Some comments on Lagrangean duality, optimality conditions and convexity

In the classical literature of NLP, the Lagrangian duals (minimax and Wolfe’s) are lengthily discussed under various assumptions of convexity and/or regularity. Their relations with optimality conditions (saddle points and stationary points) are also very much exploited. This paper exploits the characteristics of the optimal value function of the dual problems, trying to clarify the essential or non-essential role of the usual assumptions, presenting some probably obvious results like the necessity and sufficiency of v(.) being convex and l.s.c. in order to to avoid duality gaps in the minimax dual, some characterizations of dual solutions, the role of weak duality theorems and the idea of f(.) uniqueness.