Write the primal problem in standard form

Oct 13, 2025 by windailys.com 79 views

Introduction

In linear programming, the dual problem is a way to transform a given linear inequality system into a new problem that is equivalent in terms of its optimal solution. The dual problem is useful for finding the optimal solution of the original problem, especially when the original problem is difficult to solve directly. In this article, we will discuss how to form the dual problem of a given linear inequality system.

The Original Problem

The original problem is given by the following linear inequality system:

{ \begin{array}{ll} 3x_1 + x_2 \geq 4 \\ -11x_1 - 2x_2 \geq 7 \\ x_1, x_2 \geq 0 \end{array} \}

This is a system of two linear inequalities with two variables, $x_1$ and $x_2$ . The objective is to find the values of $x_1$ and $x_2$ that satisfy both inequalities and maximize the objective function.

Forming the Dual Problem

To form the dual problem, we need to follow these steps:

Write the primal problem in standard form: The primal problem is already in standard form, with the inequalities written in the form $ax + by \geq c$ .
Identify the coefficients of the primal problem: The coefficients of the primal problem are the coefficients of the variables $x_1$ and $x_2$ in the inequalities.
Create the dual problem: The dual problem is a maximization problem with the same number of variables as the primal problem. The objective function of the dual problem is the sum of the products of the coefficients of the primal problem and the dual variables.
Write the constraints of the dual problem: The constraints of the dual problem are the inequalities that correspond to the primal problem.

Forming the Dual Problem of the Original Problem

Let's apply the steps above to form the dual problem of the original problem.

Write the primal problem in standard form: The primal problem is already in standard form.
Identify the coefficients of the primal problem: The coefficients of the primal problem are:

Inequality	Coefficient of $x_1$	Coefficient of $x_2$
1	3	1
2	-11	-2

Create the dual problem: The dual problem is a maximization problem with two variables, $y_1$ and $y_2$ . The objective function of the dual problem is:

$P = 3y_1 - 11y_2 + y_1 + 7y_2$

Simplifying the objective function, we get:

$P = 4y_1 - 4y_2$

Write the constraints of the dual problem: The constraints of the dual problem are the inequalities that correspond to the primal problem. The first inequality corresponds to the first constraint of the primal problem, and the second inequality corresponds to the second constraint of the primal problem.

The first constraint of the dual problem is:

$4y_1 - 4y_2 \leq 4$

The second constraint of the dual problem is:

$-4y_1 + 4y_2 \leq 7$

The Dual Problem

The dual problem is:

Maximize

$P = 4y_1 - 4y_2$

Subject to

{ \begin{array}{ll} 4y_1 - 4y_2 \leq 4 \\ -4y_1 + 4y_2 \leq 7 \\ y_1, y_2 \geq 0 \end{array} \}

This is the dual problem of the original problem.

Conclusion

In this article, we discussed how to form the dual problem of a given linear inequality system. We applied the steps to form the dual problem of the original problem, which is a system of two linear inequalities with two variables. The dual problem is a maximization problem with two variables, $y_1$ and $y_2$ , and the objective function is the sum of the products of the coefficients of the primal problem and the dual variables. The constraints of the dual problem are the inequalities that correspond to the primal problem.

Future Work

In future work, we can apply the dual problem to solve the original problem. We can use the dual problem to find the optimal solution of the original problem, especially when the original problem is difficult to solve directly.

References

[1] Linear Programming by Dantzig, G. B.
[2] Linear Programming by Chvátal, V.

Glossary

Primal problem: The original problem that is being transformed into a dual problem.
Dual problem: The transformed problem that is equivalent to the primal problem.
Standard form: The form of the primal problem with the inequalities written in the form $ax + by \geq c$ .
Coefficients: The coefficients of the variables in the inequalities of the primal problem.
Dual variables: The variables of the dual problem that correspond to the coefficients of the primal problem.
Constraints: The inequalities that correspond to the primal problem in the dual problem.

Introduction

In our previous article, we discussed how to form the dual problem of a given linear inequality system. In this article, we will answer some frequently asked questions about forming the dual problem.

Q: What is the purpose of forming the dual problem?

A: The purpose of forming the dual problem is to transform a given linear inequality system into a new problem that is equivalent in terms of its optimal solution. The dual problem is useful for finding the optimal solution of the original problem, especially when the original problem is difficult to solve directly.

Q: How do I know if the dual problem is equivalent to the primal problem?

A: The dual problem is equivalent to the primal problem if the optimal solution of the dual problem is the same as the optimal solution of the primal problem. This can be verified by solving both problems and comparing their optimal solutions.

Q: Can I use the dual problem to solve the primal problem?

A: Yes, you can use the dual problem to solve the primal problem. The dual problem can be used to find the optimal solution of the primal problem, especially when the primal problem is difficult to solve directly.

Q: How do I form the dual problem of a linear programming problem with multiple constraints?

A: To form the dual problem of a linear programming problem with multiple constraints, you need to follow the same steps as before. First, write the primal problem in standard form. Then, identify the coefficients of the primal problem. Next, create the dual problem by writing the objective function and the constraints of the dual problem. Finally, simplify the objective function and the constraints of the dual problem.

Q: Can I use the dual problem to find the optimal solution of a linear programming problem with multiple variables?

A: Yes, you can use the dual problem to find the optimal solution of a linear programming problem with multiple variables. The dual problem can be used to find the optimal solution of the primal problem, especially when the primal problem is difficult to solve directly.

Q: How do I interpret the results of the dual problem?

A: To interpret the results of the dual problem, you need to understand the meaning of the dual variables and the constraints of the dual problem. The dual variables represent the shadow prices of the primal problem, and the constraints of the dual problem represent the bounds on the dual variables.

Q: Can I use the dual problem to solve a linear programming problem with integer variables?

A: Yes, you can use the dual problem to solve a linear programming problem with integer variables. However, you need to be careful when using the dual problem to solve a linear programming problem with integer variables, as the dual problem may not be able to find the optimal solution of the primal problem.

Q: How do I know if the dual problem is a good approximation of the primal problem?

A: To know if the dual problem is a good approximation of the primal problem, you need to compare the optimal solutions of the dual problem and the primal problem. If the optimal solutions of the dual problem and the primal problem are close, then the dual problem is a good approximation of the primal problem.

Conclusion

In this article, we answered some frequently asked questions about forming the dual problem. We discussed how to form the dual problem, how to use the dual problem to solve the primal problem, and how to interpret the results of the dual problem. We also discussed some common pitfalls when using the dual problem to solve a linear programming problem.

Future Work

In future work, we can explore more advanced topics in linear programming, such as the use of the dual problem to solve linear programming problems with multiple objectives and the use of the dual problem to solve linear programming problems with uncertain parameters.

References

[1] Linear Programming by Dantzig, G. B.
[2] Linear Programming by Chvátal, V.

Glossary

Primal problem: The original problem that is being transformed into a dual problem.
Dual problem: The transformed problem that is equivalent to the primal problem.
Standard form: The form of the primal problem with the inequalities written in the form $ax + by \geq c$ .
Coefficients: The coefficients of the variables in the inequalities of the primal problem.
Dual variables: The variables of the dual problem that correspond to the coefficients of the primal problem.
Constraints: The inequalities that correspond to the primal problem in the dual problem.
Shadow prices: The dual variables that represent the shadow prices of the primal problem.
Bounds: The constraints of the dual problem that represent the bounds on the dual variables.