Duality
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Duality is a fundamental concept in mathematics, particularly in linear optimization. It involves the relationship between two linear programs, known as the primal and the dual, where the optimal solution to one program provides valuable information about the optimal solution to the other.
Origins of Duality
The concept of duality was first introduced by the renowned mathematician John von Neumann in the 1940s while working on game theory. Von Neumann recognized the close connection between linear programming and the theory of games.
Duality theory has since become a powerful tool in various fields, including economics, finance, and engineering.
Primal and Dual Linear Programs
The primal linear program attempts to minimize (or maximize) a linear objective function subject to a set of linear constraints. The dual linear program, on the other hand, seeks to maximize (or minimize) another linear function subject to a different set of linear constraints.
The duality theorem states that the optimal value of the primal linear program is equal to the optimal value of the dual linear program.
Applications of Duality
Duality has numerous applications in real-world problems. Some notable examples include:
