Feasibility

Definition

Feasibility refers to the condition of a solution or set of solutions that satisfies all constraints of an optimization problem. It is essential to determine whether a proposed solution can be realized under given restrictions, such as resource limitations and requirements for decision variables, thereby connecting the solution space to valid and practical outcomes.

5 Must Know Facts For Your Next Test

1. In the simplex algorithm, feasibility is checked at each vertex of the feasible region to find optimal solutions.
2. For a linear programming problem to have a solution, it must be both feasible and bounded; otherwise, it may lead to unbounded solutions or infeasibility.
3. In primal-dual relationships, if one problem is feasible, it often implies specific conditions about the feasibility of its dual counterpart.
4. When formulating quadratic programs, ensuring feasibility involves checking that all quadratic constraints are met by the proposed solutions.
5. In chance-constrained programming, feasibility relates not only to deterministic constraints but also to satisfying probabilistic constraints under uncertainty.

Review Questions

• How does the concept of feasibility influence the simplex algorithm in finding optimal solutions?
  • In the simplex algorithm, feasibility is crucial because the algorithm iterates through vertices of the feasible region defined by constraints. Each vertex corresponds to a potential solution, and only feasible solutions can be evaluated for optimality. The algorithm continues until it finds an optimal solution at a feasible vertex or determines that no feasible solution exists.
• Discuss how feasibility is addressed in primal-dual relationships and its implications for optimality.
  • In primal-dual relationships, feasibility plays a significant role in establishing weak and strong duality. If a primal problem has a feasible solution, then its dual must also be feasible. This relationship is essential as it allows us to derive insights about optimality; if both problems are feasible and bounded, they will yield the same objective value at optimality. Conversely, if one problem is infeasible, it affects the feasibility of its dual.
• Evaluate the role of feasibility in chance-constrained programming and its impact on modeling real-world problems.
  • In chance-constrained programming, feasibility is evaluated not just on deterministic terms but involves satisfying probabilistic constraints that account for uncertainty in parameters. This dual aspect means that a solution must not only meet hard constraints but also satisfy specific probabilities regarding constraint violations. Evaluating this feasibility impacts how real-world problems are modeled, especially in fields like finance and supply chain management where uncertainties are prevalent.

Related terms

Optimality: Optimality is the condition of being the best possible solution among feasible solutions, maximizing or minimizing an objective function.

Constraints: Constraints are the restrictions or limitations imposed on the decision variables in an optimization problem, defining the feasible region.

Infeasibility: Infeasibility occurs when no solution can satisfy all constraints of an optimization problem, leading to an empty feasible region.