Farkas-Type Results for Vector-Valued Functions with Applications

Dinh, N.; Goberna, Miguel Á.; López, Marco A.; Mo, Tran Hong

doi:10.1007/s10957-016-1055-2

Cited by 19 publications

(28 citation statements)

References 21 publications

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“…In Z • we adopt the same conventions as in (2.6). Moreover, we recall the cone of positive operators (see [1], [8]) and the cone of weak positive operators [11] respectively, as follows:…”

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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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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“…We conclude this section by recalling a result in [10] whose robust version will be used to prove the key result Theorem 3.1 below.…”

Section: Preliminariesmentioning

confidence: 99%

“…The following lemma is a robust counterpart of Proposition 2.2 in [10] and constitutes a useful technical instrument role in this paper. This lemma involves nonempty uncertainty sets V L(X; Y ) and W Y .…”

Section: Asymptotic Robust Vector Farkas Lemmasmentioning

confidence: 99%

“…Di¤erent concepts of solutions in vector optimization are discussed and analyzed in [1], [7], [8], [17], [18], [21] and [24], among others. In particular, [10] is focussed on the so-called weak minimum of f (A); which turns out to be WMin f (A) := f (A) (f (A) + int K) ; with int K denoting the interior of K; whenever f (x) 6 = 1 Y for all x 2 A and f is not identically +1 Y :…”

Section: Introductionmentioning

confidence: 99%

“…A Farkas-type result is called asymptotic whenever the characterization of A B involves the closure of certain sets, and it is called P A / P B whenever P A and P B are properties satis…ed by A and B (or by the functions involved in the systems describing them), e.g., the properties linear, a¢ ne, convex, and (possibly) non-convex are referred to the corresponding constraint functions while the adjective reverse stands for B being the inverse image by certain function of the complement of a convex set (in particular, X int K). In [10] we have provided Farkas-type results in which the system de…ning the container set B involves some vector-valued function. From such results, called vector Farkas lemmas, we have obtained characterizations of the weakly minimal elements of deterministic vector optimization problems (including scalar and multi-objective programs, for which the notion of weakly minimal element collapses to the notions of optimal solution and weakly e¢ cient solution, respectively).…”

Section: Introductionmentioning

confidence: 99%

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Robust optimization revisited via robust vector Farkas lemmas

et al. 2016

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This paper provides characterizations of the weakly minimal elements of vector optimization problems and the global minima of scalar optimization problems posed on locally convex spaces whose objective functions are deterministic while the uncertain constraints are treated under the robust (or risk-averse) approach, i.e., requiring the feasibility of the decisions to be taken for any possible scenario. To get these optimality conditions we provide Farkas-type results characterizing the inclusion of the robust feasible set into the solution set of some system involving the objective function and possibly uncertain parameters. In the particular case of scalar convex optimization problems, we characterize the optimality conditions in terms of the convexity and closedness of an associated set regarding a suitable point.

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Complete Characterizations of Robust Strong Duality for Robust Vector Optimization Problems

Dinh

Long

2018

Vietnam J. Math.

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Farkas-Type Results for Vector-Valued Functions with Applications

Cited by 19 publications

References 21 publications

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

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

Robust optimization revisited via robust vector Farkas lemmas

Complete Characterizations of Robust Strong Duality for Robust Vector Optimization Problems

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