Feasible Set

Definition

A feasible set is a collection of all possible solutions that satisfy a given set of constraints in optimization problems. It is crucial for determining which options are valid for achieving the best outcome, as it helps identify the limits within which solutions can exist based on restrictions or requirements. Understanding the feasible set allows analysts to explore viable solutions while ensuring that decisions align with specific criteria.

5 Must Know Facts For Your Next Test

1. The feasible set can be represented graphically, with regions marked by constraints showing where solutions are valid.
2. In linear programming, the feasible set is often a convex polytope, meaning it is shaped by linear inequalities.
3. When constraints are added or modified, the feasible set can change dramatically, affecting potential solutions.
4. Identifying the feasible set is a critical step in optimization as it directly influences the search for optimal solutions.
5. The size and shape of the feasible set can impact computational efficiency when solving optimization problems.

Review Questions

• How does the concept of a feasible set relate to the constraints imposed in an optimization problem?
  • The feasible set is directly determined by the constraints of an optimization problem. Each constraint defines boundaries that solutions must adhere to, and collectively these constraints form the feasible set. Understanding this relationship is key for evaluating which potential solutions can be considered valid when optimizing an objective function.
• Discuss how changes in constraints can affect the shape and size of the feasible set and its implications for solution finding.
  • Changes in constraints can either expand or shrink the feasible set significantly. For instance, adding a new constraint may eliminate previously viable solutions, while relaxing an existing constraint could open up new possibilities. This alteration has profound implications for solution finding, as it could make finding an optimal solution more challenging or simpler, depending on how it reshapes the solution landscape.
• Evaluate the importance of understanding the feasible set when applying optimization techniques in real-world scenarios.
  • Understanding the feasible set is crucial in real-world optimization because it determines what solutions are practically achievable given specific limitations. For example, in operations research, knowing the feasible set allows decision-makers to focus their efforts on realistic options rather than exploring impractical ones. This comprehension enhances efficiency and effectiveness when making strategic decisions across various fields, from economics to resource management.

Related terms

Constraint: A limitation or condition that must be satisfied in an optimization problem, affecting the feasible set.

Objective Function:

The function that is being maximized or minimized in an optimization problem, evaluated over the feasible set.

Optimal Solution:

The best solution found within the feasible set, yielding the highest or lowest value of the objective function.