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Consider the following constrained optimization problem \begin{align*} &\underset{p,\theta}{\max}~R(p,\theta)\triangleq\frac{qp\theta}{1-q}+p(1-\theta)(1-p)\\ &\mbox{s.t. }~q = 1-\frac{c+\theta p}{\theta +q(1-\theta)}\\ &~~~~p\geq \frac{c}{q(1-\theta)}\\ &~~~~\theta\in[0,1]\\ &~~~~q\in[0,1]\\ &~~~~p\geq 0 \end{align*} Let $R^*(p^*,\theta^*)$ be the optimal objective function value of this optimization problem.

How to show that $R^*(p^*,\theta^*)>(1-\sqrt{c})^2$ for $c\in[0,1/4)$?

I have done numerical analysis, and find that $R^*$ is decreasing in $c$ and when $c\rightarrow~0$, $R^*=1.25$ and when $c=0.25$, then $R^*=0.25$. However, due to the fact that $q$ does not have an explicit expression (I can solve the equation of $q$ and get two solutions, but that does not seem to help), I couldn't come up with a way to prove this claim. I can prove the case of $c\rightarrow~0$, but for general $c\in[0,1/4)$, it seems to be very challenging.

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