Today s class Constrained optimization Linear programming 1
Midterm Exam 1 Count: 26 Average: 73.2 Median: 72.5 Maximum: 100.0 Minimum: 45.0 Standard Deviation: 17.13 Numerical Methods Fall 2011 2
Optimization General optimization problem: p 3
Linear Programming General linear programming problem: 4
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
Linear Programming Example: 6
Linear Programming 7
Linear Programming 8
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
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
Possible outcomes Alternate solutions 11
Possible outcomes No feasible solution 12
Possible outcomes Unbounded problems 13
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
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
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
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
Start by setting n-m of the variables to zero. An obvious starting point is the original nonslack variables 18
Keep a table that summarizes and tracks all the variables Objective function Constraint functions 19
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
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
Numerical Methods Fall 2010 Lecture 11 22 Prof. Lei Wang ECE, UConn
Numerical Methods Fall 2010 Lecture 11 23 Prof. Lei Wang ECE, UConn
-11 Numerical Methods Fall 2010 Lecture 11 24 Prof. Lei Wang ECE, UConn
-11 Numerical Methods Fall 2010 Lecture 11 25 Prof. Lei Wang ECE, UConn
Numerical Methods Fall 2010 Lecture 11 26 Prof. Lei Wang ECE, UConn
Numerical Methods Fall 2010 Lecture 11 27 Prof. Lei Wang ECE, UConn
Numerical Methods Fall 2010 Lecture 11 28 Prof. Lei Wang ECE, UConn
Numerical Methods Fall 2010 Lecture 11 29 Prof. Lei Wang ECE, UConn
Numerical Methods Fall 2010 Lecture 11 30 Prof. Lei Wang ECE, UConn
Numerical Methods Fall 2010 Lecture 11 31 Prof. Lei Wang ECE, UConn
Numerical Methods Fall 2010 Lecture 11 32 Prof. Lei Wang ECE, UConn
Numerical Methods Fall 2010 Lecture 11 33 Prof. Lei Wang ECE, UConn
Numerical Methods Fall 2010 Lecture 11 34 Prof. Lei Wang ECE, UConn
Numerical Methods Fall 2010 Lecture 11 35 Prof. Lei Wang ECE, UConn
Numerical Methods Fall 2010 Lecture 11 36 Prof. Lei Wang ECE, UConn
Numerical Methods Fall 2010 Lecture 11 37 Prof. Lei Wang ECE, UConn
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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
Next class Numerical Integration Read Chapter PT6 and 21 42