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

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.

Automated summary

This paper introduces two new theorems on Geoffrion's properly efficient solutions and provides seven examples demonstrating their applications in linear fractional vector optimization problems with unbounded constraint sets. The first theorem gives sufficient conditions for the efficient solution set to coincide with Geoffrion's properly efficient solution set when all components of the objective function are properly fractional. The second theorem establishes sufficient conditions for an efficient solution to be a Geoffrion's properly efficient solution, even when the objective function includes some affine components.