On approximate solutions for nonsmooth robust multiobjective optimization problems

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 $\left(\mathrm{KKT}\right)$ condition and robust ϵ-weakly efficient solutions are also given.

Keywords:

• Generalized convexity of degree n
• robust ϵ-quasi-(weakly) efficient solutions
• robust optimality
• ϵ-approximate $\left(\mathrm{KKT}\right)$ condition of degree n
• ϵ-vector duality of degree n
• robust ϵ-Mond-Weir type duality of degree n
• ϵ-approximate weak vector saddle-point of degree n

Acknowledgements

The authors are grateful to reviewers and Associate Editor for valuable comments. They would like also to thank T. Jabarootian for useful discussions.

Disclosure statement

No potential conflict of interest was reported by the authors.