Optimality Conditions For E-Optimal Solutions Via Improvement Set In Vector Optimization

The results of the various notions of optimal points, the properties of the set ofoptimal solutions such as nonemptiness, boundedness and compactness, and theoptimality conditions of optimal solutions in vector optimization problems are plentiful.In2010, M. Chicco et al introduced a new definition of optimal point--E-optimal point,which unifies the definitions of various optimal solution notions such as efficientsolution, weak efficient solution and strongly efficient solution, and variousapproximate solutions, such as ε-efficient solution and weak ε-efficient solutiondefined by Kutateladze. But, unfortunately, there is no one to study the properties of theset of E-optimal solutions and the optimality conditions of E-optimal solutions.In this paper，we generate the notion of E-optimal solutions for vector optimizationproblem via improvement set in finite dimentional space into general topological vectorspace. We characterize the properties like nonemptiness, boundedness and compactnessof the set of E-optimal solutions for vector optimization problem without any convexityassumption and for a bouned/unbounded constrained convex vector optimizationproblem. Furthermore, by scalarization approach--nonlinear scalarizating functionΔ,we give some optimality conditions of E-optimal solutions and indicate that the similarresults hold in the previous literature when improvement set E degenerates into oderingset. In addition, we give a necessary and sufficient conditions of the existence ofefficient points in terms of a new definition of strong set convergence and a necessaryand sufficient conditions of the existence of weak total optimal points, respectively.