0
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.