Download presentation

Presentation is loading. Please wait.

Published byGregory Hallmark Modified over 2 years ago

1
1 If you still are having problems getting your program to work, send me e-mail for help. You will need to use it later in the term to start Programming Project #2 and to find answers to problems. Assignment #2 is now available: Due Thurs. Oct. 4 at the beginning of class. Mon. Oct. 8 is a holiday (Thanksgiving): late deadline is Tues. Oct. 9 at 3:30pm, you can slip late submissions under my office door.

2
Alan Turing Celebration Lecture Biological Evolution as a Form of Learning Leslie Valiant, CC and Applied Math., Harvard Wed. Oct 10, 3:30pm, ECS 124 CSC 445/545: This is the same time as our class and it is the class immediately before the midterm. If a vote is unanimous, we can reschedule the midterm.

3
The primal: Maximize c T x subject to A x ≤ b, x ≥ 0. Writing this out: Maximize c 1 x 1 + c 2 x 2 +.... + c n x n subject to a 11 x 1 + a 12 x 2 +... + a 1n x n ≤ b 1 a 21 x 1 + a 22 x 2 +... + a 2n x n ≤ b 2... a m1 x 1 + a m2 x 2 +... + a mn x n ≤ b m x 1, x 2,..., x n ≥ 0

5
Theorem: For every primal feasible solution x= (x 1, x 2,..., x n ) and for every dual feasible solution y= (y 1, y 2,..., y m ), c 1 x 1 + c 2 x 2 +.... + c n x n ≤ b 1 y 1 + b 2 y 2 +.... + b m y m That is, the objective function values for dual feasible solutions always give upper bounds on the objective function values for primal feasible solutions.

6
Proof: Because the constraints of the dual are (and x ≥ 0): a 11 y 1 + a 21 y 2 +... + a m1 y m ≥ c 1 a 12 y 1 + a 22 y 2 +... + a m2 y m ≥ c 2... a 1n y 1 + a 2n y 2 +... + a mn y m ≥ c n we have that S= (a 11 y 1 + a 21 y 2 +... + a m1 y m )x 1 + (a 12 y 1 + a 22 y 2 +... + a m2 y m )x 2 +... (a 1n y 1 + a 2n y 2 +... + a mn y m ) x n ≥ c 1 x 1 + c 2 x 2 +.... + c n x n

7
We have that S= (a 11 y 1 + a 21 y 2 +... + a m1 y m )x 1 + (a 12 y 1 + a 22 y 2 +... + a m2 y m )x 2 +... + (a 1n y 1 + a 2n y 2 +... + a mn y m ) x n ≥ c 1 x 1 + c 2 x 2 +.... + c n x n Regroup the terms on the LHS so they are in terms of x i ’s instead of y j ’s: S= (a 11 x 1 + a 12 x 2 +... + a 1n x n )y 1 + (a 21 x 1 + a 22 x 2 +... + a 2n x n )y 2 +... + (a m1 x 1 + a m2 x 2 +... + a mn x n ) y m

8
S= (a 11 x 1 + a 12 x 2 +... + a 1n x n )y 1 + (a 21 x 1 + a 22 x 2 +... + a 2n x n )y 2 +... + (a m1 x 1 + a m2 x 2 +... + a mn x n ) y m From the primal problem: a 11 x 1 + a 12 x 2 +... + a 1n x n ≤ b 1 a 21 x 1 + a 22 x 2 +... + a 2n x n ≤ b 2... a m1 x 1 + a m2 x 2 +... + a mn x n ≤ b m Therefore since y ≥0, S ≤ b 1 y 1 + b 2 y 2 +.... + b m y m

9
Summing it up, we have shown that c 1 x 1 + c 2 x 2 +.... + c n x n ≤ S ≤ b 1 y 1 + b 2 y 2 +.... + b m y m so by transitivity, this proves what we were trying to show: c 1 x 1 + c 2 x 2 +.... + c n x n ≤ b 1 y 1 + b 2 y 2 +.... + b m y m Important observation: it is critical for preserving the direction of the inequalities that the solutions x and y are feasible and hence non-negative.

11
A linear programming problem can have an optimal solution, or it can be infeasible or unbounded. Thought question: which combinations are possible?

Similar presentations

OK

Opener. Notes: 3.4 Linear Programming Optimization Many real-life problems involve a process called optimization. This means finding a maximum or.

Opener. Notes: 3.4 Linear Programming Optimization Many real-life problems involve a process called optimization. This means finding a maximum or.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Marketing ppt on product Ppt on do's and don'ts of group discussion games Ppt on solar energy class 10 Ppt on creating brand equity Ppt on van de graaff generator demonstrations Viewer ppt online training Ppt on water resources of maharashtra Best ppt on french revolution Ppt on online marketing trends Ppt on singular plural for grade 1