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Integer Programs - Discrete Modeling and Optimization - Lecture Slides, Slides of Discrete Structures and Graph Theory

Aliah UniversityDiscrete Structures and Graph Theory

The key points in these lecture slides, which are core of the discrete modeling and optimization are:Integer Programs, Branch-And-Bound, Binary Integer Programs, Fathoming Criteria, Optimality Test, Steps of Algorithm, Lower and Upper Bounds, Current Iteration, Heuristic Algorithm, Traveling Salesman Problem

Typology: Slides

2012/2013

Uploaded on 04/27/2013

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Branch-and-Bound
In this handout,
Summary of branch-and-bound for integer programs
•Updating the lower and upper bounds for OPT(IP)
•Summary of fathoming criteria
•Which variable to branch on?
•Which open subproblem to solve first?
•Optimality test
Branch-and-bound applied to
•binary integer programs
•mixed integer programs
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Download Integer Programs - Discrete Modeling and Optimization - Lecture Slides and more Slides Discrete Structures and Graph Theory in PDF only on Docsity!

Branch-and-Bound

In this handout,

Summary of branch-and-bound for integer programs

  • Updating the lower and upper bounds for OPT(IP)
  • Summary of fathoming criteria
  • Which variable to branch on?
  • Which open subproblem to solve first?
  • Optimality test

Branch-and-bound applied to

  • binary integer programs
  • mixed integer programs

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Examples of the main steps of the algorithm will be

given based on the following solution tree

  • Z=75. All - S
    • Z=68. - S
      • Z=70. - S - Z=60. - S - S Infeas. - Z=68. - S - Z=64. - S - Z= - S - Z=54. - S - Z=66. - S - Z= - S - Z=62. - F - F int - F - F int

Summary of fathoming criteria

Fathom a subproblem if at least one of the following is true:

 Criterion F1:

The LP optimal value of the subproblem is ≤ Z*,

where Z* is the optimal value of the current incumbent.

In our example, S8 is fathomed based on F1 (54.4 < Z*=59).

 Criterion F2: The subproblem is infeasible.

In our example, S4 is fathomed based on F2.

 Criterion F3:

The optimal solution of the subproblem is integral.

In our example, S7 and S10 are fathomed based on F3.Docsity.com

Which open (unfathomed)

subproblem to solve first?

Possible natural choices:

Option 1 : The subproblem with the best LP value.

This subproblem is the most promising one

 to create an incumbent with higher Z* value;  to contain an optimal IP solution.

Option 2 : The most recently created subproblem.

We don’t need to start the simplex method from scratch to

solve this subproblem; it is solved by reoptimizing the

solution of its parent-subproblem.

The second option is normally the preferred choice because it

makes the algorithm more time-efficient. Docsity.com

Finding a first incumbent quickly

  • Recall that having an incumbent allows us

to fathom subproblems (Criterion F1).

  • But it might take many iterations until

branch-and-bound finds a first incumbent

(a subproblem which has an integer LP solution).

  • To accelerate the process,

a first incumbent is often found

by applying a fast heuristic algorithm to the problem.

  • For example, to solve the Traveling Salesman

Problem by Branch-and-Bound,

we can start by applying the Nearest Neighbor

algorithm to find a first incumbent.Docsity.com

How the upper bound on OPT(IP) can be used

  • In practice, branch-and-bound is pretty fast most of the time.
  • But sometimes it might get really slow.

(In the worst-case scenario it is still exponential-time.)

  • What can we do if the branch-and-bound couldn’t find an optimal solution after struggling several hours on the problem?
  • Recall the tightest lower and upper bounds on OPT(IP) in the current iteration: Z* ≤ OPT(IP) ≤ Z’.
  • What can we say about the current incumbent based on these bounds? Let k = Z’ / Z*. Then

That is, the current incumbent is at most k times worse than the optimal solution. If k is close to 1, then the current incumbent will be a pretty good solution.

E.g., if Z*=100, Z’=102, then the current incumbent is

at most 1.02 times worse than the optimal solution.

* Z Z OPT IP

Z

Z

k ⋅ Z = ⋅ = ≥

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