On approximate solutions for nonsmooth robust multiobjective optimization problems

We introduce a new concept of generalized convexity of 'degree n' for a multiobjective optimization problem and is compared it to the previous notions of generalized convex functions. Some examples to justify the importance of the term 'degree n' are provided. Namely, the conclus...

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Published in:Optimization Vol. 68; no. 9; pp. 1653 - 1683
Main Authors: Fakhar, M., Mahyarinia, M.R., Zafarani, J.
Format: Journal Article
Language:English
Published: Philadelphia Taylor & Francis 02.09.2019
Taylor & Francis LLC
Subjects:
condition of degree n
Convexity
Generalized convexity of degree n
Mathematical programming
Multiple objective analysis
Nonlinear programming
Optimization
robust optimality
robust ϵ-Mond-Weir type duality of degree n
robust ϵ-quasi-(weakly) efficient solutions
Robustness
Saddle points
ϵ-approximate
ϵ-approximate weak vector saddle-point of degree n
ϵ-vector duality of degree n
ISSN:0233-1934, 1029-4945
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Abstract We introduce a new concept of generalized convexity of 'degree n' for a multiobjective optimization problem and is compared it to the previous notions of generalized convex functions. Some examples to justify the importance of the term 'degree n' are provided. Namely, the conclusions of our results may fail if this term is dropped. By applying our new definition to nonsmooth robust multiobjective optimization problems, we establish the nonsmooth robust optimality conditions and robust duality theory for robust ϵ-quasi-(weakly) efficient solutions. A robust ϵ-Mond-Weir type duality of degree n for an uncertain multi-objective optimization problem under our generalized convexity assumption is presented. Furthermore, we introduce an ϵ-approximate scalar saddle-point and an ϵ-approximate weak vector saddle-point of degree n for the robust multi-objective optimization problem. The relationships between these two concepts with robust ϵ-approximate condition and robust ϵ-weakly efficient solutions are also given.
AbstractList We introduce a new concept of generalized convexity of 'degree n' for a multiobjective optimization problem and is compared it to the previous notions of generalized convex functions. Some examples to justify the importance of the term 'degree n' are provided. Namely, the conclusions of our results may fail if this term is dropped. By applying our new definition to nonsmooth robust multiobjective optimization problems, we establish the nonsmooth robust optimality conditions and robust duality theory for robust [GREEK LUNATE EPSILON SYMBOL]-quasi-(weakly) efficient solutions. A robust [GREEK LUNATE EPSILON SYMBOL]-Mond-Weir type duality of degree n for an uncertain multi-objective optimization problem under our generalized convexity assumption is presented. Furthermore, we introduce an [GREEK LUNATE EPSILON SYMBOL]-approximate scalar saddle-point and an [GREEK LUNATE EPSILON SYMBOL]-approximate weak vector saddle-point of degree n for the robust multi-objective optimization problem. The relationships between these two concepts with robust [GREEK LUNATE EPSILON SYMBOL]-approximate [Formula omitted.] condition and robust [GREEK LUNATE EPSILON SYMBOL]-weakly efficient solutions are also given.
We introduce a new concept of generalized convexity of 'degree n' for a multiobjective optimization problem and is compared it to the previous notions of generalized convex functions. Some examples to justify the importance of the term 'degree n' are provided. Namely, the conclusions of our results may fail if this term is dropped. By applying our new definition to nonsmooth robust multiobjective optimization problems, we establish the nonsmooth robust optimality conditions and robust duality theory for robust ϵ-quasi-(weakly) efficient solutions. A robust ϵ-Mond-Weir type duality of degree n for an uncertain multi-objective optimization problem under our generalized convexity assumption is presented. Furthermore, we introduce an ϵ-approximate scalar saddle-point and an ϵ-approximate weak vector saddle-point of degree n for the robust multi-objective optimization problem. The relationships between these two concepts with robust ϵ-approximate condition and robust ϵ-weakly efficient solutions are also given.
Author Zafarani, J.
Mahyarinia, M.R.
Fakhar, M.
Author_xml – sequence: 1
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Snippet We introduce a new concept of generalized convexity of 'degree n' for a multiobjective optimization problem and is compared it to the previous notions of...
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SubjectTerms condition of degree n
Convexity
Generalized convexity of degree n
Mathematical programming
Multiple objective analysis
Nonlinear programming
Optimization
robust optimality
robust ϵ-Mond-Weir type duality of degree n
robust ϵ-quasi-(weakly) efficient solutions
Robustness
Saddle points
ϵ-approximate
ϵ-approximate weak vector saddle-point of degree n
ϵ-vector duality of degree n
Title On approximate solutions for nonsmooth robust multiobjective optimization problems
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