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Unformatted text preview: nd the
same objective function.
2. Model 1 will be solved faster because it
has fewer constraints.
3. If we removed the integrality constraints
from both models, they would become
two different linear programs.
4. Model 1 has the fewest number of
constraints for an IP with this feasible
21 Why integer programs? Advantages of restricting variables to take on
– More realistic
– More flexibility Disadvantages
– More difficult to model
– Can be much more difficult to solve 22 On computation for IPs Much, much harder than solving LPs Very good solvers can solve large problems – e.g., 50,000 columns 2 million non-zeros Hard to predict what will be solved quickly and
what will take a long time. 23 Running time to optimality (CPLEX) number of columns 1,000,000 < 1 Hour > 1 hour Not yet solved 100,000 10,000 Instances are taken
from MIP Lib
1,000 10,000 100,000 number of rows 1,000,000
24 Mental Break 25 On formulating integer programs
Consider an instance of a combinatorial optimization
When we form the integer program (IP), we usually
want the following:
3. If x is feasible for the COP, then x is feasible for the IP.
If x is feasible for the IP, then x is feasible for the COP.
If x is feasible, then its objective function value is the same
for both the IP and COP. Note: We often need to add variables to the COP
(especially 0-1 variables), when formulating integer
26 Example : Maximum Clique Problem
INPUT: a friendship network
G = (N, A). If persons i and j
are friends, then (i, j) ∈ A. Decision variables FEASIBLE SOLUTION: a set
S of people such that every
pair of persons in S are
OBJECTIVE: maximize |S| 27 The Game of Fiver. Click on a circle, and
flip its color and that
of adjacent colors.
Can you make all of
the circles red? 28 The game of fiver. Click on (3, 3) 29 The game of fiver. Click on (3, 1)
Click on (4, 4) 30 The game of fiver. Next: an
the problem in
moves. 31 On forming Integer programs
1. First select the set of decision variabl...
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