Nonlinear Separation Theorems for Co-Radiant Sets and Optimality Conditions for Approximate and Proper Approximate Solutions in Vector Optimization

This paper deals with $\varepsilon$-efficient and $\varepsilon$-properly efficient points with respect to a co-radiant set in vector optimization problems. In the first part of the paper, we establish a new nonlinear separation theorem for co-radiant sets in normed spaces. Subsequently, we obtain necessary and sufficient conditions, via scalarization, for both $\varepsilon$-efficient and $\varepsilon$-properly efficient points in a general framework, without requiring any assumptions on the co-radiant set or convexity conditions on the sets under consideration. Consequently, our results are applicable in a broader range of settings than those previously addressed in the literature.

💡 Research Summary

The paper investigates ε‑efficient and ε‑properly efficient points in vector optimization with respect to co‑radiant sets, introducing a substantially more general framework than previously available. The authors first develop a new nonlinear separation theorem for co‑radiant sets in arbitrary normed spaces. Building on the strict separation property (SSP) introduced in earlier work, they show that if the cones generated by two co‑radiant sets C and K satisfy SSP, there exist a functional f∈X* and a positive scalar α such that the inequality f(x)−α‖x‖<0 holds for all x in −C, while f(y)−α‖y‖>0 holds for all y on the boundary of −K. By employing δ‑conic neighborhoods they prove that this separation persists under small dilations of C, eliminating the need for closedness, bounded bases, or finite‑dimensionality that were required in earlier results (e.g.,