introduction to mixed integer linear programming

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?


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The variables no longer havea contiguous domain (Non-convex set)Standard linear programming is no longer applicable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


5

Example


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

6

Example


4x1 + 2x2 = 15

x1 + 2x2 = 8

x1 + x2 = 5

x1

x28

6

4

2

0 2 4 6 8

7

LP relaxation


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


4x1 + 2x2 = 15

x1 + 2x2 = 8

x1 + x2 = 5

x1

x28

6

4

2

0 2 4 6 8


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


4x1 + 2x2 = 15

x1 + 2x2 = 8

x1 + x2 = 5

x1

x28

6

4

2

0 2 4 6 8


10

Solution of the integer problem


4x1 + 2x2 = 15

x1 + 2x2 = 8

x1 + x2 = 5

x1

x28

6

4

2

0 2 4 6 8


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

11

Naïve rounding off


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


13

Another example


Relaxed LP solution: (1.75, 0.75)

14

Branch on x1


Problem 0

Problem 2Problem 1

15

Branch on x1


Problem 2Problem 1

Solution of Problem 2Solution of Problem 1

16

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

17

Branch on x2


Problem 1

Problem 3 Problem 4

18

Search Tree: second layer


No solutionNo integer solution yet

19

Branch and Bound: what next?


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


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


introduction to mixed integer linear programming

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