Lecture 24: Linear Programming and Two-Person Games
Description
This lecture focuses on several topics that are specific parts of optimization. These include linear programming (LP), the max-flow min-cut theorem, two-person zero-sum games, and duality.
Summary
Linear program: Minimize cost subject to \(Ax = b\) and \(x\geq 0\)
Inequalities make the problem piecewise linear.
Simplex method reduces cost from corner point to corner point.
Dual linear program is a maximization: Max = Min!
Game: \(X\) chooses rows of payoff matrix, \(Y\) chooses columns.
Related sections in textbook: VI.2–VI.3
Instructor: Prof. Gilbert Strang
Problems for Lecture 24
From textbook Section VI.2
1. Minimize \(F(\boldsymbol{x})=\frac{1}{2}\boldsymbol{x}^{\mathtt{T}} S\boldsymbol{x}=\frac{1}{2}x_1^2+2x_2^2\) subject to \(A^{\mathtt{T}} \boldsymbol{x}=x_1+3x_2=b\).
- What is the Lagrangian \(L(\boldsymbol{x},\lambda)\) for this problem?
- What are the three equations “derivative of \(L=\) zero” ?
- Solve those equations to find \(\boldsymbol{x}^\ast=(x_1^\ast, x_2^\ast)\) and the multiplier \(\lambda^\ast\).
- Draw Figure VI.4 for this problem with constraint line tangent to cost circle.
- Verify that the derivative of the minimum cost is \(\partial F^\ast/\partial b=-\lambda^\ast\).
From textbook Section VI.3
2. Suppose the constraints are \(x_1+x_2+2x_3=4\) and \(x_1\geq 0, x_2\geq 0, x_3\geq 0\).
Find the three corners of this triangle in R3. Which corner minimizes the cost \(\boldsymbol{c}^{\mathtt{T}} \boldsymbol{x}=5x_1+3x_2+8x_3\)?
5. Find the optimal (minimizing) strategy for \(X\) to choose rows. Find the optimal (maximizing) strategy for \(Y\) to choose columns. What is the payoff from \(X\) to \(Y\) at this optimal minimax point \(\boldsymbol{x}^\ast, \boldsymbol{y}^\ast\)?
$$
\begin{array}{c} \textbf{Payoff}\\ \textbf{matrices}\end{array}\qquad \left[\begin{matrix}1 & 2\\ 4 & 8\end{matrix}\right] \qquad \left[\begin{matrix}1 & 4\\ 8 & 2\end{matrix}\right]$$
