Feasible region

Definition

A feasible region is the set of all possible solutions to a constrained optimization problem that satisfy all given constraints. This region represents the limits within which the optimal solution must lie and is typically visualized as a bounded or unbounded area on a graph, depending on the nature of the constraints. The points within this region can be evaluated to find the best solution to the problem, often defined by an objective function.

5 Must Know Facts For Your Next Test

1. The feasible region is defined by the intersection of all constraints, which can be linear inequalities in linear programming scenarios.
2. Points on the boundary of the feasible region are critical because they are potential candidates for optimal solutions.
3. If there are no constraints, the feasible region can be unbounded, indicating that solutions can extend infinitely.
4. In linear programming, if the feasible region is empty, it means that there are no solutions that satisfy all constraints simultaneously.
5. The optimal solution for many problems in linear programming will occur at one of the vertices of the feasible region.

Review Questions

• How does the concept of a feasible region influence the search for an optimal solution in constrained optimization problems?
  • The feasible region is crucial because it delineates the area where all potential solutions must lie based on the given constraints. Since only points within this region are considered valid solutions, it directly impacts where one can search for the optimal solution. Understanding this area helps identify which combinations of variables will yield the best results according to the objective function.
• Discuss how changes to constraints affect the shape and size of the feasible region in a linear programming problem.
  • When constraints are modified, either by changing their coefficients or altering their bounds, it can lead to significant changes in the feasible region's shape and size. For example, tightening a constraint could shrink the feasible region, potentially eliminating some previously viable solutions. Conversely, loosening a constraint may expand the region. These adjustments can shift where optimal solutions are located and may even change whether an optimal solution exists at all.
• Evaluate a scenario where a feasible region is unbounded. What implications does this have for finding an optimal solution in linear programming?
  • In cases where the feasible region is unbounded, it indicates that there are potentially infinite solutions available that meet the constraints. This scenario often arises when no upper limit restricts one or more variables. While having an unbounded feasible region means there might be many solutions, it can also suggest that an optimal solution may not exist if it allows for continuous improvement of the objective function without reaching a maximum or minimum value. Therefore, identifying whether an unbounded region leads to an optimal solution requires careful analysis of how the objective function interacts with those variables.

Related terms

Objective function: The function that needs to be maximized or minimized in an optimization problem, representing a quantity such as profit, cost, or distance.

Constraints: Conditions or restrictions placed on the variables in an optimization problem that define the feasible region.

Linear programming: A mathematical method for determining a way to achieve the best outcome in a given mathematical model, often used to find the optimal solution within a feasible region.