Today s class. Constrained optimization Linear programming. Prof. Jinbo Bi CSE, UConn. Numerical Methods, Fall 2011 Lecture 12

Transcription

1 Today s class Constrained optimization Linear programming 1

2 Midterm Exam 1 Count: 26 Average: 73.2 Median: 72.5 Maximum: Minimum: 45.0 Standard Deviation: Numerical Methods Fall

3 Optimization General optimization problem: p 3

4 Linear Programming General linear programming problem: 4

5 Linear Programming Graphical solutions Limited to two or three dimensions Graph the constraints to determine feasible solution space and then map the objective function on to this space 5

6 Linear Programming Example: 6

7 Linear Programming 7

8 Linear Programming 8

9 Linear Programming Solution at x 1 = 44/9, x 2 = 35/9 Constraints 1 and 2 are binding Constraint 3 is redundant Constraints 4, 5, and 6 are nonbinding 9

10 Linear Programming Possible outcomes Unique solution Alternate solutions Possible if objective function is parallel to one of the constraints No feasible solution Problem is over-constrained Unbounded problems Problem is under-constrained 10

11 Possible outcomes Alternate solutions 11

12 Possible outcomes No feasible solution 12

13 Possible outcomes Unbounded problems 13

14 Extreme points Unique solutions occur at the extreme convex corner points of the feasible solution space The extreme points occur at the intersection of various constraints Since the optimal solution occurs just at the extreme point, we can simplify our search by just looking at extreme points Not every extreme point is feasible 14

15 Transform problem into normal form - i.e. only equality constraints and nonnegativity constraints for all decision variables Add slack variables The slack variables must be positive if the constraints are to be met 15

16 We have more unknowns than equations, so we can t use the linear algebraic methods we used before In general, there are m equations (not counting non-negativity constraints) and n unknowns We re looking for the extreme points In the m equality constraints, set n-m variables to zero, and then solve the m equation, m unknown linear system Problem is that you don t know which of the n-m variables to set to zero 16

17 You could try all the possibilities, but that can get out of hand as n and m get large There are possibilities Many of those choices may not even give you feasible extreme points The Simplex Method offers a more efficient way to quickly identify the feasible extreme points 17

18 Start by setting n-m of the variables to zero. An obvious starting point is the original nonslack variables 18

19 Keep a table that summarizes and tracks all the variables Objective function Constraint functions 19

20 First step is to take our initial starting point and increase one of the variables so that it is non-zero Pick the variable that is going to increase the Z objective the most This column will have the largest positive entry in the Z row. Increase that variable until one of the other values goes negative 20

21 Look at the entry at the intersection of the row and the column we chose above. This is called the pivot element If it is negative, increasing the chosen variable will only make the row basic variable bigger, so it is not restrictive If it is positive, increasing the chosen variable will make row variable smaller, so it is restrictive Divide the constant in the row by the pivot element and that will tell you how much you increase the variable before you hit a constraint restriction Choose the smallest of these values as your pivot element 21

22 Numerical Methods Fall 2010 Lecture Prof. Lei Wang ECE, UConn

23 Numerical Methods Fall 2010 Lecture Prof. Lei Wang ECE, UConn

24 -11 Numerical Methods Fall 2010 Lecture Prof. Lei Wang ECE, UConn

25 -11 Numerical Methods Fall 2010 Lecture Prof. Lei Wang ECE, UConn

26 Numerical Methods Fall 2010 Lecture Prof. Lei Wang ECE, UConn

27 Numerical Methods Fall 2010 Lecture Prof. Lei Wang ECE, UConn

28 Numerical Methods Fall 2010 Lecture Prof. Lei Wang ECE, UConn

29 Numerical Methods Fall 2010 Lecture Prof. Lei Wang ECE, UConn

30 Numerical Methods Fall 2010 Lecture Prof. Lei Wang ECE, UConn

31 Numerical Methods Fall 2010 Lecture Prof. Lei Wang ECE, UConn

32 Numerical Methods Fall 2010 Lecture Prof. Lei Wang ECE, UConn

33 Numerical Methods Fall 2010 Lecture Prof. Lei Wang ECE, UConn

34 Numerical Methods Fall 2010 Lecture Prof. Lei Wang ECE, UConn

35 Numerical Methods Fall 2010 Lecture Prof. Lei Wang ECE, UConn

36 Numerical Methods Fall 2010 Lecture Prof. Lei Wang ECE, UConn

37 Numerical Methods Fall 2010 Lecture Prof. Lei Wang ECE, UConn

38 38

39 39

40 40

41 Linear Programming Simplex Method is the best known algorithm for linear programming problems Can be exponential in the worst case but in practice, is usually polynomial Special cases of linear programming may have faster algorithms 41

42 Next class Numerical Integration Read Chapter PT6 and 21 42