# min(f(x)) is convex or concave based on type of f(x)

i have f(x) that is concave function. My question is g=min(f(x)) is concave or convex? And max(g) is concave or convex? there is a theorem for this?

• What is g=min(f) - a number? But it sounds like g is meant to be a function. Either way, this question is a lot closer to Math than to Computational Science. Jul 6, 2023 at 20:49
• Optimization problems that involve minimizing a convex function or maximizing a concave function are called convex optimization problems. If your problem is to maximize a convex function then it is a no convex optimization problem. Jul 7, 2023 at 5:00
• Does $f$ have some parameter $a$ so that you're maximizing $\max_{a} \min_{x} f_{a}(x)$? Jul 7, 2023 at 19:59
• If $g=min_x(f(x))$, then $g$ is just a number as pointed out above. You can't maximize it -- it's just a number, there is nothing you can vary to optimize it. To optimize something, it must depend on some kind of variable that you can choose. Jul 7, 2023 at 23:30
• @Maria Add formulas, not pictures. Explain all the symbols, especially if you have dangling variables like $t_k,p_k$. Jul 8, 2023 at 16:25