# How to convexify log(convex) function?

I have the following optimization problem: \begin{align}\max_x&\quad\log_2(1+|a+bx|^2+cx^2)\\\text{s.t.}&\quad0\le x\le1\\&\quad(1-x^2)\ge\text{constant}\end{align} where $$a$$ and $$b$$ are two complex numbers, and $$c$$ is a real positive number.

Does anyone have any idea regarding how this objective function can be converted into a convex one?

Is it true to say that the optimal value of $$x$$ is equal to $$\sqrt{1-\text{constant}}$$?

• Just pointing out that since you are maximising, maybe you want to get a concave function instead? Commented Apr 14, 2021 at 10:43

You are maximizing a convex quadratic (the monotonic log is irrelevant) so the maximum is attained at the border, i.e. either $$0$$ or $$\min(1,\sqrt{1-\text{constant}})$$.
• Thank you for your prompt reply, I have another question! what if we have the following constraint $log_2(1+|a+b*x|^2+c*x^2)\geq constant2$ as well? Do we get the previous solution still? Commented Apr 15, 2021 at 12:47