Presentation on theme: "Branch-and-Bound Technique for Solving Integer Programs"— Presentation transcript:
1 Branch-and-Bound Technique for Solving Integer Programs In this handoutBasic conceptsIntroducing the technique based on an example
2 Basic ConceptsThe basic concept underlying the branch-and-bound techniqueis to divide and conquer.Since the original “large” problem is hard to solve directly,it is divided into smaller and smaller subproblemsuntil these subproblems can be conquered.The dividing (branching) is done by partitioningthe entire set of feasible solutions into smaller and smaller subsets.The conquering (fathoming) is done partially by(i) giving a bound for the best solution in the subset;(ii) discarding the subset if the bound indicates thatit can’t contain an optimal solution.These three basic steps – branching, bounding, and fathoming –are illustrated on the following example.
4 Utilizing the information about the optimal solution of the LP-relaxation Fact: If LP-relaxation has integral optimal solution x*, then x* is optimal for IP too.In our case, (x1, x2) = (2.25, 3.75) is the optimal solution of the LP-relaxation. But, unfortunately, it is not integral.The optimal value isFact: OPT(LP-relaxation) ≥ OPT(IP)(for maximization problems)That is, the optimal value of the LP-relaxationis an upper boundfor the optimal value of the integer program.In our case, is an upper bound for OPT(IP).
5 Branching step In an attempt to find out more about the location of the IP’s optimal solution,partition the feasible region of the LP-relaxation.Choose a variable that is fractional in the optimal solution to the LP-relaxation – say, x2 . Observe that every feasible IP point must have either x2 3 or x2 ≥ 4 .With this in mind, branch on the variable x2 to create the following two subproblems:Subproblem 1 Subproblem 2Max Z = 5x1 + 8x2 Max Z = 5x1 + 8x2s.t. x1 + x2 6 s.t. x1 + x2 65x1 + 9x2 x1 + 9x2 45x2 x2 ≥ 4x1 , x2 ≥ x1 , x2 ≥ 0Solve both subproblems(note that the original optimal solution (2.25, 3.75) can’t recur)
6 Branching step (graphically) Z=41Subproblem 1: Opt. solution (3,3) with value 39Subproblem 2: Opt. solution (1.8,4) with value 415Subproblem 24(1.8, 4)3(3, 3)2Subproblem 11Z=39Z=2012345786
7 Solution treeS1: x2 3(3, 3)Z=39For each subproblem, we recordthe restriction that creates the subproblemthe optimal LP solutionthe LP optimum valueThe optimal solution for Subproblem 1 is integral: (3, 3).If further branching on a subproblem will yield no useful information, then we can fathom (dismiss) the subproblem.In our case, we can fathom Subproblem 1 because its solution is integral.The best integer solution found so far is stored as incumbent. The value of the incumbent is denoted by Z*.In our case, the first incumbent is (3, 3), and Z*=39.Z* is a lower bound for OPT(IP): OPT(IP) ≥ Z* .In our case, OPT(IP) ≥ 39. The upper bound is 41: OPT(IP) 41.int.All(2.25, 3.75)Z=41.25S2: x2 ≥ 4(1.8, 4)Z=41
8 Next branching step (graphically) - Fathom Subproblem 1.- Branch Subproblem 2 on x1 :Subproblem 3: New restriction is x1 1.Opt. solution (1, 4.44) with value 40.55Subproblem 4: New restriction is x1 ≥ 2.The subproblem is infeasible5(1, 4.44)Subpr. 4Subpr. 343Z=40.55Z=202112345786
9 Solution tree (cont.) int. S1: x2 3(3, 3)Z=39int.If a subproblem is infeasible, then it is fathomed.In our case, Subproblem 4 is infeasible; fathom it.The upper bound for OPT(IP) is updated: 39 OPT(IP) Next branch Subproblem 3 on x2 .(Note that the branching variable might recur).All(2.25, 3.75)Z=41.25S3: x1 1(1, 4.44)Z=40.55S2: x2 ≥ 4(1.8, 4)Z=41S4: x1 ≥ 2infeasible
10 Next branching step (graphically) Branch Subproblem 3 on x2 :Subproblem 5: New restriction is x2 4.Feasible region:the segment joining (0,4) and (1,4)Opt. solution (1, 4) with value 37Subproblem 6: New restriction is x2 ≥ 5.Feasible region is just one point: (0, 5)Opt. solution (0, 5) with value 40(0, 5)54(1, 4)3Z=202112345786
11 Solution tree (final) int. int. S1: x2 3(3, 3)Z=39int.S5: x2 4(1, 4)Z=37If the optimal value of a subproblem is Z*, then it is fathomed.In our case, Subproblem 5 is fathomed because 37 39 = Z*.If a subproblem has integral optimal solution x*,and its value > Z*, then x* replaces the current incumbent.In our case, Subproblem 5 has integral optimal solution, and its value >39=Z*. Thus, (0,5) is the new incumbent, and new Z*=40.If there are no unfathomed subproblems left, then the current incumbent is an optimal solution for (IP).In our case, (0, 5) is an optimal solution with optimal value 40.int.All(2.25, 3.75)Z=41.25S3: x1 1(1, 4.44)Z=40.55S2: x2 ≥ 4(1.8, 4)Z=41S6: x2 ≥ 5(0, 5)Z=40int.S4: x1 ≥ 2infeasible
12 In the next handout Summary of branch-and-bound for integer programs Which variable to branch on?Which open subproblem to solve first?Updating the bounds (lower and upper) on OPT(IP)Summary of fathoming criteriaOptimality testBranch-and-bound applied tobinary integer programsmixed integer programs