everybody. I am stuck in a problem in Convex Optimization 2nd by Boyd.
The problem is below:
The solution is below:
I am stuck in problem (d). I have no clue how the third inequality comes. I think that the general inequality means pointwise inequality. But where is the relationship between each element ? Could anyone help me out? Thanks in advance!
The equality is elementwise (since it describes a polyhedron). However, there is only one $t$ for all equations, and that's what the third condition guarantees. If $h_i>0$ then $f_i^Tz+g \leq h_i/t$ is equivalent to $(f_i^Tz+g_i)/h_i \leq 1/t$. Similarly, if $h_k<0$, $(f_k^Tz+g_k)/h_k \geq 1/t$. Combining this, we get $(f_i^Tz+g_i)/h_i \leq 1/t \leq (f_k^Tz+g_k)/h_k$.
Define: $$l = \max_{i : h_i > 0} (f_i^Tz+g_i)/h_i$$ $$u = \min_{k : h_k < 0} (f_k^Tz+g_k)/h_k$$ If $l<u$ (which is equivalent to your third requirement) Then any $t$ such that $1/t \in [l,u]$ will do.