1-duality_ex.Rmd

## Exercises {-}

1.  Write down the dual problem of
    \[\begin{array}{rrcrcl}
    \min & 4x_1 & - & 2x_2 \\
    \text{s.t.} 
     & x_1 & + & 2x_2 & \geq & 3\\
     & 3x_1 & - & 4x_2 & = & 0 \\
     &  & & x_2 & \geq & 0.
    \end{array}\]

2.  Write down the dual problem of the following:
    \[
    \begin{array}{rrcrcrcl}
    \min  &    &  &3x_2  & + & x_3 \\
    \mbox{s.t.}
     & x_1 & + & x_2 & + & 2x_3 & = & 1 \\
     & x_1 &   &     & - & 3x_3 & \leq & 0 \\
     & x_1 & , & x_2 & , & x_3 & \geq & 0.
    \end{array}
    \]

3.  Write down the dual problem of the following:
    \[
    \begin{array}{rrcrcrcl}
    \min  & x_1 &  &  & - & 9x_3 \\
    \mbox{s.t.}
     & x_1 & - & 3x_2 &  + & 2x_3 & = & 1 \\
     & x_1 & &  & &  & \leq & 0 \\
     & &  & x_2 & &  &  & \mbox{free} \\
     & &  & &  & x_3 & \geq & 0.
    \end{array}\]

4.  Determine all values \(c_1,c_2\) such that the linear programming problem
\[\begin{array}{rl}
\min & c_1 x_1 + c_2 x_2 \\
\text{s.t.} & 2x_1 + x_2 \geq 2 \\
& x_1 + 3x_2 \geq 1.
\end{array}
\]
has an optimal solution. Justify your answer