1. I.3 Introduction to the theory of convex conjugated function

2. Epigraph E : set Epigraph,is the set

4. lower semicontinuous Assume that E is a topological space. Define is called lower semicontinuous (l.s.c) at x if

5. i.e. for any there is a neighborhood N of x such that f is l.s.c on E if f is l.s.c at each point of E

6. Exercise is l.s.c on E if and only if is open is closed

7. is l.s.c on E if and only if is closed in E x R

8. is l.s.c on E if and only if is closed

9. are l.s.c then so is Ifand

10. is l.s.c t h e n t h e u p p e r e n v e l o p e o f If is a family of l.s.c functions on E i.e. the functiondefinded by is l.s.c

11. is attained If E is compact and is l.s.c on E,then i.e.

12. Convex is convex if D e f Suppose E is a vector space (real)

13. is convex if and only if is convex in E x R

14. is a convex function then is convex. If forthe set Converse statement is not true in genernal see next page

15. counter example

16. are convex then so is Ifand

17. is convex t h e n t h e u p p e r e n v e l o p e o f If is a family of convex functions on E

18. Conjugated function such that G i v e n Assume that E is a real n.v.s D e f i n e t h e c o n j u g a t e d f u n c t i o n o f b y

19. proposition 1-9 then i s c o n v e x, l. s. c Suppose

22. Def

23. Theorem I.10 (Fenchel-Moreau) then i s c o n v e x, l. s. c Suppose

29. Example

32. Lemma I.4 then i s c o n v e x, Let then IntC is convex If

33. Theorem I.11 are convex and suppose that a n d such that Suppose there is and is continuous at see next page

35. Observe usually appears for constrain (1) (2) see next page

36. The proof of Thm I.11 see next page

41. Example

44. Exercise

45. Example Let be nonempty, close and convex. Put

46. Let