# Is dynamic programming suitable for a specific optimization problem?

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.