Second order necessary and sufficient conditions for convex composite NDO

In convex composite NDO one studies the problem of minimizing functions of the formF:=h ○f whereh:ℝm → ℝ is a finite valued convex function andf:ℝn → ℝm is continuously differentiable. This problem model has a wide range of application in mathematical programming since many important problem classes can be cast within its framework, e.g. convex inclusions, minimax problems, and penalty methods for constrained optimization. In the present work we extend the second order theory developed by A.D. Ioffe in [11, 12, 13] for the case in whichh is sublinear, to arbitrary finite valued convex functionsh. Moreover, a discussion of the second order regularity conditions is given that illuminates their essentially geometric nature.