Duality in Vector Optimization

Boţ, Radu Ioan; Grad, Sorin‐Mihai; Wanka, Gert

doi:10.1007/978-3-642-02886-1

Cited by 181 publications

(93 citation statements)

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“…Assume now that WSup M 6 = f+1 Y g. Applying (2.11) to the set WSup M (note that 1 Y = 2 WSup M ) we get and dropping 1 Y and +1 Y in both sides we get (2.16), together with the conclusion that the three sets in the right-hand side are disjoint (see again Proposition 7.4.1(d) in [3]). …”

Section: Preliminariesmentioning

confidence: 88%

“…Now we extend the above concepts to vector-valued functions as it is done in [3] and [25]. Let Y be a second lcHtvs, with origin 0 Y and topological dual space Y .…”

Section: Preliminariesmentioning

confidence: 99%

“…Conjugate maps and Farkas-type results are crucial in any duality theory, and this is why they constitute the main tool and the main objective, respectively, of our research. The state of the art in vector optimization is described, e.g., in [3], [16], [18], [23], and references therein.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Farkas-Type Results for Vector-Valued Functions with Applications

Dinh

Goberna

López

et al. 2017

J Optim Theory Appl

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The main purpose of this paper consists of providing characterizations of the inclusion of the solution set of a given conic system posed in a real locally convex topological space into a variety of subsets of the same space de…ned by means of vector-valued functions. These Farkas-type results are used to derive characterizations of the weak solutions of vector optimization problems (including multiobjective and scalar ones), vector variational inequalities, and vector equilibrium problems.

show abstract

Section: Preliminariesmentioning

confidence: 88%

“…Now we extend the above concepts to vector-valued functions as it is done in [3] and [25]. Let Y be a second lcHtvs, with origin 0 Y and topological dual space Y .…”

Section: Preliminariesmentioning

confidence: 99%

See 1 more Smart Citation

Farkas-Type Results for Vector-Valued Functions with Applications

Dinh

Goberna

López

et al. 2017

J Optim Theory Appl

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show abstract

“…Next properties of sets of weak suprema and weak minima were quoted from [8, Proposition 2.1], [7,Proposition 7.4.3], and [20,Proposition 2.4].…”

Section: It Follows From the Definition Of Wsupmentioning

confidence: 99%

Sectional Convexity of Epigraphs of Conjugate Mappings with Applications to Robust Vector Duality

Dinh

Long

2019

Acta Math Vietnam

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This paper concerns the robust vector problemswhere X, Y, Z are locally convex Hausdorff topological vector spaces, K is a closed and convex cone in Y with nonempty interior, and S is a closed, convex cone in Z, U is an uncertainty set, F :It is well-known that the epigraph of the conjugate mapping (F + I A ) * , in general, is not a convex set. We show that, however, it is "k-sectionally convex" in the sense that each section form by the intersection of epi(F + I A ) * and any translation of a "specific k-direction-subspace" is a convex subset, for any k taking from int K.The key results of the paper are the representations of the epigraph of the conjugate mapping (F + I A ) * via the closure of the k-sectionally convex hull of a union of epigraphs of conjugate mappings of mappings from a family involving the data of the problem (RVP). The results are then given rise to stable robust vector/convex vector Farkas lemmas which, in turn, are used to establish new results on robust strong stable duality results for (RVP). It is shown at the end of the paper that, when specifying the result to some concrete classes of scalar robust problems (i.e., when Y = R), our results cover and extend several corresponding known ones in the literature.

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“…An insufficient number of Pareto optimal points will be found if the tournament size is too small, and premature convergence may result if the tournament size is too large [7] [10].…”

Section: Introduction and Backgroundsmentioning

confidence: 99%

Three-Objective Programming with Continuous Variable Genetic Algorithm

Fita¹

2014

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The subject area of multiobjective optimization deals with the investigation of optimization problems that possess more than one objective function. Usually, there does not exist a single solution that optimizes all functions simultaneously; quite the contrary, we have solution set that is called nondominated set and elements of this set are usually infinite. It is from this set decision made by taking elements of nondominated set as alternatives, which is given by analysts. Since it is important for the decision maker to obtain as much information as possible about this set, our research objective is to determine a well-defined and meaningful approximation of the solution set for linear and nonlinear three objective optimization problems. In this paper a continuous variable genetic algorithm is used to find approximate near optimal solution set. Objective functions are considered as fitness function without modification. Initial solution was generated within box constraint and solutions will be kept in feasible region during mutation and recombination.

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Duality in Vector Optimization

Cited by 181 publications

References 0 publications

Farkas-Type Results for Vector-Valued Functions with Applications

Farkas-Type Results for Vector-Valued Functions with Applications

Sectional Convexity of Epigraphs of Conjugate Mappings with Applications to Robust Vector Duality

Three-Objective Programming with Continuous Variable Genetic Algorithm

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