Download presentation

Presentation is loading. Please wait.

Published byMerryl Porter Modified over 2 years ago

1
Part II General Integer Programming II.1 The Theory of Valid Inequalities 1

2
Let S = {x Z + n : Ax b} P = {x R + n : Ax b} S = P Z n Have max{cx: x S} = max{cx: x conv(S)}. How can we construct inequalities describing conv(S)? Use integrality and valid inequalities for P to construct valid inequalities for S. Def: Valid inequalities x 0 and x 0 are said to be equivalent if ( , 0 ) = ( , 0 ) for some > 0. x 0 dominates or is stronger than x 0 if they are not equivalent and there exists > 0 such that and 0 0. A maximal valid inequality is one that is not dominated by any other inequality. A maximal inequality for S defines a nonempty face of conv(S), but not conversely. Integer Programming 2011 2

3
3

4
4

5
Integer Rounding Integer Programming 2011 5

6
6

7
Chvatal-Gomory (C-G) Rounding Method Integer Programming 2011 7

8
Optimizing over the First Chvátal closure Integer Programming 2011 8

9
9

10
If we find good but not necessarily optimal solutions to the MIP, we find very effective valid inequalities. Also heuristic methods to find good feasible solutions to the MIP are helpful. MIP model may not be intended as computational tools to solve real problems. But we can examine the strength of rank-1 C-G inequalities to describe the convex hull of S for various problems. For some structured problems, e.g. knapsack problem, the separation problem for the first Chvatal closure may have some structure which enables us to handle the problem more effectively. Integer Programming 2011 10

11
Modular Arithmetic Integer Programming 2011 11

12
Disjunctive Constraints Integer Programming 2011 12

13
Integer Programming 2011 13

14
Integer Programming 2011 14

15
Boolean Implications Integer Programming 2011 15

16
Geometric or Combinatorial Implication Integer Programming 2011 16 1 3 2 4 5 76

Similar presentations

Presentation is loading. Please wait....

OK

Operations Research Models

Operations Research Models

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Free ppt on rocks and minerals Body systems for kids ppt on batteries Ppt on number patterns in maths Convert pps file to ppt online Ppt on preservation of public property definition Flat panel display ppt on tv Ppt on group development Ppt on area of parallelogram and triangles worksheets Free video backgrounds for ppt on social media Ppt on fast food industry in india