Basic variables

Definition

Basic variables are the set of variables in a linear programming problem that correspond to the basic feasible solution. In the context of equality constrained optimization, these variables are essential for expressing the solution to the optimization problem in terms of the constraints and the objective function. They play a crucial role in determining the feasible region and ultimately influence the optimal solution by defining which constraints are active.

5 Must Know Facts For Your Next Test

1. In a linear programming problem with equality constraints, basic variables can be identified from the matrix representation of the system, where they correspond to pivot columns.
2. The number of basic variables in a solution must equal the number of constraints in the system for the solution to be considered basic feasible.
3. Basic variables can take on non-zero values while non-basic variables are typically set to zero in basic feasible solutions.
4. Changing a basic variable can lead to a new vertex of the feasible region, which may yield a different optimal solution.
5. In simplex methods for solving linear programming problems, identifying and updating basic variables is a key step in moving towards optimality.

Review Questions

• How do basic variables contribute to identifying feasible solutions in equality constrained optimization?
  • Basic variables are critical in identifying feasible solutions because they determine which constraints are active and directly influence the shape of the feasible region. When solving equality constrained optimization problems, a basic feasible solution is formed when the values of these variables satisfy all given constraints. This relationship allows us to pinpoint specific solutions that are permissible under the defined constraints.
• Discuss how the selection of basic versus non-basic variables affects the outcome of an optimization problem.
  • The selection between basic and non-basic variables significantly impacts the outcome of an optimization problem. Basic variables, which are allowed to take on non-zero values, are directly involved in forming solutions that satisfy all constraints. In contrast, non-basic variables are typically set to zero and do not contribute to the immediate solution. The interplay between these variable sets can affect whether an optimal solution is reached or if further iterations are needed in methods like simplex.
• Evaluate the role of basic variables in transforming an initial feasible solution into an optimal one through iterative methods.
  • Basic variables play a pivotal role in transforming an initial feasible solution into an optimal one by allowing for systematic adjustments during iterative methods like simplex. As iterations progress, changing the values of basic variables can lead to exploring new vertices of the feasible region, each potentially yielding better objective function values. This process continues until no further improvements can be made, indicating that an optimal solution has been reached based on the defined constraints and objective function.

Related terms

feasible region: The set of all possible points that satisfy the constraints of an optimization problem, including both equality and inequality constraints.

non-basic variables: Variables in a linear programming problem that are not part of the basic feasible solution, typically set to zero in a basic solution.

optimal solution: The best possible outcome of an optimization problem, achieved when the objective function is maximized or minimized under given constraints.