New results on proper efficiency for a class of vector optimization problems

ABSTRACT

This paper presents two new theorems on Geoffrion's properly efficient solutions and seven examples illustrating their applications to linear fractional vector optimization problems with unbounded constraint sets. Provided that all the components of the objective function are properly fractional, the first theorem gives sufficient conditions for the efficient solution set to coincide with the Geoffrion properly efficient solution set. Admitting that the objective function can have some affine components, in the second theorem we give sufficient conditions for an efficient solution to be a Geoffrion's properly efficient solution. The recession cone of the constraint set, the derivatives of the scalar objective functions, but no tangent cone to the constraint set at the efficient point, are used in the second theorem.

Keywords:

• Linear fractional vector optimization
• efficient solution
• Geoffrion's properly efficient solution
• coincidence of solution sets
• recession cone

2010 Mathematics Subject Classifications:

• 90C31
• 90C29
• 90C33
• 90C32
• 90C47

Acknowledgements

We would like to thank the anonymous referee for insightful comments and valuable suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors were supported respectively by National Foundation for Science & Technology Development (Vietnam) under grant number 101.01-2018.306 and Le Quy Don Technical University (Vietnam), China Medical University (Taiwan), and Institute of Mathematics (VAST, Vietnam).