(4a) If [rho] > 0, then f is called strongly incave (TIV) at u;
(4b) If [rho] = 0, then f is called incave (IV) at u;
(4c) If [rho] < 0, then f is called weakly incave (sIV) at u;
(4d) [for all]x [member of] A, x [not equal to] u: f(x) - f(u) < [f'.sub.+](u; [eta](x,u), then f is called strictly incave (SIV) at u.
If f ([u.sub.1],...,[u.sub.m]) is a convex [strictly convex] function and in addition, f is increasing [strictly increasing] function with respect to each invex [strictly invex] component, [u.sub.i], or f is decreasing [strictly decreasing] with respect to each incave [strictly incave] component [u.sub.j], then F is invex [strictly invex] on X.