Necessary Optimality Conditions for Robust Nonsmooth Multiobjective Optimization Problems

Nazih Abderrazzak Gadhi; Mohamed Ohda

doi:10.2478/candc-2022-0018

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Necessary Optimality Conditions for Robust Nonsmooth Multiobjective Optimization Problems

Control and Cybernetics

Volume 51 (2022): Issue 3 (September 2022)

By: Nazih Abderrazzak Gadhi and Mohamed Ohda

Open Access

|Mar 2023

Abstract

This paper deals with a robust multiobjective optimization problem involving nonsmooth/nonconvex real-valued functions. Under an appropriate constraint qualification, we establish necessary optimality conditions for weakly robust efficient solutions of the considered problem. These optimality conditions are presented in terms of Karush-Kuhn-Tucker multipliers and convexificators of the related functions. Examples illustrating our findings are also given.

References

Apostol, T.M. (2004) Mathematical Analysis, Second Edition. Pearson Education Asia Limited and China Machine.
Search in Google Scholar Back to article
Babahadda, H. and Gadhi, N.Ä. (2006) Necessary optimality conditions for bilevel optimization problems using convexificators. Journal of Global Optimization 34, 535–549.
Search in Google Scholar Back to article
Ben-Tal, Ä. and Nemirovski, Ä. (2000) Robust solutions of linear programming problems contaminated with uncertain data. Mathematical Programming 88, 411-424.
Search in Google Scholar Back to article
Bertsimas, D and Sim, M. (2004) The price of robustness. Operations Research 52, 35–53.
Search in Google Scholar Back to article
Bertsimas, D. and Sim, M. (2006) Tractable approximations to robust conic optimization problems. Mathematical Programming 107, 5–36.
Search in Google Scholar Back to article
Bokrantz, R. and Fredriksson, Ä. (2017) Necessary and sufficient conditions for Pareto efficiency in robust multiobjective optimization. European Journal of Operational Research 262, 682–692.
Search in Google Scholar Back to article
Chen, J., Köbis, E. and Yao, J.C. (2019) Optimality conditions and duality for robust nonsmooth multiobjective optimization problems with constraints. Journal of Optimization Theory and Applications 181, 411–436.
Search in Google Scholar Back to article
Chuong, T.D. (2016) Optimality and duality for robust multiobjective optimization problems. Nonlinear Analysis 134, 127–143.
Search in Google Scholar Back to article
Clarke, F.C. (1983) Optimization and Nonsmooth Analysis. New York, Wiley-Interscience.
Search in Google Scholar Back to article
Craven, B.D. and Islam, S.M. (2012) Linear programming with uncertain data: Some extensions to robust optimization. Journal of Optimization Theory and Applications 155, 673–679.
Search in Google Scholar Back to article
Demyanov, V.F. and Jeyakumar, V. (1997) Hunting for a smaller convex subdifferential. Journal of Global Optimization 10, 305–326.
Search in Google Scholar Back to article
Dutta, J. and Chandra, S. (2002) Convexifactors, generalized convexity, and optimality conditions. Journal of Optimization Theory and Applications 113, 41–64.
Search in Google Scholar Back to article
Dutta, J. and Chandra, S. (2004) Convexifactors, generalized convexity and vector optimization. Optimization 53, 77–94.
Search in Google Scholar Back to article
Ewing, G.M. (1977) Sufficient conditions for global minima of suitably convex functionals from variational and control theory. SIAM Review 19, 202– 220.
Search in Google Scholar Back to article
Gadhi, N.A. (2021) Comments on “A Note on the Paper “Optimality Conditions for Optimistic Bilevel Programming Problem Using Convexifactors”“. Accepted for publication in Journal of Optimization Theory and Applications.
Search in Google Scholar Back to article
Gadhi, N. A. and Ichatouhane, A. (2021) A note on the paper “Optimality conditions for nonsmooth interval-valued and multiobjective semi-infinite programming”. RAIRO-Oper. Res. 55, 13–22.
Search in Google Scholar Back to article
Hiriart-Urruty, J.B. and Lemarechal, C. (2001) Fundamentals of Convex Analysis. Springer-Verlag, Berlin Heidelberg.
Search in Google Scholar Back to article
Jeyakumar, V. and Luc, D.T. (1999) Nonsmooth calculus, minimality, and monotonicity of convexificators. Journal of Optimization Theory and Applications 101, 599–621.
Search in Google Scholar Back to article
Jeyakumar, V., Li, G. and Lee, G. M. (2012) Robust duality for generalized convex programming problems under data uncertainty. Nonlinear Analysis 75, 1362–1373.
Search in Google Scholar Back to article
Kabgani, A. and Soleimani-damaneh, M. (2017) Characterization of (weakly/properly/robust) efficient solutions in nonsmooth semiinfinite multiobjective optimization using convexificators. Optimization 67, 217–235.
Search in Google Scholar Back to article
Kabgani, A. and Soleimani-damaneh, M. (2019) Constraint qualifications and optimality conditions in nonsmooth locally star-shaped optimization using convexificators. Pacific Journal of Optimization 15, 399–413.
Search in Google Scholar Back to article
Kanzi N. (2011) Necessary optimality conditions for nonsmooth semi-infinite programming problems. Journal of Global Optimization 49, 713–725.
Search in Google Scholar Back to article
Kaur S. (1983) Theoretical studies in mathematical programming. PhD thesis. University of Delhi.
Search in Google Scholar Back to article
Kuroiwa, D. and Lee, G. M. (2012) On robust multiobjective optimization. Vietnam Journal of Mathematics 40, 305–317.
Search in Google Scholar Back to article
Lee, G. M. and Son, P.T. (2014) On nonsmooth optimality theorems for robust optimization problems. Bulletin of the Korean Mathematical Society 51, 287–301.
Search in Google Scholar Back to article
Li, X. F. and Zhang, J. Z. (2006) Necessary optimality conditions in terms of convexificators in Lipschitz optimization. Journal of Optimization Theory and Applications 131, 429–452.
Search in Google Scholar Back to article
Li, C., Ng, K.F. and Pong T.K. (2008) Constraint qualifications for convex inequality systems with applications in constrained optimization. SIAM Journal on Optimization 19, 163–187.
Search in Google Scholar Back to article
Li, W., Nahak, C. and Singer, I. (2000) Constraint qualifications for semiinfinite systems of convex inequalities. SIAM J Optim. 11, 31–52.
Search in Google Scholar Back to article
Mordukhovich, B.S. and Shao, Y. (1995) On nonconvex subdifferential calculus in Banach spaces. Journal of Convex Analysis 2, 211–227.
Search in Google Scholar Back to article
Saadati, M. and Oveisiha, M. (2021) Optimality conditions for robust nonsmooth multiobjective optimization problems in Asplund spaces. arXiv preprint arXiv:2105.14366.
Search in Google Scholar Back to article
Tung, L.T. (2020) Karush-Kuhn-Tucker optimality conditions and duality for convex semi-infinite programming with multiple interval-valued objective functions. Journal of Applied Mathematics and Computing 62, 67–91.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/candc-2022-0018 | Journal eISSN: 2720-4278 | Journal ISSN: 0324-8569

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Language: English

Page range: 289 - 302

Submitted on: Jul 1, 2021

Accepted on: Jun 1, 2022

Published on: Mar 22, 2023

Published by: Systems Research Institute Polish Academy of Sciences

In partnership with: Paradigm Publishing Services

Publication frequency: 4 issues per year

Keywords:

convexificator,

directional constraint qualification,

efficient solution,

optimality conditions,

robust multiobjective optimization

Related subjects:

Computer sciences,

Computer sciences, other,

Engineering,

Electrical engineering,

Fundamentals of electrical engineering,

Mechanical engineering,

Fundamentals of mechanical engineering,

Mathematics,

General mathematics

© 2023 Nazih Abderrazzak Gadhi, Mohamed Ohda, published by Systems Research Institute Polish Academy of Sciences
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 51 (2022): Issue 3 (September 2022)