Abstract
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.
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Borwein, J.M., Burachik, R.S., Yao, L.: Conditions for zero duality gap in convex programming. J. Nonlinear Convex Anal. 15, 167–190 (2014)
Dinh, N., Goberna, M.A., López, M.A., Volle, M.: A unifying approach to robust convex infinite optimization duality. J. Optim. Theor. Appl. 174, 650–685 (2017)
Dinh, N., Goberna, M.A., López, M.A., Volle, M.: Characterizations of robust and stable duality for linearly perturbed uncertain optimization problems. http://arxiv.org/abs/1803.04673
Dinh, N., López, M.A., Volle, M.: Functional inequalities in the absence of convexity and lower semicontinuity with applications to optimization. SIAM J. Optim. 20, 2540–2559 (2010)
Dinh, N., Mo, T.H., Vallet, G.V., Volle, M.: A unified approach to robust Farkas-type results with applications to robust optimization problems. SIAM J. Optim. 27, 1075–1101 (2017)
Ernst, E., Volle, M.: Zero duality gap and attainment with possibly non-convex data. J. Convex Anal. 23, 615–629 (2016)
Fajardo, M.D., Vidal, J.: Stable strong Fenchel and Lagrange duality for evenly convex optimization problems. Optimization 65, 1675–1691 (2016)
Ghate, A.: Robust optimization in countably infinite linear programs. Optim. Lett. 10, 847–863 (2016)
Hantoute, A., López, M.A., Zălinescu, C.: Subdifferential calculus rules in convex analysis: a unifying approach via pointwise supremum functions. SIAM J. Optim. 19, 863–882 (2008)
Hiriart-Urruty, J.-B., Moussaoui, M., Seeger, A., Volle, M.: Subdifferential calculus without qualification conditions, using approximate subdifferentials: a survey. Nonlinear Anal. 24, 1727–1754 (1995)
Jeyakumar, V., Li, G.Y.: Stable zero duality gaps in convex programming: complete dual characterisations with applications to semidefinite programs. J. Math. Anal. Appl. 360, 156–167 (2009)
Lee, J., Jiao, L.: On quasi \(\varepsilon \)-solution for robust convex optimization problems. Optim. Lett. 11, 1609–1622 (2017)
Li, Ch., Fang, D., López, G., López, M.A.: Stable and total Fenchel duality for convex optimization problems in locally convex spaces. SIAM J. Optim. 20, 1032–1051 (2009)
Li, G.Y., Jeyakumar, V., Lee, G.M.: Robust conjugate duality for convex optimization under uncertainty with application to data classification. Nonlinear Anal. 74, 2327–2341 (2011)
Mordukhovich, B.S., Nam, N.M.: Extremality of convex sets with some applications. Optim. Lett. 11, 1201–1215 (2017)
Sun, X., Peng, Z.-Y., Guo, X-Le: Some characterizations of robust optimal solutions for uncertain convex optimization problems. Optim. Lett. 10, 1463–1478 (2016)
Acknowledgements
This research was supported by the Ministry of Economy and Competitiveness of Spain and the European Regional Development Fund (ERDF) of the European Commission, Project MTM2014-59179-C2-1-P, by the Australian Research Council, Project DP160100854, and by Vietnam National University - HCM city, Vietnam, project “Generalized scalar and vector Farkas-type results with applications to optimization theory”.
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Dinh, N., Goberna, M.A., López, M.A. et al. Convexity and closedness in stable robust duality. Optim Lett 13, 325–339 (2019). https://doi.org/10.1007/s11590-018-1311-5
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DOI: https://doi.org/10.1007/s11590-018-1311-5