Optimization of convex functions with fenchel biconjugation and duality

Analysis of conjugation operations to induce a bijection between proper closed convex functions and to discuss the problems of boundedness of closed convex proper functions using continuity of conjugates. Present study shows a great contribution of biconjugation of convex functions in optimization. Fenchel biconjugation describes the relation of duality in optimization. Problems of finite dimensional Lagrangian convex duality theory and problems on duality gap are comparative of primal and dual solutions in convex optimization.