A unifying approach to robust convex infinite optimization duality
- Authors: Dinh, Nguyen , Goberna, Miguel , López, Marco , Volle, Michel
- Date: 2017
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 174, no. 3 (2017), p. 650-685
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: This paper considers an uncertain convex optimization problem, posed in a locally convex decision space with an arbitrary number of uncertain constraints. To this problem, where the uncertainty only affects the constraints, we associate a robust (pessimistic) counterpart and several dual problems. The paper provides corresponding dual variational principles for the robust counterpart in terms of the closed convexity of different associated cones.
Characterizations of robust and stable duality for linearly perturbed uncertain optimization problems
- Authors: Dinh, Nguyen , Goberna, Miguel , López, Marco , Volle, Michel
- Date: 2020
- Type: Text , Conference paper
- Relation: Jonathan Borwein Commemorative Conference, JBCC 2017 Vol. 313, p. 43-74
- Relation: http://purl.org/au-research/grants/arc/DP180100602
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- Description: We introduce a robust optimization model consisting in a family of perturbation functions giving rise to certain pairs of dual optimization problems in which the dual variable depends on the uncertainty parameter. The interest of our approach is illustrated by some examples, including uncertain conic optimization and infinite optimization via discretization. The main results characterize desirable robust duality relations (as robust zero-duality gap) by formulas involving the epsilon-minima or the epsilon-subdifferentials of the objective function. The two extreme cases, namely, the usual perturbational duality (without uncertainty), and the duality for the supremum of functions (duality parameter vanishing) are analyzed in detail. © Springer Nature Switzerland AG 2020.
Convexity and closedness in stable robust duality
- Authors: Dinh, Nguyen , Goberna, Miguel , López, Marco , Volle, Michel
- Date: 2019
- Type: Text , Journal article
- Relation: Optimization Letters Vol. 13, no. 2 (2019), p. 325-339
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: The paper deals with optimization problems with uncertain constraints and linear perturbations of the objective function, which are associated with given families of perturbation functions whose dual variable depends on the uncertainty parameters. More in detail, the paper provides characterizations of stable strong robust duality and stable robust duality under convexity and closedness assumptions. The paper also reviews the classical Fenchel duality of the sum of two functions by considering a suitable family of perturbation functions.
Farkas-type results for vector-valued functions with applications
- Authors: Dinh, Nguyen , Goberna, Miguel , López, Marco , Mo, T. H.
- Date: 2017
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 173, no. 2 (2017), p. 357-390
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- Description: 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 defined 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.
From the Farkas lemma to the Hahn-Banach theorem
- Authors: Dinh, Nguyen , Goberna, Miguel , López, Marco , Mo, T. H.
- Date: 2014
- Type: Text , Journal article
- Relation: SIAM Journal on Optimization Vol. 24, no. 2 (2014), p. 678-701
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- Description: This paper provides new versions of the Farkas lemma characterizing those inequalities of the form f(x) ≥ 0 which are consequences of a composite convex inequality (S ° g)(x) ≤ 0 on a closed convex subset of a given locally convex topological vector space X, where f is a proper lower semicontinuous convex function defined on X, S is an extended sublinear function, and g is a vector-valued S-convex function. In parallel, associated versions of a stable Farkas lemma, considering arbitrary linear perturbations of f, are also given. These new versions of the Farkas lemma, and their corresponding stable forms, are established under the weakest constraint qualification conditions (the so-called closedness conditions), and they are actually equivalent to each other, as well as quivalent to an extended version of the so-called Hahn-Banach-Lagrange theorem, and its stable version, correspondingly. It is shown that any of them implies analytic and algebraic versions of the Hahn-Banach theorem and the Mazur-Orlicz theorem for extended sublinear functions.
New Farkas-type results for vector-valued functions : A non-abstract approach
- Authors: Dinh, Nguyen , Goberna, Miguel , Long, Dang , Lopez-Cerda, Marco
- Date: 2019
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 182, no. 1 (2019), p. 4-29
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- Description: This paper provides new Farkas-type results characterizing the inclusion of a given set, called contained set, into a second given set, called container set, both of them are subsets of some locally convex space, called decision space. The contained and the container sets are described here by means of vector functions from the decision space to other two locally convex spaces which are equipped with the partial ordering associated with given convex cones. These new Farkas lemmas are obtained via the complete characterization of the conic epigraphs of certain conjugate mappings which constitute the core of our approach. In contrast with a previous paper of three of the authors (Dinh et al. in J Optim Theory Appl 173:357-390, 2017), the aimed characterizations of the containment are expressed here in terms of the data.
Relaxed lagrangian duality in convex infinite optimization : reducibility and strong duality
- Authors: Dinh, Nguyen , Goberna, Miguel , López-Cerdá, Marco , Volle, Michel
- Date: 2023
- Type: Text , Journal article
- Relation: Optimization Vol. 72, no. 1 (2023), p. 189-214
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- Description: We associate with each convex optimization problem, posed on some locally convex space, with infinitely many constraints indexed by the set T, and a given non-empty family (Formula presented.) of finite subsets of T, a suitable Lagrangian-Haar dual problem. We obtain necessary and sufficient conditions for (Formula presented.) -reducibility, that is, equivalence to some subproblem obtained by replacing the whole index set T by some element of (Formula presented.). Special attention is addressed to linear optimization, infinite and semi-infinite, and to convex problems with a countable family of constraints. Results on zero (Formula presented.) -duality gap and on (Formula presented.) -(stable) strong duality are provided. Examples are given along the paper to illustrate the meaning of the results. © 2022 Informa UK Limited, trading as Taylor & Francis Group.
Relaxed Lagrangian duality in convex infinite optimization: Reverse strong duality and optimality
- Authors: Dinh, Nguyen , Goberna, Miguel , Lopez, Marco , Volle, Michel
- Date: 2021
- Type: Text , Journal article
- Relation: Journal of Applied and Numerical Optimization Vol. , no. (2021), p.
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- Description: We associate with each convex optimization problem posed on some locally convex space with an infinite index set T, and a given non-empty family H formed by finite subsets of T, a suitable Lagrangian-Haar dual problem. We provide reverse H-strong duality theorems, H-Farkas type lemmas and optimality theorems. Special attention is addressed to infinite and semi-infinite linear optimization problems.