Gomory cuts

Definition

Gomory cuts are a specific type of cutting plane used in integer programming to help solve linear programming problems that have integer constraints. They are derived from the concept of adding linear inequalities to eliminate fractional solutions from the feasible region, thus guiding the search for integer solutions more efficiently. By strategically cutting off parts of the feasible region, Gomory cuts enhance the ability to find optimal solutions in mixed-integer linear programming problems.

5 Must Know Facts For Your Next Test

1. Gomory cuts were introduced by Ralph Gomory in the early 1960s as a way to solve integer programming problems more effectively.
2. These cuts can be generated from a basic feasible solution of the linear relaxation of an integer programming problem by analyzing the tableau.
3. Gomory cuts are particularly useful because they can provide valid inequalities that tighten the feasible region without excluding any integer feasible solutions.
4. In practice, using Gomory cuts can significantly reduce the number of nodes explored in a branch-and-bound algorithm for integer programming.
5. They are one of several types of cutting planes used in conjunction with other techniques to improve the efficiency of solving integer programs.

Review Questions

• How do Gomory cuts improve the process of finding optimal solutions in integer programming?
  • Gomory cuts enhance the search for optimal solutions by eliminating fractional solutions from the feasible region, thereby tightening the boundaries within which an integer solution can exist. This is accomplished through the addition of linear inequalities that cut off non-integer points while preserving all valid integer points. As a result, Gomory cuts help to focus the search process, leading to potentially quicker convergence on an optimal integer solution.
• Discuss the importance of Gomory cuts in relation to other methods used in integer programming, such as branch-and-bound.
  • Gomory cuts play a significant role when used alongside methods like branch-and-bound in solving integer programming problems. While branch-and-bound systematically explores possible solutions by dividing them into smaller subproblems, Gomory cuts reduce the size of these subproblems by eliminating non-integer solutions from consideration. This combination allows for a more efficient exploration of the solution space, improving computational performance and effectiveness in reaching optimal solutions.
• Evaluate the implications of using Gomory cuts on computational efficiency and solution quality in complex integer programming scenarios.
  • Using Gomory cuts can lead to substantial improvements in both computational efficiency and solution quality when dealing with complex integer programming problems. By narrowing down the feasible region and reducing the number of candidate solutions that need to be evaluated, Gomory cuts minimize computation time and resource usage. Additionally, these cuts ensure that only relevant integer solutions are considered, which enhances solution quality by focusing on viable options and ultimately leading to faster convergence towards optimal results.

Related terms

Integer Programming: A mathematical optimization technique where some or all of the variables are restricted to be integers.

Cutting Plane Method: An optimization technique that iteratively refines the feasible region of a linear programming problem by adding linear constraints (cuts).

Linear Programming: A method for optimizing a linear objective function, subject to linear equality and inequality constraints.