APPROXIMATE PROPER EFFICIENCY ON REAL LINEAR VECTOR SPACES

The aim of the paper is presenting necessary and sufficient conditions to characterize the approximate (weak/proper) efficient solutions of vector optimization problems under real linear vector spaces without any particular topology. To this end, we use different scalarization approaches based upon the dual cone, Gerstewitz's scalarization function, and the Lagrangian mapping notions. Since there is not any topology here, we utilize some algebraic concepts instead of topological (relative) interior, closure, and dual cone. Also, some separation and alternative theorems play a fundamental role in establishing the main results.