Integer Programming. Wolfram Wiesemann. December 6, 2007

Transcription

1 Integer Programming Wolfram Wiesemann December 6, 2007

2 Contents of this Lecture Revision: Mixed Integer Programming Problems Branch & Bound Algorithms: The Big Picture Solving MIP s: Complete Enumeration Divide and Conquer Principle Branch & Bound Algorithm for MIP s Example

3 Revision: Mixed Integer Programming Problems Mixed Integer Programming (MIP) Problem: min x 0 = c T x subject to Ax = b x j 0 for j N = {1,...,n} x j Z for j Z N. Note: x j N \ Z are continuous, x j Z are integral. Additional assumption this lecture: Finite bounds x j, x j for j Z: x j { x j,x j + 1,...,x j }.

4 Contents of this Lecture Revision: Mixed Integer Programming Problems Branch & Bound Algorithms: The Big Picture Solving MIP s: Complete Enumeration Divide and Conquer Principle Branch & Bound Algorithm for MIP s Example

5 Solving MIP s: Complete Enumeration Idea: Loop through all possible values of the integer variables (without loss of generality, {x 1,...,x z } with z n) and solve LP problems in the remaining (continuous) variables: for x 1 {x 1, x 1 + 1,..., x 1 } do for x 2 {x 2, x 2 + 1,..., x 2 } do... for x z {x z, x z + 1,..., x z } do Solve LP in x k+1,...,x n with x 1,...,x k fixed. Update tentative optimal solution if necessary. end for... end for end for Print (final) optimal solution or report infeasibility. Complexity?

6 Contents of this Lecture Revision: Mixed Integer Programming Problems Branch & Bound Algorithms: The Big Picture Solving MIP s: Complete Enumeration Divide and Conquer Principle Branch & Bound Algorithm for MIP s Example

7 Divide and Conquer Principle Need to structure search so that we touch only few solutions. One Approach: Divide and Conquer Divide a large problem into several smaller ones. Conquer by working on the smaller problems. Branch & Bound: Solve continuous relaxation of original problem P 0 x (P 0 ). Divide (Branch): Choose p Z with xp / Z. Create two subproblems, P 1 and P 2, with added constraints x p xp and x p xp, respectively. Conquer (Bound/Fathom): If optimal solution of continous relaxation of P i is worse than any known feasible solution for P 0, disregard P i. Note: Any solution to P 0 is also feasible for either P 1 or P 2. Hence, by solving P 1 and P 2, we solve P 0.

8 Divide and Conquer Principle Recursive application of divide and conquer principle leads to binary tree: Terminal nodes = problems that remain to be solved.

9 Contents of this Lecture Revision: Mixed Integer Programming Problems Branch & Bound Algorithms: The Big Picture Solving MIP s: Complete Enumeration Divide and Conquer Principle Branch & Bound Algorithm for MIP s Example

10 Branch & Bound Algorithm for MIP s Preliminaries: P 0 denotes original problem: subject to Ax = b x j 0 min x 0 = c T x for j N = {1,...,n} x j Z for j Z N. For any problem P, x (P) denotes optimal solution for continuous relaxation of P. OPT denotes objective function value of best feasible solution (for P 0 ) found so far. At beginning, OPT = or based on a priori knowledge (heuristic).

11 Branch & Bound Algorithm for MIP s Algorithm: 1. Initialization. Set list of problems to {P0 }. Initialize OPT. Solve LP relaxation of P0 x (P 0 ). If x (P 0 ) feasible for P 0, OPT = c T x (P 0 ) and stop. 2. Problem Selection. Choose a problem P from list whose x (P) has c T x (P) < OPT. If no such P exists, stop. 3. Variable Selection. Choose x p Z with x p(p) / Z. 4. Branching. Create two new problems P and P with x p x p (P) and x p x p (P), respectively. Solve continuous relaxations of P and P x (P ), x (P ). Update OPT: If P feasible, x (P ) feasible for P 0 and c T x (P ) < OPT OPT = c T x (P ). Same for P. Further Inspection: If P feasible and c T x (P ) < OPT add P to list of problems. Same for P. Afterwards, go back to (2).

12 Branch & Bound Algorithm for MIP s Output: OPT = : P 0 is infeasible. OPT < : P 0 is feasible. OPT = optimal objective value. Optimal Solution: Obtained via slight modification Store vector x for best feasible solution (for P 0 ) found so far. Whenever OPT is updated (Steps 1+4), also update x. Termination: Under assumption of finite bounds x j, x j for j Z, algorithm terminates in finitely many steps.

13 Contents of this Lecture Revision: Mixed Integer Programming Problems Branch & Bound Algorithms: The Big Picture Solving MIP s: Complete Enumeration Divide and Conquer Principle Branch & Bound Algorithm for MIP s Example

14 Example Assume the following problem is given: max 2x 1 + 3x 2 + x 3 + 2x 4 subject to 5x 1 + 2x 2 + x 3 + x x 1 + 6x x 3 + 8x 4 60 x 1 + x 2 + x 3 + x 4 8 2x 1 + 2x 2 + 3x 3 + 3x The bounds are x 1 [0,3], x 2 [0,7], x 3 [0,5] and x 4 [0,5]. Furthermore, x j Z for all j = 1,...,4.

15 Example Change to minimization objective (not necessary!): min 2x 1 3x 2 x 3 2x 4 subject to 5x 1 + 2x 2 + x 3 + x x 1 + 6x x 3 + 8x 4 60 x 1 + x 2 + x 3 + x 4 8 2x 1 + 2x 2 + 3x 3 + 3x x 1 [0,3], x 2 [0,7], x 3 [0,5] and x 4 [0,5]. x j Z for all j = 1,...,4.

16 Example 1. Initialization. Set list of problems to {P0 }. Initialize OPT. Solve LP relaxation of P0 x (P 0 ). If x (P 0 ) feasible for P 0, OPT = c T x (P 0 ) and stop. Problem list: {P 0 }, OPT =.

17 Example 1. Problem Selection. Choose a problem P from list whose x (P) has c T x (P) < OPT. If no such P exists, stop. 2. Variable Selection. Choose x p Z with xp (P) / Z. 3. Branching. Create two new problems P and P with x p xp (P) and x p xp (P), respectively. Problem list: {P 1,P 2 }, OPT =.

18 Example 1. Branching. Solve continuous relaxations of P and P x (P ), x (P ). Update OPT: If P feasible, x (P ) feasible for P 0 and c T x (P ) < OPT OPT = c T x (P ). Same for P. Further Inspection: If P feasible and c T x (P ) < OPT add P to list of problems. Same for P. Problem list: {P 1 }, OPT = 18.

19 Example 1. Problem Selection. Choose a problem P from list whose x (P) has c T x (P) < OPT. If no such P exists, stop. 2. Variable Selection. Choose x p Z with x p(p) / Z. 3. Branching. Create two new problems P and P with x p x p (P) and x p x p (P), respectively. Problem list: {P 3,P 4 }, OPT = 18.

20 Example 1. Branching. Solve continuous relaxations of P and P x (P ), x (P ). Update OPT: If P feasible, x (P ) feasible for P 0 and c T x (P ) < OPT OPT = c T x (P ). Same for P. Further Inspection: If P feasible and c T x (P ) < OPT add P to list of problems. Same for P. Problem list: {P 3,P 4 }, OPT = 18.

21 Example 1. Problem Selection. Choose a problem P from list whose x (P) has c T x (P) < OPT. If no such P exists, stop. 2. Variable Selection. Choose x p Z with xp (P) / Z. 3. Branching. Create two new problems P and P with x p xp (P) and x p xp (P), respectively. Problem list: {P 4,P 5,P 6 }, OPT = 18.

22 Example 1. Branching. Solve continuous relaxations of P and P x (P ), x (P ). Update OPT: If P feasible, x (P ) feasible for P 0 and c T x (P ) < OPT OPT = c T x (P ). Same for P. Further Inspection: If P feasible and c T x (P ) < OPT add P to list of problems. Same for P. Problem list: {P 4 }, OPT = 21.

23 Example 1. Problem Selection. Choose a problem P from list whose x (P) has c T x (P) < OPT. If no such P exists, stop. 2. Variable Selection. Choose x p Z with xp (P) / Z. 3. Branching. Create two new problems P and P with x p xp (P) and x p xp (P), respectively. Problem list: {P 7,P 8 }, OPT = 21.

24 Example 1. Branching. Solve continuous relaxations of P and P x (P ), x (P ). Update OPT: If P feasible, x (P ) feasible for P 0 and c T x (P ) < OPT OPT = c T x (P ). Same for P. Further Inspection: If P feasible and c T x (P ) < OPT add P to list of problems. Same for P. Problem list: {}, OPT = 21. Done; x = (0,7,0,0).