2. Notation OR-1 2015 1

3. 2

4. 3

5. 4 Backgrounds

6. OR-1 2015 5 Convex sets Nonconvex set

7. OR-1 2015 6 x y z

8. 7

9. 8 Convex function

10. OR-1 2015 9

11. 10 : extreme points

12. OR-1 2015 11 Review-Linear Algebra

13. OR-1 2015 12  Submatrices multiplication

14. OR-1 2015 13  submatrix multiplications which will be frequently used.

15. OR-1 2015 14

16. OR-1 2015 15

17. OR-1 2015 16 Simultaneous Linear Equations

18. OR-1 2015 17

19. OR-1 2015 18

20. OR-1 2015 19

21. OR-1 2015 20 Solving systems of linear equations (Gauss-Jordan Elimination, 변수의 치환 ) (will be used in the simplex method to solve LP problems)

22. OR-1 2015 21  Infinitely many solutions case

23. OR-1 2015 22

24. OR-1 2015 23

25. OR-1 2015 24  Elementary row operations are equivalent to premultiplying a nonsingular square matrix to both sides of the equations

26. OR-1 2015 25

27. OR-1 2015 26

28. OR-1 2015 27  So if we multiply all elementary row operation matrices, we get the matrix having the information about the elementary row operations we performed

29. OR-1 2015 28

30. OR-1 2015 29