Constrained optimization problems of the form (1) minimize f(x) subject to , g(x), with differentiable functions f, g f type I or type II are considered: The functions f, g are called of type I with respect to a vector function (x) at if the relations
hold for all feasible x of the problem (1).
Similarly f, g are called of type II with respect to x at , if
are satisfied for all feasible solutions of the problem (1). Various sufficient conditions, under which the functions f, g are of type I or II are given. Sufficient optimality conditions for the problem (1), in which f, g are of type I or II are proved.