A theorem in convex programming

An optimization problem which frequently arises in applications of mathematical programming is the following: Where fi are convex functions. In this paper, the function F is studied and shown to satisfy F(A, B) = M (A) + N(B), where M and N are increasing and decreasing convex functions, respectively. Also, the functional equation F (A, C) = F(A, B) + F(B, C) − F(B, B) is established. These results generalize to the continuous case F(A, B)=min ∫ OT f(t, x(t))dt, with x(t) increasing and A ≤ x (0) ≤ x (T) ≤ B. The results obtained in this paper are useful for reducing an optimization problem with many variables to one with fewer variables.