2 model 1 will be solved faster because it has fewer

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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 region. 21 Why integer programs? Advantages of restricting variables to take on integer values – 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 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 problem (COP). When we form the integer program (IP), we usually want the following: 1. 2. 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 programs. 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 friends. 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 optimization problem whose solution solves the problem in the fewest moves. 31 On forming Integer programs 1. First select the set of decision variabl...
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