Multiobjective approximate gradient projection method for constrained vector optimization: sequential optimality conditions without constraint qualifications (Q2122022)

This paper extends the concepts related to multiobjective optimization (MOP) problems to multiobjective approximate gradient projection (MAGP) optimality conditions using the scalarization technique to so-called linear multiobjective approximate gradient projection (LMAGP) in which additional conditions are imposed on linear constraints compared to MAGP. The outline of this article is as follows: Section 2, some preliminaries related to the basic properties of a set-valued mapping as the sequential Painlevé-Kuratowski outer/upper limit of a map \(F(x)\), are given. Section 3 is devoted to MAGP optimality conditions of multiobjective optimization problems and their associated constraint qualification. In Section 4, the behavior of LMAGP optimality conditions and the LMAGP-regularity property are discussed. The relationship between the introduced constraint qualifications is presented in Section 5 (``MAGP regularity implies LMAGP regularity, but converse may not be true'', ``LMAGP regularity implies Abadie's constraint qualification, but converse is not true'' or ``MAGP -- regularity implies Abadie's constraints qualification, but converse is not true''). Section 6 is dedicated to concluding remarks and future directions such'' MAGP-type optimality conditions can be derived for semi-infinite optimization problems (SIOP)''.