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.


Your Answer

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

Browse other questions tagged or ask your own question.