Valid inequalities

Definition

Valid inequalities are constraints added to a mathematical optimization problem that do not exclude any feasible solutions of the original problem, thus maintaining its feasible region while potentially improving the efficiency of the solution process. They play a crucial role in enhancing linear programming formulations and are particularly significant in cutting plane methods, where they help to tighten the linear relaxation of integer programming problems by cutting off non-integer solutions.

5 Must Know Facts For Your Next Test

1. Valid inequalities can help reduce the size of the solution space, making it easier and faster to find optimal solutions in optimization problems.
2. They can be derived from valid linear combinations of the original constraints or from other properties of the feasible region.
3. In cutting plane methods, valid inequalities are systematically introduced to enhance the current formulation without excluding any feasible points.
4. Adding valid inequalities does not change the optimal solution of the original problem; it only tightens the bounds on the solution space.
5. The effectiveness of valid inequalities can significantly improve the performance of branch-and-bound algorithms in integer programming.

Review Questions

• How do valid inequalities contribute to the efficiency of solving optimization problems?
  • Valid inequalities improve the efficiency of solving optimization problems by tightening the feasible region without excluding any potential solutions. This means that when additional constraints are added, they help direct the search towards optimal solutions more quickly. By reducing the number of unnecessary candidate solutions, valid inequalities enable algorithms like branch-and-bound to operate more efficiently and effectively.
• Discuss how valid inequalities relate to cutting plane methods and their application in integer programming.
  • In cutting plane methods, valid inequalities are essential as they serve as additional constraints that refine the feasible region while preserving all feasible solutions. This is particularly important in integer programming, where finding optimal solutions can be challenging due to discrete variables. By strategically adding these inequalities, cutting planes can eliminate non-integer solutions from consideration, leading to a more manageable problem that focuses only on viable candidates for integer solutions.
• Evaluate the role of valid inequalities in enhancing linear relaxations and their impact on computational results in optimization.
  • Valid inequalities play a crucial role in enhancing linear relaxations by creating tighter bounds on feasible regions which leads to improved computational results. When valid inequalities are integrated into a linear relaxation, they significantly reduce the search space for potential solutions, thereby increasing the likelihood of quickly converging on an optimal solution. Their impact is evident in practical applications where complex integer programming problems require efficient resolution strategies, demonstrating how effective formulations can lead to faster and more reliable computational outcomes.

Related terms

Cutting Plane: A method used in optimization that involves adding linear constraints (cuts) to eliminate infeasible solutions while preserving all feasible solutions.

Linear Relaxation: The process of relaxing an integer programming problem to a linear programming problem by allowing integer variables to take on continuous values.

Integer Programming:

A type of optimization problem where some or all decision variables are required to take on integer values, often used in combinatorial optimization.