Feasible Set

Definition

The feasible set is the collection of all possible solutions that satisfy the given constraints in an optimization problem. This set is crucial as it defines the boundary of acceptable solutions from which the optimal solution can be determined. Understanding the feasible set helps in analyzing how different constraints interact and shape the solution space, ultimately guiding the search for optimality.

5 Must Know Facts For Your Next Test

1. The feasible set can be represented graphically in two or three dimensions, showing all points that meet the constraints.
2. In quadratic programming, the feasible set may take the shape of a convex region, especially if the constraints are linear.
3. Finding the feasible set is often a critical step in optimization as it defines where we can look for solutions to the problem.
4. If there are no solutions that satisfy all constraints, the feasible set is empty, indicating an infeasible problem.
5. Techniques like graphical methods, simplex method, or interior-point methods are commonly used to identify and analyze the feasible set.

Review Questions

• How does the feasible set influence the process of finding an optimal solution in an optimization problem?
  • The feasible set directly influences the search for an optimal solution because it defines the boundaries within which all potential solutions must lie. When seeking an optimal point, only those solutions that belong to this set can be considered valid. If constraints are altered, it can change the size and shape of the feasible set, which in turn affects where the optimal solution might be found.
• Compare and contrast a feasible set with an optimal solution in terms of their roles in quadratic programming.
  • While the feasible set encompasses all possible solutions that meet specified constraints, the optimal solution is a single point within this set that optimally satisfies the objective function. In quadratic programming, one must first identify the feasible set through its constraints and then evaluate these solutions to find which one provides the best outcome for the objective function. Hence, without a feasible set, there can be no optimal solution.
• Evaluate how changes in constraints might affect both the feasible set and the process of optimization in quadratic programming scenarios.
  • When constraints in a quadratic programming problem change, they can either enlarge or shrink the feasible set. For example, adding new constraints may limit the number of possible solutions available, potentially making it more challenging to find an optimal solution. Conversely, relaxing existing constraints can expand the feasible set, potentially leading to a broader range of solutions and possibly a better optimal solution. Therefore, understanding how these changes impact both the feasible set and optimization is critical for effectively solving such problems.

Related terms

Constraints: Conditions or limitations that a solution to an optimization problem must satisfy, such as equality or inequality equations.

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

Convex Set: A set where any line segment connecting two points within the set lies entirely inside the set, often associated with problems that are easier to optimize.