Linear Programming Relaxation

Definition

Linear programming relaxation is a technique used to simplify integer programming problems by allowing the decision variables that are normally constrained to take on integer values to instead take on continuous values. This transformation makes the problem easier to solve since linear programs can be solved efficiently using polynomial-time algorithms. The solution obtained from the relaxed model provides a bound for the original integer problem, helping to inform the search for the optimal integer solution in more complex optimization frameworks.

5 Must Know Facts For Your Next Test

1. Linear programming relaxation transforms an integer programming problem into a linear programming problem by relaxing the integer constraints on the decision variables.
2. The optimal solution from the relaxed problem is often used as a benchmark for evaluating the quality of solutions found in the original integer programming problem.
3. When using branch and bound, solving the linear programming relaxation at each node helps identify whether to continue exploring that branch or prune it.
4. Relaxation can lead to fractional values for decision variables, which means additional steps may be needed to round these values to find a feasible integer solution.
5. The gap between the objective function value of the relaxed problem and that of the original integer problem is known as the 'relaxation gap,' which can indicate how far off the solution is.

Review Questions

• How does linear programming relaxation help in solving integer programming problems more effectively?
  • Linear programming relaxation helps in solving integer programming problems by allowing decision variables to take continuous values instead of being restricted to integers. This change simplifies the problem, making it possible to use efficient linear programming algorithms that can find optimal solutions quickly. The results from this relaxed model serve as bounds for the original problem, guiding further search strategies such as branch and bound.
• What role does linear programming relaxation play within the branch and bound algorithm?
  • Within the branch and bound algorithm, linear programming relaxation is crucial for determining whether to explore or prune specific branches of the solution tree. By solving the relaxed problem at each node, we can assess its potential for yielding an optimal integer solution. If the relaxed solution provides a lower bound that does not improve upon existing known solutions, that branch can be pruned, thus optimizing computational efforts and focusing resources on more promising branches.
• Evaluate the importance of understanding relaxation gaps when applying linear programming relaxation to solve optimization problems.
  • Understanding relaxation gaps is vital when applying linear programming relaxation because it quantifies how much room there is between the optimal solution of the relaxed problem and that of the original integer program. A smaller gap indicates that solutions from the relaxed model are close to what could be achieved with integer constraints, suggesting that finding an optimal integer solution will be more feasible. Conversely, a larger gap might signal that additional strategies, such as rounding or heuristic methods, may be necessary to approach a satisfactory solution.

Related terms

Integer Programming: A mathematical optimization technique where some or all of the decision variables are required to take on integer values.

Branch and Bound: An algorithmic technique for solving optimization problems, which involves partitioning the feasible region into smaller subproblems and systematically exploring them.

Feasible Region: The set of all possible solutions that satisfy the constraints of a linear programming problem.