Brief explaination on convex optimization problem

Question

I have following type optimization problem (I transformed original max-min problem into this kind), and I can show that all $g_1(l_1),\cdots,g_M(l_1,\cdots,l_M)$ functions are concave. $$\max_{\substack{l_1 \in [0, 1],\cdots,l_{M} \in [0,1]}} T$$ $$g_1(l_1) \geq T,\cdots,g_M(l_1,\cdots,l_M) \geq T$$

Can someone please explain plainly (or provide me a helpful short doc to read) why this kind of problem is a convex optimization problem? I am confusing with $T$ which now appears in objective and all constraints.

Answer 1

One should view $T$ as a variable as well.

Maximizing $T$ is equivalent to minimizing $-T$, which is linear. Hence, the objective function is convex.

Define $h_i(l_1,\ldots,l_i,T)=T-g_i(l_1,\ldots,l_i)$, Since $g_i$ is concave and $T$ is linear, we can conclude that $h_i$ is convex.

Hence the problem is now of the form of

min $-T$ subject to

$h_i(l_1,\ldots,l_i,T) \leq 0 , \forall i=1,\ldots,M$.

The new formulation is in the standard form of a convex optimization problem.