Exercise 1.6 (Rocket control)

First model. The first option described in the Section 1.3 is to minimize the total fuel consumption:

As shown in the previous exercise, for each $t=0,\dots ,T$, we can equivalently introduce nonnegative dummies $a_t^{+},a_t^{-}$ and replace every occurrence of $a_t$ with $a_t^{+}+a_t^{-}$ and every occurrence of $|a_t|$ with $a_t^{+}-a_t^{-}$. The resulting linear optimization problem is then the following:

Second model. The second option suggested in the book is to minimize the maximum thrust that is required for the rocket:

We can move the maximum function to the constraints section by writing

In a manner similar to the previous exercise, we replace $z\ge |a_t|$ by the double condition $z\ge a_t$ and $z\ge -a_t$: