Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces

In this article, we work on vector optimization problems in linear topological spaces. Our vector optimization problems have weakened convex inequality constraints and weakened affine equality constraints. Our inequalities are given by partial orders that are induced by pointed convex cones. We prove a Farkas–Minkowski-type theorem of alternative and obtain some optimality conditions through the discussions of vector saddle points and scalar saddle points.