I am learning Robust Optimization and been stuck on this example. I've brushed up on my knowledge of Lagrange duality and referred to a couple of textbooks on Linear Programming but not able to understand how the author arrived at the following robust counterpart.
Here's the link to the complete description of the robust optimization example:Hands-On Mathematical Optimization
Original problem:
\begin{split} \begin{array}{rrcrcl} \max & 12 x_1 & + & 9 x_2 \\ \text{s.t.} & x_1 & & & \leq & 1000 \\ & & & x_2 & \leq & 1500 \\ & x_1 & + & x_2 & \leq & 1750 \\ & z_1 x_1 & + & z_2 x_2 & \leq & 4800 & \forall \, y \in \mathbb{R}^2 \,:\, z_j=\bar{z}_j+\delta y_j, \, \|y\|_\infty \leq 1, \, \|y\|_1\leq \Gamma \\ & x_1 & , & x_2 & \geq & 0 \\ \end{array} \end{split}
Robust Counterpart:
\begin{split} \begin{array}{rrcrcrcrcrcrcl} \max & 12 x_1 & + & 9 x_2 \\ \text{s.t.} & x_1 & & & & & & & & & \leq & 1000 \\ & & & x_2 & & & & & & & \leq & 1500 \\ & x_1 & + & x_2 & & & & & & & \leq & 1750 \\ & \bar{z}_1 x_1 & + & \bar{z}_2 x_2 & + & \lambda\Gamma & + & t_1 & + & t_2 & \leq & 4800 \\ &-\delta x_1 & & & + & \lambda & + & t_1 & & & \geq & 0 \\ & & &-\delta x_2 & + & \lambda & & & + & t_2 & \geq & 0 \\ &\delta x_1 & & & + & \lambda & + & t_1 & & & \geq & 0 \\ & & &\delta x_2 & + & \lambda & & & + & t_2 & \geq & 0 \\ & x_1 & , & x_2 & , & \lambda & , & t_1 & , & t_2 & \geq & 0 \\ \end{array} \end{split}
Any help to understand this and references to further learn would be greatly appreciated!
Thanks!