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.
Similar content being viewed by others
References
Alghamdi, M.A., Alotaibi, A., Combettes, P.L., Shahzad, N.: A primal-dual method of partial inverses for composite inclusions. Optim. Lett. 8, 2271–2284 (2014)
Anderson, E.J., Nash, P.: Linear Programming in Infinite-Dimensional Spaces. Wiley, New York (1987)
Bae, J., Rathinam, S.: A primal-dual approximation algorithm for a two depot heterogeneous traveling salesman problem. Optim. Lett. 10, 1269–1285 (2016)
Bank, B., Guddat, J., Klatte, D., Kummer, B., Tammer, K.: Nonlinear Parametric Optimization. Birkhäuser, Boston (1983)
Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2010)
Cánovas, M.J., López, M.A., Mordukhovich, B.S., Parra, J.: Variational analysis in semi-infinite and infinite programming. I. Stability of linear inequality systems of feasible solutions. SIAM J. Optim. 20, 1504–1526 (2009)
Daniilidis, A., Goberna, M.A., Lopez, M.A., Lucchetti, R.: Lower semicontinuity of the solution set mapping of linear systems relative to their domains. Set Valued Var. Anal. 21, 67–92 (2013)
Dantzig, G.B.: Linear Programming and Extensions. Princeton University Press, Princeton (1963)
Dinh, N., Goberna, M.A., López, M.A.: From linear to convex systems: consistency, Farkas’ lemma and applications. J. Convex Anal. 13, 279–290 (2006)
Dunford, N., Schwartz, J.T.: Linear Operators Part I: General Theory. Wiley, Chichester (1972)
Dür, M., Jargalsaikhan, B., Still, G.: Genericity results in linear conic programming—a tour d’horizon. Math. Oper. Res. 42, 77–94 (2017)
Fan, K.: Existence theorems and extreme solutions for inequalities concerning convex functions or linear transformations. Math. Z. 68, 205–217 (1957)
Goberna, M.A., López, M.A.: Linear Semi-infinite Optimization. Wiley, Chichester (1998)
Goberna, M.A., López, M.A.: Post-optimal Analysis in Linear Semi-infinite Optimization. Springer, New York (2014)
Goberna, M.A., López, M.A.: Recent contributions to linear semi-infinite optimization: an update. Ann. Oper. Res. 271, 237–278 (2018)
Goberna, M.A., López, M.A., Volle, M.: New glimpses on convex infinite optimization duality. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 109, 431–450 (2015)
Goberna, M.A., Ridolfi, A.B., Vera de Serio, V.N.: Stability of the duality gap in linear optimization. Set Valued Var. Anal. 25, 617–636 (2017)
Holmes, R.B.: Geometric Functional Analysis and Its Applications. Springer, New York (1975)
Li, B., Dam, H.H., Cantoni, A., Teo, K.L.: A primal-dual interior point method for optimal zero-forcing beamformer design under per-antenna power constraints. Optim. Lett. 8, 1829–1843 (2014)
López, M.A., Ridolfi, A.B., Vera de Serio, V.N.: On coderivatives and Lipschitzian properties of the dual pair in optimization. Nonlinear Anal. 75, 1461–1482 (2012)
Ochoa, P.D., Vera de Serio, V.N.: Stability of the primal-dual partition in linear semi-infinite programming. Optimization 61, 1449–1465 (2012)
Stefanov, S.M.: Well-posedness and primal-dual analysis for some convex separable optimization problems. Adv. Oper. Res. Art. ID 279030 (2013)
Vinh, N.T., Kim, D.S., Tam, N.N., Yen, N.D.: Duality gap function in infinite dimensional linear programming. J. Math. Anal. Appl. 437, 1–15 (2016)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Goberna, M.A., López, M.A., Ridolfi, A.B. et al. A note on primal-dual stability in infinite linear programming. Optim Lett 14, 2247–2263 (2020). https://doi.org/10.1007/s11590-020-01549-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-020-01549-4