Second-order strong Karush/Kuhn-Tucker conditions for proper efficiencies in multiobjective optimization (Q2420814)

The authors consider the following multiobjective problem: \[ \min~ f(x):= (f_1(x), \dots, f_p(x))\text{ subject to }g_j(x) \leq 0,\, j = 1, \dots m, \] where \(f_i: \Omega \rightarrow R\), \(i = 1, \dots , p\), \(g_j:\Omega \rightarrow R\), \(\Omega \subset R\), \(j = 1, \dots, m\). The following concepts known from the literature are recalled: Geofrion properly efficient solution (GPE), Borwein properly efficient solution (BPE), and strong Karush/Kuhn-Tucker conditions (KKT). A new secod-order regularity condition of Abadie-type (SOARC) is introduced. It is proved that SOARC implies the validity of the second-order strong KKT conditions at every Borwein-properly efficient solution. The second-order KKT-type sufficient solution for local GPE is expressed. Directions of further research are briefly outlined in the conclusions of the paper.