Robust optimization revisited via robust vector Farkas lemmas

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

doi:10.1080/02331934.2016.1152272

Cited by 13 publications

(19 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Some of these results cover or extend some in [11], [20]. In Section 5, from the duality results in Section 4, we derive variants of stable robust Farkas lemmas for linear infinite systems with uncertainty which cover the ones in [12], [16] while the others are new. In Section 6, as an extension/application of the approach, we get robust strong duality results for linear problems with sub-affine constraints.…”

Section: Introductionmentioning

confidence: 92%

“…Observe that N 3 is M ℓf in [12], and N 3 and N 6 were introduced in [20] and known as "robust moment cone" and "characteristic cone", respectively.…”

Section: Robust Stable Strong Duality For (Rlip C )mentioning

confidence: 99%

“…The robust linear infinite problems of the model (RLIP c ) together with their duality were considered in several works in the literature such as, [6], [12], [16], [20], [23]. There are variants of duality results for robust convex problems (see [4], [5], [14], [15], [11], [18], [16], [22], [24] and references therein), and also for robust vector optimization/multiobjective problems (see, e.g., [7], [12], [13], [21]). In the mentioned papers, results for robust strong duality are established for classes of problems from linear to convex, non-convex, and vector problems, under various constraint qualification conditions (or qualification conditions).…”

Section: Introductionmentioning

confidence: 99%

“…The model of Problem (RLP c ) was considered in[12] where some characterizations of its solutions were proposed.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Duality for Robust Linear Infinite Programming Problems Revisited

2020

Self Cite

View full text Add to dashboard Cite

In this paper, we consider the robust linear infinite programming problem (RLIP c ) defined bywhere X is a locally convex Hausdorff topological vector space, T is an arbitrary (possible infinite) index set, c ∈ X * , and U t ⊂ X * × R, t ∈ T are uncertainty sets.We propose an approach to duality for the robust linear problems with convex constraints (RP c ) and establish corresponding robust strong duality and also, stable robust strong duality, i.e., robust strong duality holds "uniformly" with all c ∈ X * . With the different choices/ways of setting/arranging data from (RLIP c ), one gets back to the model (RP c ) and the (stable) robust strong duality for (RP c ) applies. By such a way, nine versions of dual problems for (RLIP c ) are proposed. Necessary and sufficient conditions for stable robust strong duality of these pairs of primal-dual problems are given, for which some cover several known results in the literature while the others, due to the best knowledge of the authors, are new. Moreover, as by-products, we obtained from the robust strong duality for variants pairs of primal-dual problems, several robust Farkas-type results for linear infinite systems with uncertainty. Lastly, as extensions/applications, we extend/apply the results obtained to robust linear problems with sub-affine constraints, and to linear infinite problems (i.e., (RLIP c ) with the absence of uncertainty). It is worth noticing even for these cases, we are able to derive new results on (robust/stable robust) duality for the mentioned classes of problems and new robust Farkas-type results for sub-linear systems, and also for linear infinite systems in the absence of uncertainty.Key words: Linear infinite programming problems, robust linear infinite problems, stable robust strong duality for robust linear infinite problems, Farkas-type results for infinite linear systems with uncertainty, Farkas-type results for sub-affine systems with uncertainty.Mathematics Subject Classification: 39B62, 49J52, 46N10, 90C31, 90C25.

show abstract

Section: Introductionmentioning

confidence: 92%

“…Observe that N 3 is M ℓf in [12], and N 3 and N 6 were introduced in [20] and known as "robust moment cone" and "characteristic cone", respectively.…”

Section: Robust Stable Strong Duality For (Rlip C )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The model of Problem (RLP c ) was considered in[12] where some characterizations of its solutions were proposed.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Duality for Robust Linear Infinite Programming Problems Revisited

2020

Self Cite

View full text Add to dashboard Cite

show abstract

“…Concerning the problem(RVP), we recall the qualifying set [9] and the weak qualifying set [10] defined respectively as follows:…”

Section: Epigraphs Of Conjugate Mappings Via Sectionally Convex Hullsmentioning

confidence: 99%

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

Dinh

Long

2019

Acta Math Vietnam

View full text Add to dashboard Cite

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.

show abstract

Complete Characterizations of Robust Strong Duality for Robust Vector Optimization Problems

Dinh

Long

2018

Vietnam J. Math.

View full text Add to dashboard Cite

Robust optimization revisited via robust vector Farkas lemmas

Cited by 13 publications

References 23 publications

Duality for Robust Linear Infinite Programming Problems Revisited

Duality for Robust Linear Infinite Programming Problems Revisited

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

Complete Characterizations of Robust Strong Duality for Robust Vector Optimization Problems

Contact Info

Product

Resources

About