0. Introduction to Mixed Integer Linear Programming

1. LP formulation of Economic Dispatchx1
P1MAX x2
P2MAX x3
P3MAX
P1MIN
P2MIN
P3MIN 1 2 3 L
Objective function is linear
All constraints are linear
All variables are real
Problem can be solved using standard linear programming
© 2011 D. Kirschen & the University of Washington

2. Can we use LP for unit commitment?x1
P1MAX x2
P2MAX x3
P3MAX
P1MIN
P2MIN
P3MIN
The variables no longer have a contiguous domain
(Non-convex set)
Standard linear programming
is no longer applicable
© 2011 D. Kirschen & the University of Washington

3. 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
© 2011 D. Kirschen & the University of Washington

4. Example
Except for the fact that the variables are integer, this looks very much like a linear programming problem.
© 2011 D. Kirschen & the University of Washington

5. Example x2 x1 8 6 4 2 4x1 + 2x2 = 15 x1 + 2x2 = 8 x1 + x2 = 5
4x1 + 2x2 = 15
x1 + x2 = 5
x1 + 2x2 = 8
© 2011 D. Kirschen & the University of Washington

6. LP relaxation x1 x2 8 6 4 2
Let us relax the constraint that the variables must be integer.
4x1 + 2x2 = 15
The problem is then a regular LP
x1 + x2 = 5
Solution of the relaxed LP
x1 = 2.5; x2 = 2.5; Objective = 12.5
x1 + 2x2 = 8
© 2011 D. Kirschen & the University of Washington

7. LP relaxation x1 x2 8 6 4 2
The solution of the relaxed problem is always better than the solution of original problem!
(Lower objective for minimization problem, higher for maximization)
4x1 + 2x2 = 15
x1 + x2 = 5
Solution of the relaxed LP
x1 = 2.5; x2 = 2.5; Objective = 12.5
x1 + 2x2 = 8
© 2011 D. Kirschen & the University of Washington

8. Solution of the integer problemx1 x2 8 6 4 2
4x1 + 2x2 = 15
x1 + x2 = 5
Solution of the relaxed LP
x1 = 2.5; x2 = 2.5; Objective = 12.5
x1 + 2x2 = 8
© 2011 D. Kirschen & the University of Washington

9. Solution of the integer problemx1 x2 8 6 4 2
4x1 + 2x2 = 15
Solution of the original problem
x1 = 2; x2 = 3; Objective = 12.0
x1 + x2 = 5
Solution of the relaxed LP
x1 = 2.5; x2 = 2.5; Objective = 12.5
x1 + 2x2 = 8
© 2011 D. Kirschen & the University of Washington

10. Naïve rounding off x2 x1 LP solution IP 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
© 2011 D. Kirschen & the University of Washington

11. 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

12. Another example Relaxed LP solution: (1.75, 0.75)
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13. Branch on x1 Problem 0 Problem 2 Problem 1
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14. Branch on x1 Problem 2 Problem 1 Solution of Problem 1
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15. Search Tree: first layer
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
© 2011 D. Kirschen & the University of Washington

16. Branch on x2 Problem 1 Problem 3 Problem 4
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17. Search Tree: second layer
No solution
No integer solution yet
© 2011 D. Kirschen & the University of Washington

18. Branch and Bound: what next?
Optimal solution
Solution of relaxed problem 4 is bounded by solution of problem 2. No point in going further
Can’t go any further in this direction
No solution
© 2011 D. Kirschen & the University of Washington

19. 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

20. 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