Corner points

Definition

Corner points are specific points on the boundary of a feasible region in linear programming, where the constraints intersect. These points are crucial because they represent potential optimal solutions to a linear programming problem. Since the optimal solution to a linear programming problem will always occur at one of these corner points, understanding how to identify and analyze them is essential for solving optimization problems effectively.

5 Must Know Facts For Your Next Test

1. Corner points are formed by the intersection of two or more constraints, which can be represented as lines in a graph.
2. To find corner points, one typically sets the equations of the constraints equal to each other and solves for the variables.
3. In cases where the feasible region is unbounded, corner points still exist at the intersection of the constraints that form the region's boundaries.
4. Evaluating the objective function at each corner point allows you to determine which point gives the best result, whether maximizing or minimizing.
5. Graphing tools or software can help visualize corner points and feasible regions, making it easier to solve complex linear programming problems.

Review Questions

• How do corner points relate to finding optimal solutions in linear programming?
  • Corner points are critical in determining optimal solutions in linear programming because they are the only locations where these solutions can exist. Since the feasible region is defined by constraints, each corner point represents an intersection of these constraints. When evaluating the objective function at these points, one can find out which corner point provides the maximum or minimum value, thus identifying the best solution.
• What methods can be used to identify corner points when dealing with multiple constraints in a linear programming problem?
  • To identify corner points when multiple constraints are present, you can solve pairs of constraint equations simultaneously to find their intersection. This process involves setting two equations equal to each other and solving for their respective variables. After finding these intersection points, it's essential to check whether they fall within the bounds of the feasible region defined by all constraints. Utilizing graphing methods can also assist in visually locating these corner points.
• Evaluate how changing a constraint in a linear programming problem affects the location and number of corner points.
  • Altering a constraint in a linear programming problem can significantly impact both the location and number of corner points within the feasible region. For instance, if a constraint is relaxed (made less restrictive), it may allow for new corner points to emerge, potentially expanding the feasible region. Conversely, tightening a constraint could eliminate existing corner points and change the optimal solution. By analyzing these changes, one can gain insight into how sensitive optimal solutions are to variations in constraints, an important aspect of optimization.

Related terms

Feasible Region:

The set of all possible points that satisfy the constraints of a linear programming problem, represented graphically.

Objective Function:

The mathematical expression that defines the goal of the linear programming problem, usually maximizing or minimizing a value.

Constraints:

Inequalities or equations that restrict the values that the variables in a linear programming problem can take.