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

This paper deals with $\epsilon$-efficient and $\epsilon$-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 $\epsilon$-efficient and $\epsilon$-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.