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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 $\sqrt{1-\text{constant}}$?

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  • $\begingroup$ Just pointing out that since you are maximising, maybe you want to get a concave function instead? $\endgroup$
    – Nikos Kazazakis
    Apr 14, 2021 at 10:43

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You are maximizing a convex quadratic (the monotonic log is irrelevant) so the maximum is attained at the border, i.e. either $0$ or $\min(1,\sqrt{1-\text{constant}})$.

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  • $\begingroup$ 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? $\endgroup$
    – Shayan zargari
    Apr 15, 2021 at 12:47
  • $\begingroup$ If you pick the maximum, and it does not satisfy that constraint, then the problem is obviously not solvable. $\endgroup$
    – Johan Löfberg
    Apr 15, 2021 at 12:50
  • $\begingroup$ So, I cannot derive a close-form solution! Shall I resort to SCA approach? $\endgroup$
    – Shayan zargari
    Apr 15, 2021 at 13:02
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    $\begingroup$ I have already given you the closed-form solution, and the added constraint does not change anything as it either will be a redundant constraint (the maximal solution satisfies it) or render the problem infeasible (the maximal solution does not satisfy it) $\endgroup$
    – Johan Löfberg
    Apr 15, 2021 at 13:05
  • $\begingroup$ I get it, Thank you so much! $\endgroup$
    – Shayan zargari
    Apr 15, 2021 at 13:15

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