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A note on primal-dual stability in infinite linear programming

Abstract

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.

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Acknowledgements

This research was partially supported by PGC2018-097960-B-C22 of the Ministerio de Ciencia, Innovación y Universidades (MCIU), the Agencia Estatal de Investigación (AEI), and the European Regional Development Fund (ERDF); by the Australian Research Council, Project DP180100602; by CONICET, Argentina, Res D N\(^{\circ }\) 4198/17; and by Universidad Nacional de Cuyo, Secretaría de Investigación, Internacionales y Posgrado (SIIP), Res. 3922/19-R, Cod.06/D227, Argentina.

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Correspondence to Miguel A. Goberna.

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Goberna, M.A., López, M.A., Ridolfi, A.B. et al. A note on primal-dual stability in infinite linear programming. Optim Lett (2020). https://doi.org/10.1007/s11590-020-01549-4

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Keywords

  • Linear programming
  • Infinite dimensions
  • Primal-dual stability
  • Consistency
  • Inconsistency