Dynamic programming's principle of optimality as an abstract construct
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In dynamic programming, the principle of optimality (refer to Bertsekas's Optimal Control, volume 1, page 18) is a statement that says:
For any optimal policy $pi$ , we always have a suboptimal policy; which solves the corresponding sub-problem.
I want to reformulate this problem as an abstract construct, using the tools of category theory. First of all, I tried to formulate the set of all Paths flowing from A to B as a set $S_{AB}$. Similarly, I can do it for all the pairs of points in the space .This can be viewed as: a filter on the set of all the possible paths; that start and end at the same points. There are, $C(n,2)$ such sets.(does it sound some what like homotopy or something from algebraic topology?)
Now, from each of these sets, I can consider a shortest path as: some index on these objects. Then, I realize that: the principal of optimality, reduces to saying, such an index category is a monoid.
Now, how do I formulate this entire thing as an universal property?
I am trying to make a generic statement; which goes like: "The optimal policy is a monoid on index category" or some other universal language. Can some one help me find, what i am looking for?
optimization category-theory control-theory optimal-control dynamic-programming
edited Oct 27 '18 at 14:46
Rodrigo de Azevedo
12.8k41855
asked Oct 11 '18 at 5:43
Sasikanth RaghavaSasikanth Raghava
557
$endgroup$
add a comment |
$begingroup$
In dynamic programming, the principle of optimality (refer to Bertsekas's Optimal Control, volume 1, page 18) is a statement that says:
For any optimal policy $pi$ , we always have a suboptimal policy; which solves the corresponding sub-problem.
I want to reformulate this problem as an abstract construct, using the tools of category theory. First of all, I tried to formulate the set of all Paths flowing from A to B as a set $S_{AB}$. Similarly, I can do it for all the pairs of points in the space .This can be viewed as: a filter on the set of all the possible paths; that start and end at the same points. There are, $C(n,2)$ such sets.(does it sound some what like homotopy or something from algebraic topology?)
Now, from each of these sets, I can consider a shortest path as: some index on these objects. Then, I realize that: the principal of optimality, reduces to saying, such an index category is a monoid.
Now, how do I formulate this entire thing as an universal property?
I am trying to make a generic statement; which goes like: "The optimal policy is a monoid on index category" or some other universal language. Can some one help me find, what i am looking for?
optimization category-theory control-theory optimal-control dynamic-programming
edited Oct 27 '18 at 14:46
Rodrigo de Azevedo
12.8k41855
asked Oct 11 '18 at 5:43
Sasikanth RaghavaSasikanth Raghava
557
$endgroup$
$begingroup$
0 OK matroids are what I might be looking at, there is an entire category of matroids . though, the universal properties do not look as beautiful as I would've expected. But of all the universal properties of matroids, in this particular case, for the purpose of dynamic programming you need just the hereditary property.
$endgroup$
– Sasikanth Raghava
Jan 11 at 6:28
add a comment |
$begingroup$
In dynamic programming, the principle of optimality (refer to Bertsekas's Optimal Control, volume 1, page 18) is a statement that says:
For any optimal policy $pi$ , we always have a suboptimal policy; which solves the corresponding sub-problem.
I want to reformulate this problem as an abstract construct, using the tools of category theory. First of all, I tried to formulate the set of all Paths flowing from A to B as a set $S_{AB}$. Similarly, I can do it for all the pairs of points in the space .This can be viewed as: a filter on the set of all the possible paths; that start and end at the same points. There are, $C(n,2)$ such sets.(does it sound some what like homotopy or something from algebraic topology?)
Now, from each of these sets, I can consider a shortest path as: some index on these objects. Then, I realize that: the principal of optimality, reduces to saying, such an index category is a monoid.
Now, how do I formulate this entire thing as an universal property?
I am trying to make a generic statement; which goes like: "The optimal policy is a monoid on index category" or some other universal language. Can some one help me find, what i am looking for?
optimization category-theory control-theory optimal-control dynamic-programming
edited Oct 27 '18 at 14:46
Rodrigo de Azevedo
12.8k41855
asked Oct 11 '18 at 5:43
Sasikanth RaghavaSasikanth Raghava
557
$endgroup$
In dynamic programming, the principle of optimality (refer to Bertsekas's Optimal Control, volume 1, page 18) is a statement that says:
For any optimal policy $pi$ , we always have a suboptimal policy; which solves the corresponding sub-problem.
I want to reformulate this problem as an abstract construct, using the tools of category theory. First of all, I tried to formulate the set of all Paths flowing from A to B as a set $S_{AB}$. Similarly, I can do it for all the pairs of points in the space .This can be viewed as: a filter on the set of all the possible paths; that start and end at the same points. There are, $C(n,2)$ such sets.(does it sound some what like homotopy or something from algebraic topology?)
Now, from each of these sets, I can consider a shortest path as: some index on these objects. Then, I realize that: the principal of optimality, reduces to saying, such an index category is a monoid.
Now, how do I formulate this entire thing as an universal property?
I am trying to make a generic statement; which goes like: "The optimal policy is a monoid on index category" or some other universal language. Can some one help me find, what i am looking for?
optimization category-theory control-theory optimal-control dynamic-programming
optimization category-theory control-theory optimal-control dynamic-programming
edited Oct 27 '18 at 14:46
Rodrigo de Azevedo
12.8k41855
asked Oct 11 '18 at 5:43
Sasikanth RaghavaSasikanth Raghava
557
edited Oct 27 '18 at 14:46
Rodrigo de Azevedo
12.8k41855
asked Oct 11 '18 at 5:43
Sasikanth RaghavaSasikanth Raghava
557
edited Oct 27 '18 at 14:46
Rodrigo de Azevedo
12.8k41855
edited Oct 27 '18 at 14:46
Rodrigo de Azevedo
12.8k41855
edited Oct 27 '18 at 14:46
Rodrigo de Azevedo
12.8k41855
12.8k41855
asked Oct 11 '18 at 5:43
Sasikanth RaghavaSasikanth Raghava
557
asked Oct 11 '18 at 5:43
Sasikanth RaghavaSasikanth Raghava
557
asked Oct 11 '18 at 5:43
Sasikanth RaghavaSasikanth Raghava
557
557
$begingroup$
0 OK matroids are what I might be looking at, there is an entire category of matroids . though, the universal properties do not look as beautiful as I would've expected. But of all the universal properties of matroids, in this particular case, for the purpose of dynamic programming you need just the hereditary property.
$endgroup$
– Sasikanth Raghava
Jan 11 at 6:28
add a comment |
$begingroup$
0 OK matroids are what I might be looking at, there is an entire category of matroids . though, the universal properties do not look as beautiful as I would've expected. But of all the universal properties of matroids, in this particular case, for the purpose of dynamic programming you need just the hereditary property.
$endgroup$
– Sasikanth Raghava
Jan 11 at 6:28
$begingroup$
0 OK matroids are what I might be looking at, there is an entire category of matroids . though, the universal properties do not look as beautiful as I would've expected. But of all the universal properties of matroids, in this particular case, for the purpose of dynamic programming you need just the hereditary property.
$endgroup$
– Sasikanth Raghava
Jan 11 at 6:28
$begingroup$
0 OK matroids are what I might be looking at, there is an entire category of matroids . though, the universal properties do not look as beautiful as I would've expected. But of all the universal properties of matroids, in this particular case, for the purpose of dynamic programming you need just the hereditary property.
$endgroup$
– Sasikanth Raghava
Jan 11 at 6:28
add a comment |
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0 OK matroids are what I might be looking at, there is an entire category of matroids . though, the universal properties do not look as beautiful as I would've expected. But of all the universal properties of matroids, in this particular case, for the purpose of dynamic programming you need just the hereditary property.
$endgroup$
– Sasikanth Raghava
Jan 11 at 6:28