Introduction to Mixed Integer Linear Programming

2

LP formulation of Economic Dispatch

© 2011 D. Kirschen & the University of Washington

1 2 3 L

x1P1MAX

x2P2MAX

x3P3MAX

P1MIN

P2MIN

P3MIN

• Objective function is linear • All constraints are linear• All variables are real• Problem can be solved using

standard linear programming

3

Can we use LP for unit commitment?

© 2011 D. Kirschen & the University of Washington

x1P1MAX

x2P2MAX

x3P3MAX

P1MIN

P2MIN

P3MIN

The variables no longer havea contiguous domain (Non-convex set)Standard linear programming is no longer applicable

4

Mixed Integer Linear Programming (MILP)

• Some decision variables are integers– Special case: binary variables {0,1}

• Other variables are real• Objective function and constraints are linear

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5

Example

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Except for the fact that the variables are integer, this looksvery much like a linear programming problem.

6

Example

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4x1 + 2x2 = 15

x1 + 2x2 = 8

x1 + x2 = 5

x1

x28

6

4

2

0 2 4 6 8

7

LP relaxation

© 2011 D. Kirschen & the University of Washington

4x1 + 2x2 = 15

x1 + 2x2 = 8

x1 + x2 = 5

x1

x28

6

4

2

0 2 4 6 8

Let us relax the constraint that thevariables must be integer.

The problem is then a regular LP

Solution of the relaxed LPx1 = 2.5; x2 = 2.5; Objective = 12.5

8

LP relaxation

© 2011 D. Kirschen & the University of Washington

4x1 + 2x2 = 15

x1 + 2x2 = 8

x1 + x2 = 5

x1

x28

6

4

2

0 2 4 6 8

Solution of the relaxed LPx1 = 2.5; x2 = 2.5; Objective = 12.5

The solution of the relaxed problemis always better than the solution oforiginal problem!(Lower objective for minimizationproblem, higher for maximization)

9

Solution of the integer problem

© 2011 D. Kirschen & the University of Washington

4x1 + 2x2 = 15

x1 + 2x2 = 8

x1 + x2 = 5

x1

x28

6

4

2

0 2 4 6 8

Solution of the relaxed LPx1 = 2.5; x2 = 2.5; Objective = 12.5

10

Solution of the integer problem

© 2011 D. Kirschen & the University of Washington

4x1 + 2x2 = 15

x1 + 2x2 = 8

x1 + x2 = 5

x1

x28

6

4

2

0 2 4 6 8

Solution of the relaxed LPx1 = 2.5; x2 = 2.5; Objective = 12.5

Solution of the original problemx1 = 2; x2 = 3; Objective = 12.0

11

Naïve rounding off

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x1

x2

LP solutionIP solution

The optimal integer solution can be far away from the LP solution“Far away” can be difficult to find when there are many dimensions

12

Finding the integer solution

• Large number of integer variables• Vast number of possible integer solutions• Need a systematic procedure to search this

solution space• Fix the variables to the nearest integer one at

a time• “Branch and Bound” algorithm

© 2011 D. Kirschen & the University of Washington

13

Another example

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Relaxed LP solution: (1.75, 0.75)

14

Branch on x1

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Problem 0

Problem 2Problem 1

15

Branch on x1

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Problem 2Problem 1

Solution of Problem 2Solution of Problem 1

16

Search Tree: first layer

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Solution of Problem 1:• x1 integer• x2 real• Not a feasible solution yet• Can still branch on x2

Solution of Problem 2:• x1 & x2 integer• Feasible solution of

the original problem• Bound on the optimum• Best solution so far

17

Branch on x2

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Problem 1

Problem 3 Problem 4

18

Search Tree: second layer

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No solutionNo integer solution yet

19

Branch and Bound: what next?

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No solution

Can’t go any further in this direction

Solution of relaxed problem 4 is bounded by solution of problem 2. No point in going further

Optimal solution

20

Comments on Branch and Bound

• Search tree gets very big if there are more than a few integer or binary variables

• Even with the bounds provided by the relaxed solutions, exploring the tree usually takes a ridiculous amount of time

• Clever mathematicians have developed techniques to identify “cuts”– Constraints based on the structure of the problem

that eliminate part of the search tree– “Branch and Cut” algorithm

© 2011 D. Kirschen & the University of Washington

21

Duality Gap

• Finding the optimal solution for a large problem can take too much time even with branch and cut

• Best solution of relaxed problem provides a bound on the solution

• Duality gap: Difference between best solutions of relaxed problem and actual problem

• Stop searching the tree if duality gap is sufficiently small

© 2011 D. Kirschen & the University of Washington