2. Notation OR-1 2016 1

3. 2

4. 3

5. 4 Backgrounds

6. OR-1 2016 5 Convex sets Nonconvex set

7. OR-1 2016 6 x y z

8. 7

9. 8 Convex function

10. OR-1 2016 9

11. 10 : extreme points

12. OR-1 2016 11 Review-Linear Algebra

13. OR-1 2016 12  Submatrices multiplication

14. OR-1 2016 13  submatrix multiplications which will be frequently used.

15. OR-1 2016 14

16. OR-1 2016 15

17. OR-1 2016 16 Simultaneous Linear Equations

18. OR-1 2016 17

19. OR-1 2016 18

20. OR-1 2016 19

21. OR-1 2016 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 2016 21  Infinitely many solutions case

23. OR-1 2016 22

24. OR-1 2016 23 We have different basic solutions depending on which variables are considered as basic.

25. OR-1 2016 24  Elementary row operations are equivalent to premultiplying a nonsingular square matrix to both sides of the equations

26. OR-1 2016 25

27. OR-1 2016 26

28. OR-1 2016 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 2016 28

30. OR-1 2016 29