Feasible Set

Definition

A feasible set is the collection of all possible solutions to an optimization problem that satisfy the given constraints. This set is crucial as it defines the limits within which any optimal solution must fall, emphasizing the importance of both equality and inequality constraints in shaping the solution space.

5 Must Know Facts For Your Next Test

1. The feasible set is formed by combining all the constraints of an optimization problem, determining which solutions are permissible.
2. If there are no feasible solutions, the optimization problem is considered infeasible and cannot yield a valid result.
3. Graphically, the feasible set can often be represented as a region in a coordinate system, bounded by lines or curves corresponding to the constraints.
4. The feasible set can be empty, meaning that no combinations of variables satisfy all constraints imposed on the problem.
5. Identifying the feasible set is a critical first step in solving an optimization problem, as it allows for understanding where potential solutions can exist.

Review Questions

• How do constraints shape the feasible set in an optimization problem?
  • Constraints directly define the boundaries of the feasible set by specifying conditions that potential solutions must meet. Each inequality or equality adds restrictions on the variables, limiting which combinations can be considered valid solutions. Therefore, understanding how each constraint interacts is essential for visualizing and determining the shape and size of the feasible set.
• Discuss why identifying the feasible set is essential before attempting to find an optimal solution in nonlinear optimization.
  • Identifying the feasible set is essential because it narrows down the search space for potential optimal solutions. Without knowing what solutions are valid, any attempt to optimize an objective function may lead to irrelevant results. The feasible set acts as a guide for where to focus efforts when solving for the optimal solution, ensuring that only permissible combinations are evaluated.
• Evaluate how changes in constraints can impact the size and shape of the feasible set and ultimately affect optimization outcomes.
  • Changes in constraints can significantly alter the size and shape of the feasible set, which may lead to different optimal solutions. For instance, tightening a constraint may reduce the feasible region, potentially eliminating previously valid solutions and changing where the optimal point lies. Conversely, relaxing constraints might expand the feasible set, allowing for more options. This dynamic illustrates how sensitive optimization outcomes are to constraint modifications, emphasizing the need for careful consideration during problem formulation.

Related terms

Constraints: Restrictions or conditions that solutions must satisfy in an optimization problem, which can be equalities or inequalities.

Objective Function: The function that needs to be maximized or minimized in an optimization problem, which evaluates the quality of each feasible solution.

Optimal Solution: The best possible solution from the feasible set that either maximizes or minimizes the objective function.