Linear Optimization

What is linear optimization?

Linear optimization, also known as linear programming, is a mathematical method for optimizing a linear objective function subject to linear constraints. It is a powerful tool for decision-making in various fields, such as engineering, economics, and business. In linear optimization, we aim to find the values of decision variables that maximize or minimize the objective function while satisfying the constraints.

The general form of a linear optimization problem is:

Subject to $\textbf{A}\textbf{x}≤\textbf{b}$, where:

• $\textbf{x}$ is the vector of decision variables.

• $\textbf{c}$ is the vector of objective function coefficients.

• $\textbf{A}$ is the constraint matrix.

• $\textbf{b}$ is the vector of constraint bounds.

Linear optimization can be solved using various algorithms, such as the simplex method, interior-point methods, and network flow algorithms. In Python, popular optimization libraries such as SciPy, CVXPY, and PuLP provide support for linear optimization.

A very intuitive way to understand this is with the graphical method. The graphical method is not suitable for more than three variables because it is difficult to visualize data in more dimensions, but the overall algorithm is the same. Let’s consider a very basic example:

Maximize $P(x,y)= 3x+4y$

Subject to:

• $x + y <= 20$

• $y-$ ...