Dual Problem: Another Perspective on Linear Programming, Expanding Horizons, Optimizing Decisions

# 1. Introduction to Linear Programming

Linear programming is a mathematical optimization technique used to find the best set of variables in a given set of constraints to maximize or minimize the objective function. A linear programming model consists of an objective function and a set of linear constraints, where the objective function is a linear combination of variables, and the constraints are linear inequalities or equalities of the variables.

Linear programming has extensive applications in the real world, such as resource allocation, network flow optimization, and supply chain management. By solving linear programming models, decision-makers can determine the best resource allocation strategies, optimize network traffic, or develop effective supply chain plans.

# 2. Theoretical Foundations of Duality

### 2.1 Standard Form and Duality Form of Linear Programming

**Standard Form of Linear Programming**

A linear programming problem can be represented in the following standard form:

min z = c^T x
s.t. Ax ≤ b, x ≥ 0

Where:

* x is an n-dimensional decision variable vector
* c is an n-dimensional objective function coefficient vector
* A is an m×n constraint matrix
* b is an m-dimensional constraint value vector

**Duality Form**

The duality form of a linear programming problem in standard form is as follows:

max w = b^T y
s.t. A^T y ≥ c, y ≥ 0

Where:

* y is an m-dimensional dual variable vector
* w is the dual objective function value

### 2.2 Duality Theorem and Its Applications

**Duality Theorem**

The duality theorem states that the optimal objective value of the original linear programming problem is equal to the optimal objective value of the dual linear programming problem, that is:

z* = w*

**Applications**

The duality theorem has important applications in the following areas:

***Sensitivity Analysis:** By solving the dual problem, the impact of changes in parameters in the original problem on the optimal solution can be analyzed.
***Algorithm Design:** The dual problem can serve as an alternative solution method for the original problem, especially when the original problem is difficult to solve.

### 2.3 Strong Duality and Weak Duality

**Strong Duality**

If both the original linear programming problem and the dual linear programming problem have feasible solutions, and their optimal solutions are equal, then the problem is said to have strong duality.

**Weak Duality**

If the original linear programming problem has a feasible solution, then the dual linear programming problem also has a feasible solution, and its optimal objective value is less than or equal to the optimal objective value of the original problem.

**Strong Duality Theorem**

If the original linear programming problem satisfies the following conditions, then it has strong duality:

* Both the original problem and the dual problem have feasible solutions
* The feasible domain of the original problem is a polyhedron
* The feasible domain of the dual problem is a polyhedron

**Weak Duality Theorem**

Any linear programming problem satisfies weak duality.

# 3.1 Application of Duality in Resource Allocation

In resource allocation problems, the dual problem can help decision-makers optimize resource allocation plans to maximize the value of the objective function.

**Problem Description:**

Suppose a company has various resources, such as machines, manpower, and capital, that need