Problems for Lecture 24
From textbook Section VI.2

1. Minimize $F(\mathit{x})=\frac{1}{2}{\mathit{x}}^{\mathtt{T}}S\mathit{x}=\frac{1}{2}{x}_1^2+2x_2^2$ subject to $A^{\mathtt{T}}\mathit{x}=x_1+3x_2=b$.

1. What is the Lagrangian $L(\mathit{x},\lambda )$ for this problem?
2. What are the three equations “derivative of $L=$ zero” ?
3. Solve those equations to find ${\mathit{x}}^{\ast }=(x_1^{\ast },x_2^{\ast })$ and the multiplier ${\lambda }^{\ast }$.
4. Draw Figure VI.4 for this problem with constraint line tangent to cost circle.
5. 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 $x_1\ge 0,x_2\ge 0,x_3\ge 0$.
Find the three corners of this triangle in R³. Which corner minimizes the cost ${\mathit{c}}^{\mathtt{T}}\mathit{x}=5x_1+3x_2+8x_3$?

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 ${\mathit{x}}^{\ast },{\mathit{y}}^{\ast }$?

$$\begin{array}{c}\mathbf{\text{Payoff}}\\ \mathbf{\text{matrices}}\end{array}\left[\begin{array}{cc}1& 2\\ 4& 8\end{array}\right]\left[\begin{array}{cc}1& 4\\ 8& 2\end{array}\right]$$