R n → R m ≥ (with ν continuous and ν(x) = 0 R m if and only if x belongs to the feasible set S) is used, i.e., the authors study a vector-valued penalisation approach f + α ν.…”

Section: Introductionmentioning

confidence: 99%

Vectorial penalisation in vector optimisation in real linear-topological spaces

2023

J. Nonlinear Var. Anal.

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The aim of this paper is to present a vectorial penalisation approach for vector optimisation problems in which the vector-valued objective function acts between real linear-topological spaces X and Y , where the image space Y is partially ordered by a pointed convex cone. In essence, the approach replaces the original constrained vector optimisation problem (with not necessarily convex feasible set) by two unconstrained vector optimisation problems, where in one of the two problems a penalisation term (function) with respect to the original feasible set is added to the vector objective function. To derive our main results, we use a generalised convexity (quasiconvexity) notion for vector functions in the sense of Jahn. Our results extend/generalise known results in the context of vectorial penalisation in multiobjective/vector optimisation. We put a special emphasis on the construction of appropriate penalisation functions for several popular classes of (vector) optimisation problems (e.g., semidefinite/copositive programming, second-order cone programming, optimisation in function spaces).

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Section: Introductionmentioning

confidence: 99%

Vectorial penalisation in vector optimisation in real linear-topological spaces

2023

J. Nonlinear Var. Anal.

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“…The negative polar of A is A − := {x * ∈ X * | x * (a) ≤ 0, ∀a ∈ A} . 2 The concepts under study Let K ⊂ Y be a proper (that is, K = {0}, K = Y ) convex cone (we do not suppose that K is pointed, in general). For such a cone, its positive dual cone is K + := {y * ∈ Y * | y * (y) ≥ 0, ∀y ∈ K} .…”

Section: Introduction and Notationmentioning

confidence: 99%

Directional Pareto Efficiency: Concepts and Optimality Conditions

Chelmuş

Durea

Florea

2019

J Optim Theory Appl

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We introduce and study a notion of directional Pareto minimality with respect to a set that generalizes the classical concept of Pareto efficiency. Then we give separate necessary and sufficient conditions for the newly introduced efficiency and several situations concerning the objective mapping and the constraints are considered. In order to investigate different cases, we adapt some well-known constructions of generalized differentiation and the connections with some recent directional regularities come naturally into play. As a consequence, several techniques from the study of genuine Pareto minima are considered in our specific situation.1 means of an inverse image of a cone through another set-valued map. For the study of this general case, we introduce an adapted tangent cone, along with several directional regularity properties of the involved maps, and this approach allows us to derive necessary optimality conditions that, in turn, generalize the prototype of Fermat Theorem at an endpoint of an interval. Furthermore, we present as well optimality conditions in terms of tangent limiting cones and coderivatives. Both on primal and dual spaces we have under consideration several situations concerning the objective and constraint mappings with their specific techniques of study, among which we mention generalized constraint qualification conditions, Gerstewitz scalarization, openness vs. minimality paradigm, Clarke penalization, extremal principle. Some results are dedicated to the sufficient optimality conditions under convexity assumptions. Finally, we consider as well the situation of minimality for sets and a brief discussion of this concept reveals the similarities and the differences with respect to the known situation of Pareto efficiency.The paper is organized as follows. First of all, we introduce the notation we use and then we present the concepts of directional minimality we study in this work. The definitions of these notions along with some comparisons and examples are the subjects of the second section. The main section of the paper is the third one, and it deals with optimality conditions for the above introduced concepts, being, in turn, divided into two subsections. Firstly, we derive optimality conditions using tangent cones and to this aim we adapt a classical concept of the Bouligand tangent cone and Bouligand derivative of a set-valued map. Using some directional metric regularities, we get several assertions concerning these objects and this allows us to present necessary optimality conditions for a wide range of situations going from problems governed by set-valued mappings having generalized inequalities constraints to fully smooth constrained problems. Secondly, we deal with optimality conditions using normal limiting cones and, again, we consider several types of problems. In this process of getting necessary optimality conditions we adapt several techniques from classical vector optimization. Moreover, some generalized convex cases are considered in order to obtain sufficient opti...

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Relationships between constrained and unconstrained multi-objective optimization and application in location theory

Günther

Tammer

2016

Math Meth Oper Res

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A new penalization tool in scalar and vector optimizations

Cited by 10 publications

References 25 publications

Vectorial penalisation in vector optimisation in real linear-topological spaces

Vectorial penalisation in vector optimisation in real linear-topological spaces

Directional Pareto Efficiency: Concepts and Optimality Conditions

Relationships between constrained and unconstrained multi-objective optimization and application in location theory

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