Properly efficient solutions for vectorial norm approximation

In the present paper a general vectorial best approximation problem using vectorial norms with respect to properly efficient solutions is considered. Necessary and sufficient optimality conditions for such solutions are derived. This is done on base of scalarization and studying a corresponding dual problem to the scalar optimization problem. Also optimality conditions in sub-differential form are formulated. For the necessity of the optimality conditions especially convexity assumptions are essential. But under only very weak supposition these conditions are sufficient.