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.
New glimpses on convex infinite optimization duality
- Authors: Goberna, Miguel , López, Marco , Volle, Michel
- Date: 2015
- Type: Text , Journal article
- Relation: Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas Vol. 109, no. 2 (2015), p. 431-450
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- Description: Given a convex optimization problem (P) in a locally convex topological vector space X with an arbitrary number of constraints, we consider three possible dual problems of (P), namely, the usual Lagrangian dual (D), the perturbational dual (Q), and the surrogate dual (Δ), the last one recently introduced in a previous paper of the authors (Goberna et al., J Convex Anal 21(4), 2014). As shown by simple examples, these dual problems may be all different. This paper provides conditions ensuring that (P)=max(D), (P)=max(Q), and (P)=max(Δ) (dual equality and existence of dual optimal solutions) in terms of the so-called closedness regarding to a set. Sufficient conditions guaranteeing (P)=sup(Q) (dual equality and existence of primal optimal solutions) are also provided, for the nominal problems and also for their perturbational relatives. The particular cases of convex semi-infinite optimization problems (in which either the number of constraints or the dimension of X, but not both, is finite) and linear infinite optimization problems are analyzed. Finally, some applications to the feasibility of convex inequality systems are described. © 2014, Springer-Verlag Italia.
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.
Subdifferential of the closed convex hull of a function and integration with nonconvex data in general normed spaces
- Authors: López, Marco , Volle, Michel
- Date: 2012
- Type: Text , Journal article
- Relation: Journal of Mathematical Analysis and Applications Vol. 390, no. 1 (2012), p. 307-312
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: In this paper we approach the study of the subdifferential of the closed convex hull of a function and the related integration problem without the usual assumption of epi-pointedness. The key tool is, as in Hiriart-Urruty et al. (2011) [7], the concept of ε-subdifferential. Some other assumptions which are standard in the literature are also removed.