Linear Programming from Fibonacci to Farkas

Area of Science:

• Mathematics
• History of Science

Background:

• The historical roots of mixture problems trace back to Fibonacci's work in the 13th century, utilizing Hindu-Arabic arithmetic.
• Early arithmetic texts codified these methods under the name 'alligation', often presented in descriptive, less accessible formats.

Discussion:

• Thomas Harriot's introduction of algebraic notation around 1600 significantly clarified alligation, linking it to Diophantine problems studied by 17th-century mathematicians.
• Joseph Fourier's contributions to mechanics involved developing methods for linear inequalities, incorporating logic, algebra, and geometric representations of solution sets.

Key Insights:

• Gyula Farkas's 1898 theorem on linear inequalities, inspired by Fourier, established a foundational result in optimization.
• The evolution from alligation to Linear Programming demonstrates a progression towards rigorous mathematical frameworks for solving complex problems.

Outlook:

• Linear Programming has become a cornerstone of modern applied mathematics with wide-ranging applications.
• Farkas's theorem continues to be pivotal, notably influencing advancements in Game Theory.