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Linear Programming - Deep Dive

Mathematical Foundations​

Rigorous treatment of linear programming and duality theory.

Fundamental Theorems​

Existence of Optimal Vertices​

Theorem: If an LP has an optimal solution, it has an optimal vertex.

Proof: (To be completed)

Simplex Method​

Algorithm: (Detailed steps to be completed)

Theorem: The simplex method terminates in finite steps (assuming non-degeneracy)

Proof: (To be completed)

Duality Theory​

Duality Theorems​

Weak Duality: If x is primal feasible and y is dual feasible, then cᵀx ≥ bᵀy

Proof: (To be completed)

Strong Duality: If the primal has optimal solution x*, the dual has optimal solution y* with cᵀx* = bᵀy*

Proof: (To be completed)

Complementary Slackness​

Theorem: x* and y* are optimal if and only if complementary slackness conditions hold

Proof: (To be completed)

Network Flow Theory​

Max Flow Min Cut​

Theorem: Maximum flow equals minimum cut capacity

Proof: (To be completed)

Network Simplex​

(To be completed)

Game Theory​

Minimax Theorem​

Theorem (von Neumann): Every finite zero-sum game has a value

Proof using LP: (To be completed)

Nash Equilibrium​

(To be completed)

Exercises​

(Advanced problems to be completed)