What does mean to minimize objective function with "less than" inequality constraints? Aren't you suppose to minimize with "greater than" constraints, like in example 1?
Example 1 (understand this) $$\begin{align} \text{max } x_1 & +5 x_2 \\ \text{s.t. } x_1 & \le 150 \\ x_2 &\le 350 \\ x_1+x_2 &\le 400 \\ x_1 , x_2 &\ge 0 \end{align}$$
I get this: the objective function wants to be as large as possible, but the constraints put an upper bound on $x_1, x_2$.
Example 2 (don't understand this) $$\begin{align} \text{min } & f_0(x) \\ \text{s.t. } & f_i \le 0 \text{ } i=1,...,m \\ \end{align}$$ where $f_0,..,f_m$ are convex functions. (Eq. 4.15 Convex Optimization)
This seems unbounded below. But since it's convex, it's bounded below? So, you are minimizing a convex function that satisfies a bunch of other convex functions.
Am I understanding this correctly? What's the point of this? Can someone provide a numerical example?
Thanks in advance!
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Follow up question:
Generally, $f_i(x)$ need not to be convex for $i=1,...,m$ $$\begin{align} \text{min } & f_0(x) \\ \text{s.t. } & f_i \le 0 \text{ } i=1,...,m \\ \end{align}$$
This is unbounded below. Isn't the solution $-\infty$?