Feasible solution

Definition

A feasible solution is a set of values for the decision variables that satisfies all the constraints of an optimization problem. It plays a crucial role in finding optimal solutions, as only those solutions that meet the constraints can be considered valid candidates. In combinatorial optimization, feasible solutions help define the search space within which optimal solutions are sought.

5 Must Know Facts For Your Next Test

1. Feasible solutions can be found using various methods, such as graphical methods for two-variable problems or algorithms for larger problems.
2. In linear programming, feasible solutions are typically represented as points within a geometric region defined by the constraints.
3. Not all feasible solutions are optimal; some may be far from the best solution that maximizes or minimizes the objective function.
4. The set of all feasible solutions is called the feasibility region, which is critical for understanding where optimal solutions lie.
5. In branch and bound methods, feasible solutions help determine which branches of the solution tree should be explored further.

Review Questions

• How does a feasible solution relate to constraints in an optimization problem?
  • A feasible solution directly depends on the constraints set forth in an optimization problem. It represents any combination of decision variable values that meets all these constraints. If even one constraint is violated, that combination cannot be considered feasible. Therefore, identifying feasible solutions helps delineate the acceptable region for potential optimal outcomes.
• In what ways can feasible solutions be utilized within branch and bound techniques to find optimal solutions?
  • In branch and bound techniques, feasible solutions are critical for determining which branches of the decision tree are worth exploring. As each node in the tree represents a subset of potential solutions, checking whether a solution is feasible helps eliminate large portions of the search space. By focusing on branches that contain feasible solutions, the algorithm can efficiently zero in on the optimal solution without wasting time on infeasible areas.
• Evaluate how the concept of feasible solutions affects the efficiency of solving combinatorial optimization problems.
  • The concept of feasible solutions significantly impacts the efficiency of solving combinatorial optimization problems by narrowing down the vast search space to manageable regions. By concentrating on only those combinations of variables that adhere to constraints, algorithms can streamline their processes and avoid unnecessary computations on infeasible options. This targeted approach enhances performance and reduces the time required to reach an optimal solution, particularly in complex problems with numerous variables and constraints.

Related terms

Optimal solution: The best feasible solution that minimizes or maximizes the objective function in an optimization problem.

Constraints: Conditions or limitations imposed on the decision variables in an optimization problem that must be satisfied by feasible solutions.

Search space: The set of all possible solutions, including both feasible and infeasible solutions, from which an optimal solution is identified.