Is dynamic programming suitable for a specific optimization problem?
$begingroup$
Let $c,,mathcal{P}_0,,mathcal{P}_1,,mathcal{P}_2,ldots$ be a sequence of positive real numbers.
Let $Nin{1,,2,,3,ldots}$ and let $tin{0,,1,,2,ldots}$, with $N$ and $t$ fixed.
For all $jin{1,ldots,,N}$ and $tin{0,,1,,2,ldots,}$, let $q_j^{(t)}=p_j^{(t)}cdot(1-p_j^{(t-1)})cdot(1-p_j^{(t-2)})cdotldotscdot(1-p_j^{(0)})$, with $p_j^{(k)}in(0,,1)$ and $p_j^{(k)}$ is a function of $mathcal{P}_k$, for all $kin{0,,1,,2,ldots,,t}$.
For all $tin{0,,1,,2,ldots}$ and with $deltain(0,,1]$ known, let
$$f(mathcal{P}_0,,mathcal{P}_1,ldots,,mathcal{P}_t)=left{prod_{j=1}^{N}(1-q_j^{(t)})right}cdotleft{delta^0cdot(mathcal{P}_0-c)cdotsum_{j=1}^{N}q_j^{(0)}+cdots+delta^tcdot(mathcal{P}_t-c)cdotsum_{j=1}^{N}q_j^{(t)}right}$$
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 $tin{0,,1,,2,ldots}$.
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.
optimization convex-optimization nonlinear-optimization numerical-optimization optimal-control
asked Dec 6 '18 at 16:19
Student1981Student1981
164
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add a comment |
$begingroup$
Let $c,,mathcal{P}_0,,mathcal{P}_1,,mathcal{P}_2,ldots$ be a sequence of positive real numbers.
Let $Nin{1,,2,,3,ldots}$ and let $tin{0,,1,,2,ldots}$, with $N$ and $t$ fixed.
For all $jin{1,ldots,,N}$ and $tin{0,,1,,2,ldots,}$, let $q_j^{(t)}=p_j^{(t)}cdot(1-p_j^{(t-1)})cdot(1-p_j^{(t-2)})cdotldotscdot(1-p_j^{(0)})$, with $p_j^{(k)}in(0,,1)$ and $p_j^{(k)}$ is a function of $mathcal{P}_k$, for all $kin{0,,1,,2,ldots,,t}$.
For all $tin{0,,1,,2,ldots}$ and with $deltain(0,,1]$ known, let
$$f(mathcal{P}_0,,mathcal{P}_1,ldots,,mathcal{P}_t)=left{prod_{j=1}^{N}(1-q_j^{(t)})right}cdotleft{delta^0cdot(mathcal{P}_0-c)cdotsum_{j=1}^{N}q_j^{(0)}+cdots+delta^tcdot(mathcal{P}_t-c)cdotsum_{j=1}^{N}q_j^{(t)}right}$$
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 $tin{0,,1,,2,ldots}$.
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.
optimization convex-optimization nonlinear-optimization numerical-optimization optimal-control
asked Dec 6 '18 at 16:19
Student1981Student1981
164
$endgroup$
add a comment |
$begingroup$
Let $c,,mathcal{P}_0,,mathcal{P}_1,,mathcal{P}_2,ldots$ be a sequence of positive real numbers.
Let $Nin{1,,2,,3,ldots}$ and let $tin{0,,1,,2,ldots}$, with $N$ and $t$ fixed.
For all $jin{1,ldots,,N}$ and $tin{0,,1,,2,ldots,}$, let $q_j^{(t)}=p_j^{(t)}cdot(1-p_j^{(t-1)})cdot(1-p_j^{(t-2)})cdotldotscdot(1-p_j^{(0)})$, with $p_j^{(k)}in(0,,1)$ and $p_j^{(k)}$ is a function of $mathcal{P}_k$, for all $kin{0,,1,,2,ldots,,t}$.
For all $tin{0,,1,,2,ldots}$ and with $deltain(0,,1]$ known, let
$$f(mathcal{P}_0,,mathcal{P}_1,ldots,,mathcal{P}_t)=left{prod_{j=1}^{N}(1-q_j^{(t)})right}cdotleft{delta^0cdot(mathcal{P}_0-c)cdotsum_{j=1}^{N}q_j^{(0)}+cdots+delta^tcdot(mathcal{P}_t-c)cdotsum_{j=1}^{N}q_j^{(t)}right}$$
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 $tin{0,,1,,2,ldots}$.
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.
optimization convex-optimization nonlinear-optimization numerical-optimization optimal-control
asked Dec 6 '18 at 16:19
Student1981Student1981
164
$endgroup$
Let $c,,mathcal{P}_0,,mathcal{P}_1,,mathcal{P}_2,ldots$ be a sequence of positive real numbers.
Let $Nin{1,,2,,3,ldots}$ and let $tin{0,,1,,2,ldots}$, with $N$ and $t$ fixed.
For all $jin{1,ldots,,N}$ and $tin{0,,1,,2,ldots,}$, let $q_j^{(t)}=p_j^{(t)}cdot(1-p_j^{(t-1)})cdot(1-p_j^{(t-2)})cdotldotscdot(1-p_j^{(0)})$, with $p_j^{(k)}in(0,,1)$ and $p_j^{(k)}$ is a function of $mathcal{P}_k$, for all $kin{0,,1,,2,ldots,,t}$.
For all $tin{0,,1,,2,ldots}$ and with $deltain(0,,1]$ known, let
$$f(mathcal{P}_0,,mathcal{P}_1,ldots,,mathcal{P}_t)=left{prod_{j=1}^{N}(1-q_j^{(t)})right}cdotleft{delta^0cdot(mathcal{P}_0-c)cdotsum_{j=1}^{N}q_j^{(0)}+cdots+delta^tcdot(mathcal{P}_t-c)cdotsum_{j=1}^{N}q_j^{(t)}right}$$
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 $tin{0,,1,,2,ldots}$.
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.
optimization convex-optimization nonlinear-optimization numerical-optimization optimal-control
optimization convex-optimization nonlinear-optimization numerical-optimization optimal-control
asked Dec 6 '18 at 16:19
Student1981Student1981
164
asked Dec 6 '18 at 16:19
Student1981Student1981
164
asked Dec 6 '18 at 16:19
Student1981Student1981
164
asked Dec 6 '18 at 16:19
Student1981Student1981
164
asked Dec 6 '18 at 16:19
Student1981Student1981
164
164
add a comment |
add a comment |
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