Feasible Set in Linear Programming

In linear programming, a feasible set represents the set of all possible solutions that satisfy the given constraints of the problem. Let's understand the concept of a feasible set in detail.

- Feasible Region: The feasible region is the set of all feasible solutions that satisfy all the constraints of the linear programming problem. It is represented by shaded regions in the graph of the problem.
- Feasible Set: The feasible set is the set of all feasible solutions that satisfy some, but not necessarily all, of the constraints of the linear programming problem. It is represented by a set of points in the graph of the problem.

Convex Set

A convex set is a set of points in which any two points can be connected by a straight line that lies entirely within the set. In other words, a set is convex if the line segment between any two points in the set lies entirely within the set.

For example, a circle is a convex set because any two points on its circumference can be connected by a straight line that lies entirely within the circle.

Feasible Set is Convex

The feasible set in a linear programming problem is always a convex set. This is because the constraints of the problem are linear, and the intersection of any two linear constraints is always a line or a plane, which is a convex set.

Moreover, the feasible set is always bounded, which means that it has a finite extent in all directions. This is because the objective function of a linear programming problem is always a linear function, and a linear function is unbounded only in one direction.

Conclusion

In conclusion, the feasible set in a linear programming problem is always a convex set. This is because the constraints of the problem are linear, and the intersection of any two linear constraints is always a convex set.