Robustness in nonsmooth nonlinear multi-objective programming

•Defining robust efficiency for nonlinear multi-objective optimization problems.•Showing that, robust efficient solutions are proper solutions (under some assumptions).•Providing a problem for calculating a robustness radius.•Comparing the newly-defined robustness notion with two existing ones.•Char...

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Published in:	European journal of operational research Vol. 247; no. 2; pp. 370 - 378
Main Authors:	Zamani, Moslem, Soleimani-damaneh, Majid, Kabgani, Alireza
Format:	Journal Article
Language:	English
Published:	Amsterdam Elsevier B.V 01.12.2015 Elsevier Sequoia S.A
Subjects:	Clarke generalized gradient Comparative analysis Convex analysis Linear programming Mathematical functions Mathematical problems Multiple objective programming Nonlinear programming Nonsmooth optimization Optimization techniques Proper efficiency Robust solution Studies Robust solution Nonsmooth optimization Proper efficiency Multiple objective programming Clarke generalized gradient
ISSN:	0377-2217, 1872-6860
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Abstract	•Defining robust efficiency for nonlinear multi-objective optimization problems.•Showing that, robust efficient solutions are proper solutions (under some assumptions).•Providing a problem for calculating a robustness radius.•Comparing the newly-defined robustness notion with two existing ones.•Characterizing robust solutions and studying alterations of objectives. Recently Georgiev, Luc, and Pardalos (2013), [Robust aspects of solutions in deterministic multiple objective linear programming, European Journal of Operational Research, 229(1), 29–36] introduced the notion of robust efficient solutions for linear multi-objective optimization problems. In this paper, we extend this notion to nonlinear case. It is shown that, under the compactness of the feasible set or convexity, each robust efficient solution is a proper efficient solution. Some necessary and sufficient conditions for robustness, with respect to the tangent cone and the non-ascent directions, are proved. An optimization problem for calculating a robustness radius followed by a comparison between the newly-defined robustness notion and two existing ones is presented. Moreover, some alterations of objective functions preserving weak/proper/robust efficiency are studied.
AbstractList	Recently Georgiev, Luc, and Pardalos (2013), [Robust aspects of solutions in deterministic multiple objective linear programming, European Journal of Operational Research, 229(1), 29-36] introduced the notion of robust efficient solutions for linear multi-objective optimization problems. In this paper, we extend this notion to nonlinear case. It is shown that, under the compactness of the feasible set or convexity, each robust efficient solution is a proper efficient solution. Some necessary and sufficient conditions for robustness, with respect to the tangent cone and the non-ascent directions, are proved. An optimization problem for calculating a robustness radius followed by a comparison between the newly-defined robustness notion and two existing ones is presented. Moreover, some alterations of objective functions preserving weak/proper/robust efficiency are studied. •Defining robust efficiency for nonlinear multi-objective optimization problems.•Showing that, robust efficient solutions are proper solutions (under some assumptions).•Providing a problem for calculating a robustness radius.•Comparing the newly-defined robustness notion with two existing ones.•Characterizing robust solutions and studying alterations of objectives. Recently Georgiev, Luc, and Pardalos (2013), [Robust aspects of solutions in deterministic multiple objective linear programming, European Journal of Operational Research, 229(1), 29–36] introduced the notion of robust efficient solutions for linear multi-objective optimization problems. In this paper, we extend this notion to nonlinear case. It is shown that, under the compactness of the feasible set or convexity, each robust efficient solution is a proper efficient solution. Some necessary and sufficient conditions for robustness, with respect to the tangent cone and the non-ascent directions, are proved. An optimization problem for calculating a robustness radius followed by a comparison between the newly-defined robustness notion and two existing ones is presented. Moreover, some alterations of objective functions preserving weak/proper/robust efficiency are studied.
Author	Zamani, Moslem Soleimani-damaneh, Majid Kabgani, Alireza
Author_xml	– sequence: 1 givenname: Moslem surname: Zamani fullname: Zamani, Moslem organization: School of Mathematics, Statistics and Computer Science,College of Science, University of Tehran, Enghelab Avenue, Tehran, Iran – sequence: 2 givenname: Majid surname: Soleimani-damaneh fullname: Soleimani-damaneh, Majid email: soleimani@khayam.ut.ac.ir organization: School of Mathematics, Statistics and Computer Science,College of Science, University of Tehran, Enghelab Avenue, Tehran, Iran – sequence: 3 givenname: Alireza surname: Kabgani fullname: Kabgani, Alireza organization: School of Mathematics, Statistics and Computer Science,College of Science, University of Tehran, Enghelab Avenue, Tehran, Iran
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SubjectTerms	Clarke generalized gradient Comparative analysis Convex analysis Linear programming Mathematical functions Mathematical problems Multiple objective programming Nonlinear programming Nonsmooth optimization Optimization techniques Proper efficiency Robust solution Studies
Title	Robustness in nonsmooth nonlinear multi-objective programming
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