1. Lecture 1: Introduction to the Course of Optimization 主講人 : 虞台文

2. Content What is Optimization? Categorization of Optimization Problems Some Optimization Problems

3. Lecture 1: Introduction to the Course of Optimization What is Optimization?

4. To Win a Game

5. To Survive

6. To Make Things Done

7. To Make a Decision

8. What is Optimization? Objective Function – A function to be minimized or maximized Unknowns or Variables – Affect the value of the objective function Constraints – Restrict unknowns to take on certain values but exclude others

9. What is Optimization? Objective Function – A function to be minimized or maximized Unknowns or Variables – Affect the value of the objective function Constraints – Restrict unknowns to take on certain values but exclude others The optimization problem is then: Find values of the variables that minimize or maximize the objective function while satisfying the constraints. The optimization problem is then: Find values of the variables that minimize or maximize the objective function while satisfying the constraints.

10. Example: 0-1 Knapsack Problem Which boxes should be chosen to maximize the amount of money while still keeping the overall weight under 15 kg ?

11. Example: 0-1 Knapsack Problem Objective Function Unknowns or Variables Constraints

12. Example: 0-1 Knapsack Problem Is the problem difficult? What is the complexity in your approach?

13. Lecture 1: Introduction to the Course of Optimization Categorization of Optimization Problems

14. Ingredients of Optimization Problems Objective Function Unknowns or Variables Constraints Are All these ingredients necessary?

15. Optimization Problems w/o Objective Function In some cases, the goal is to find a set of variables that satisfies some constraints only, e.g., – circuit layouts – n-queen – Sudoku This type of problems is usually called a feasibility problem or constraint satisfaction problem. Objective Function Unknowns or Variables Constraints

16. Optimization Problems w/ Multiple Objective Functions Sometimes, we need to optimize a number of different objectives at once, e.g., – In the panel design problem, it would be nice to minimize weight and maximize strength simultaneously. Usually, the different objectives are not compatible – The variables that optimize one objective may be far from optimal for the others. In practice, problems with multiple objectives are reformulated as single-objective problems by either forming a weighted combination of the different objectives or else replacing some of the objectives by constraints. Objective Function Unknowns or Variables Constraints

17. Variables Objective Function Unknowns or Variables Constraints Variables are essential. – Without variables, we cannot define the objective function and the problem constraints. Continuous Optimization – all the variables are allowed to take values from subintervals of the real line; Discrete Optimization – require some or all of the variables to have integer values.

18. Constraints Objective Function Unknowns or Variables Constraints Constraints are not essential. Unconstrained optimization – A large and important one for which a lot of algorithms and software are available. However, almost all problems really do have constraints, e.g., – Any variable denoting the “number of objects” in a system can only be useful if it is less than the number of elementary particles in the known universe! – In practice, though, answers that make good sense in terms of the underlying physical or economic problem can often be obtained without putting constraints on the variables.

19. Optimization Tree

20. Lecture 1: Introduction to the Course of Optimization Some Optimization Problems

21. Shortest Path Problems

22. Some Well-Known Algorithms for Shortest Path Problems Dijkstra's algorithm — solves single source problem if all edge weights are greater than or equal to zero. Dijkstra's algorithm Bellman-Ford algorithm — solves single source problem if edge weights may be negative. Bellman-Ford algorithm A* search algorithm — solves for single source shortest paths using heuristics to try to speed up the search. A* search algorithm Floyd-Warshall algorithm — solves all pairs shortest paths. Floyd-Warshall algorithm Johnson's algorithm — solves all pairs shortest paths, may be faster than Floyd-Warshall on sparse graphs. Johnson's algorithm Perturbation theory — finds the locally shortest path. Perturbation theory

23. Maximal Flow Problems

25. Hitchcock Transportation Problems SupplyDemand 1 1 2 2 3 3 4 4 1 1 2 2 3 3 4 4 5 5 5 10 8 7 2 12 5 9 2 Demand Supply Transportation Cost

26. Hitchcock Transportation Problems SupplyDemand 1 1 2 2 3 3 4 4 1 1 2 2 3 3 4 4 5 5 5 10 8 7 2 12 5 9 2 Demand Supply Transportation Cost ? Find a minimal cost transportation.

27. Hitchcock Transportation Problems Find a minimal cost transportation. A problem’s argument Supply Demand Supply Demand An answer

28. Hitchcock Transportation Problems Find a minimal cost transportation. A problem’s argument Supply Demand Supply Demand An answer What constraint we should have?

29. Hitchcock Transportation Problems Subject to

30. Task Assignment Problems Example: Machineco has four jobs to be completed. Each machine must be assigned to complete one job. The time required to setup each machine for completing each job is shown in the table below. Machinco wants to minimize the total setup time needed to complete the four jobs. Problem Instance Time (Hours) Job1Job2Job3Job4 Machine 114587 Machine 221265 Machine 37839 Machine 424610

31. Task Assignment Problems Problem Instance Time (Hours) Job1Job2Job3Job4 Machine 114587 Machine 221265 Machine 37839 Machine 424610 Answer Time (Hours) Job1Job2Job3Job4 Machine 10100 Machine 21000 Machine 30010 Machine 40001

32. Answer Time (Hours) Job1Job2Job3Job4 Machine 10100 Machine 21000 Machine 30010 Machine 40001 Task Assignment Problems Problem Instance Time (Hours) Job1Job2Job3Job4 Machine 114587 Machine 221265 Machine 37839 Machine 424610 Answer Time (Hours) Job1Job2Job3Job4 Machine 1 Machine 2 Machine 3 Machine 4 x 11 x 12 x 13 x 14 x 21 x 22 x 23 x 24 x 31 x 32 x 33 x 34 x 41 x 42 x 43 x 44 x ij  {0,1}

33. Answer Time (Hours) Job1Job2Job3Job4 Machine 10100 Machine 21000 Machine 30010 Machine 40001 Task Assignment Problems Problem Instance Time (Hours) Job1Job2Job3Job4 Machine 114587 Machine 221265 Machine 37839 Machine 424610 Answer Time (Hours) Job1Job2Job3Job4 Machine 1 Machine 2 Machine 3 Machine 4 x 11 x 12 x 13 x 14 x 21 x 22 x 23 x 24 x 31 x 32 x 33 x 34 x 41 x 42 x 43 x 44 x ij  {0,1} Problem Instance Time (Hours) Job1Job2Job3Job4 Machine 1 Machine 2 Machine 3 Machine 4 c 11 c 12 c 13 c 14 c 21 c 22 c 23 c 24 c 31 c 32 c 33 c 34 c 41 c 42 c 43 c 44

34. Task Assignment Problems Subject to

35. Traveling Salesman Problem (TSP)

36. How many feasible paths? n cities

37. Example

44. Bin-Packing Problems

46. Determine how to put the given objects in the least number of fixed space bins. There are many variants, such as, 3D, 2D, linear, pack by volume, pack by weight, minimize volume, maximize value, fixed shape objects, etc.

47. Bin-Packing Problems Example: suppose we need a number of pipes of different, specific lengths to plumb a house and we can buy pipe stock in 5 meter lengths. How can we cut the 5 meter pipes to waste as little as possible, i.e., to minimize the cost of pipe?