# What are the Difference between primal and dual LPP?

Difference between Primal and dual LPP is, they are two related optimization problems that are used to solve a wide range of problems in operations research, economics, and engineering.

__DIFFERENCE BETWEEN PRIMAL AND DUAL LPP TABLE__:-

Aspect | Primal LPP | Dual LPP |

Objective Function | Minimize or Maximize | Minimize or Maximize |

Variables | Decision variables in the primal problem | Dual variables in the dual problem |

Constraints | Constraints in the primal problem | Variables in the dual problem |

Objective Coefficients | Coefficients of the objective function in the primal problem | Coefficients of the constraint equations in the dual problem |

Feasible Region | Set of feasible solutions for the primal problem | Set of feasible solutions for the dual problem |

Optimality Condition | Satisfying the complementary slackness condition | Satisfying the complementary slackness condition |

Weak Duality | The optimal value of the dual problem is always a lower bound on the optimal value of the primal problem | The optimal value of the primal problem is always an upper bound on the optimal value of the dual problem |

Strong Duality | If the primal problem has an optimal solution, the dual problem has an optimal solution as well, and the optimal values are equal | If the dual problem has an optimal solution, the primal problem has an optimal solution as well, and the optimal values are equal |

Difference between primal and dual LPP

## The main Difference between primal and dual LPP is in the objective function and the constraints.

**Objective function: **

The objective function of a primal LPP is to maximize or minimize a linear function of the decision variables subject to linear constraints.

The objective function of a dual LPP is to minimize or maximize a linear function of the dual variables subject to linear constraints.

**Constraints:**

The constraints in a primal LPP represent the limits or requirements of the problem, and they are expressed as linear inequalities or equations.

The constraints in a dual LPP are also expressed as linear inequalities or equations, but they represent the relationship between the decision variables and the objective function of the primal problem.

**The number of variables and constraints: **

The number of decision variables in a primal LPP is equal to the number of constraints in the dual LPP, and vice versa.

The number of constraints in a primal LPP is equal to the number of decision variables in the dual LPP, and vice versa.

**Feasible solutions and optimal solutions: **

A feasible solution to a primal LPP satisfies the constraints of the problem, while a feasible solution to a dual LPP satisfies the conditions of the dual problem.

An optimal solution to a primal LPP provides the maximum or minimum value of the objective function, while an optimal solution to a dual LPP provides the minimum or maximum value of the objective function of the dual problem.

** In summary, the primal and dual linear programming problems are two sides of the same coin, **and they are both important tools for solving optimization problems.

The primal problem is used to find the optimal solution to a given problem, while the dual problem provides additional insights into the structure of the problem and can be used to check the feasibility of the primal solution.Â

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