ε-Optimality Conditions for Vector Optimization Problems with Set-Valued Maps

Abstract

In this paper, I introduce two kinds of ε-subgradients—ε-Pareto subgradients and ε-Benson proper subgradients—for set-valued maps. I give sufficient conditions for the existence of such ε-subgradients. A generalized ε-Moreau–Rockafellar type theorem for ε-Benson proper subdifferentials of set-valued maps is formulated and applied to establish ε-optimality conditions for ε-Benson proper efficient solutions of vector optimization problems with set-valued maps. Scalarization theorems for such a problem are also obtained.

Keywords:

• ε-Benson proper efficient solution
• ε-Benson proper subdifferentials
• ε-Benson proper subgradient
• ε-Efficient solution
• ε-Optimality condition
• ε-Subgradient
• Set-valued map
• Vector optimization problem

AMS Subject Classification:

• 90C26
• 90C29
• 90C46

ACKNOWLEDGMENT

The support of The National Foundation for Science and Technology Development, Vietnam, is gratefully acknowledged.