1. Convex Optimization Chapter 1 Introduction

2. What, Why and How  What is convex optimization  Why study convex optimization  How to study convex optimization

3. What is Convex Optimization?

4. Mathematical Optimization Convex Optimization Least-squaresLP Nonlinear Optimization

5. Mathematical Optimization

6. Convex Optimization

7. Least-squares

8. Analytical Solution of Least-squares f 0 ( x ) = jj A x ¡ b jj 2 2 = ( A x ¡ b ) > ( A x ¡ b ) x = ( A > A ) ¡ 1 A > b @f 0 ( x ) @ x = 2 A > ( A x ¡ b ) = 0

9. Linear Programming (LP)

10. Why Study Convex Optimization?

11. Mathematical Optimization Convex Optimization Least-squaresLP Solving Optimization Problems Nonlinear Optimization

12. Analytical solution Good algorithms and software High accuracy and high reliability Time complexity: Mathematical Optimization Convex Optimization Least-squares LP Nonlinear Optimization A mature technology!

13. No analytical solution Algorithms and software Reliable and efficient Time complexity: Mathematical Optimization Convex Optimization Least-squares LP Nonlinear Optimization Also a mature technology!

14. Mathematical Optimization Convex Optimization Nonlinear Optimization Almost a mature technology! Least-squares LP No analytical solution Algorithms and software Reliable and efficient Time complexity (roughly)

15. Mathematical Optimization Convex Optimization Nonlinear Optimization Far from a technology! (something to avoid) Least-squares LP Sadly, no effective methods to solve Only approaches with some compromise Local optimization: “more art than technology” Global optimization: greatly compromised efficiency Help from convex optimization 1) Initialization 2) Heuristics 3) Bounds

16. Why Study Convex Optimization If not, …… -- Section 1.3.2, p8, Convex Optimization there is little chance you can solve it.

17. How to Study Convex Optimization?

18. Two Directions  As potential users of convex optimization  As researchers developing convex programming algorithms

19. Recognizing least-squares problems  Straightforward: verify the objective to be a quadratic function the quadratic form is positive semidefinite  Standard techniques increase flexibility Weighted least-squares Regularized least-squares

20. Recognizing LP problems  Example: Sum of residuals approximation Chebyshev or minimax approximation t = max i j a > i x ¡ b i j t i = j r i j

21. Recognizing Convex Optimization Problems

22. An Example

25. 8 f j 1 ; j 2 ; ¢¢¢ ; j 10 g P 10 k = 1 p j k · 1 2 P m j = 1 p j Adding linear constraints????? C 10 m

26. Summary From the book, we expect to learn  To recognize convex optimization problems  To formulate convex optimization problems  To (know what can) solve them!