A note on primal-dual stability in infinite linear programming
- Authors: Goberna, Miguel , López, Marco , Ridolfi, Andrea , Vera de Serio, Virginia
- Date: 2020
- Type: Text , Journal article
- Relation: Optimization Letters Vol. 14, no. 8 (2020), p. 2247-2263
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- Description: In this note we analyze the simultaneous preservation of the consistency (and of the inconsistency) of linear programming problems posed in infinite dimensional Banach spaces, and their corresponding dual problems, under sufficiently small perturbations of the data. We consider seven different scenarios associated with the different possibilities of perturbations of the data (the objective functional, the constraint functionals, and the right hand-side function), i.e., which of them are known, and remain fixed, and which ones can be perturbed because of their uncertainty. The obtained results allow us to give sufficient and necessary conditions for the coincidence of the optimal values of both problems and for the stability of the duality gap under the same type of perturbations. There appear substantial differences with the finite dimensional case due to the distinct topological properties of cones in finite and infinite dimensional Banach spaces. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.
- Description: Funding details: Australian Research Council, ARC, DP180100602: http://purl.org/au-research/grants/arc/DP180100602
Comparative study of RPSALG algorithm for convex semi-infinite programming
- Authors: Auslender, Alfred , Ferrer, Albert , Goberna, Miguel , López, Marco
- Date: 2014
- Type: Text , Journal article
- Relation: Computational Optimization and Applications Vol. 60, no. 1 (2014), p. 59-87
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- Description: The Remez penalty and smoothing algorithm (RPSALG) is a unified framework for penalty and smoothing methods for solving min-max convex semi-infinite programing problems, whose convergence was analyzed in a previous paper of three of the authors. In this paper we consider a partial implementation of RPSALG for solving ordinary convex semi-infinite programming problems. Each iteration of RPSALG involves two types of auxiliary optimization problems: the first one consists of obtaining an approximate solution of some discretized convex problem, while the second one requires to solve a non-convex optimization problem involving the parametric constraints as objective function with the parameter as variable. In this paper we tackle the latter problem with a variant of the cutting angle method called ECAM, a global optimization procedure for solving Lipschitz programming problems. We implement different variants of RPSALG which are compared with the unique publicly available SIP solver, NSIPS, on a battery of test problems.
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: 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.