Let $c,\,\mathcal{P}_0,\,\mathcal{P}_1,\,\mathcal{P}_2,\ldots$ be a sequence of positive real numbers.

Let $N\in\{1,\,2,\,3,\ldots\}$ and let $t\in\{0,\,1,\,2,\ldots\}$, with $N$ and $t$ fixed.

For all $j\in\{1,\ldots,\,N\}$ and $t\in\{0,\,1,\,2,\ldots,\}$, let $q_j^{(t)}=p_j^{(t)}\cdot(1-p_j^{(t-1)})\cdot(1-p_j^{(t-2)})\cdot\ldots\cdot(1-p_j^{(0)})$, with $p_j^{(k)}\in(0,\,1)$ and $p_j^{(k)}$ is a function of $\mathcal{P}_k$, for all $k\in\{0,\,1,\,2,\ldots,\,t\}$.

For all $t\in\{0,\,1,\,2,\ldots\}$ and with $\delta\in(0,\,1]$ known, let $$f(\mathcal{P}_0,\,\mathcal{P}_1,\ldots,\,\mathcal{P}_t)=\left\{\prod_{j=1}^{N}(1-q_j^{(t)})\right\}\cdot\left\{\delta^0\cdot(\mathcal{P}_0-c)\cdot\sum_{j=1}^{N}q_j^{(0)}+\cdots+\delta^t\cdot(\mathcal{P}_t-c)\cdot\sum_{j=1}^{N}q_j^{(t)}\right\}$$

Problem: Determine the values of $\mathcal{P}_0,\,\mathcal{P}_1,\,\mathcal{P}_2,\ldots$ that maximize $$F(\mathcal{P}_0,\,\mathcal{P}_1,\,\mathcal{P}_2,\ldots)=\sum_{t=0}^{+\infty}f(\mathcal{P}_0,\,\mathcal{P}_1,\ldots,\,\mathcal{P}_t),$$ subject to $\mathcal{P}_t>c$, for all $t\in\{0,\,1,\,2,\ldots\}$.

To solve this optimization problem, I am trying to use dynamic programming. However, I do not know whether I can apply dynamic programming in this optimization problem. I was wondering if you could tell me whether I can use this technique and how I can do that, please. After knowing that, I can write the respective program in ${\tt R}$-software.

If you have any question, please let me know. Thank you very much for your help and suggestions.

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.