Algorithm for finding sequence verifying a floor equation
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We are looking for an algorithm solving the following problem.
Given a sequence $ 0 < x_1< dots < x_n $ find a sequence $0 < y_1 < dots < y_n$ such that $forall j in {2, dots, n-1}, i in {1, dots, j-1}, y in [y_j, y_{j+1}[ quad leftlfloor frac{y}{y_i} rightrfloor= leftlfloor frac{y_{j+1}}{y_i} rightrfloor ,$
while minimizing $sum_{i=1}^n a_i |y_i - x_i| $ with $a_1,dots,a_n in mathbb{R}^+$.
The distance may be replaced by another of the same spirit if it allows for a nice solution.
optimization algorithms floor-function
asked Nov 20 at 10:58
Alfred M.
65
|
show 2 more comments
up vote
1
down vote
favorite
We are looking for an algorithm solving the following problem.
Given a sequence $ 0 < x_1< dots < x_n $ find a sequence $0 < y_1 < dots < y_n$ such that $forall j in {2, dots, n-1}, i in {1, dots, j-1}, y in [y_j, y_{j+1}[ quad leftlfloor frac{y}{y_i} rightrfloor= leftlfloor frac{y_{j+1}}{y_i} rightrfloor ,$
while minimizing $sum_{i=1}^n a_i |y_i - x_i| $ with $a_1,dots,a_n in mathbb{R}^+$.
The distance may be replaced by another of the same spirit if it allows for a nice solution.
optimization algorithms floor-function
asked Nov 20 at 10:58
Alfred M.
65
2
What do you mean by "almost surely"? I don't see how the formal meaning would apply here.
– Todor Markov
Nov 22 at 17:04
what is $n$, typically?
– LinAlg
Nov 25 at 20:03
@LinAlg n is typically between 5 and 20
– Alfred M.
Nov 26 at 9:18
@TodorMarkov: True, this was ambiguous. I changed the formulation.
– Alfred M.
Nov 26 at 9:19
Thank for you significantly modifying the question four days after a comment. The new formulation does not make much sense to me since you do not need the index $j$ as you require $leftlfloor frac{y}{y_i} rightrfloor= leftlfloor frac{y_{i+1}}{y_i} rightrfloor$ for all $i$ and for all $y$ in $[y_1,y_i)$
– LinAlg
Nov 26 at 13:48
|
show 2 more comments
up vote
1
down vote
favorite
up vote
1
down vote
favorite
We are looking for an algorithm solving the following problem.
Given a sequence $ 0 < x_1< dots < x_n $ find a sequence $0 < y_1 < dots < y_n$ such that $forall j in {2, dots, n-1}, i in {1, dots, j-1}, y in [y_j, y_{j+1}[ quad leftlfloor frac{y}{y_i} rightrfloor= leftlfloor frac{y_{j+1}}{y_i} rightrfloor ,$
while minimizing $sum_{i=1}^n a_i |y_i - x_i| $ with $a_1,dots,a_n in mathbb{R}^+$.
The distance may be replaced by another of the same spirit if it allows for a nice solution.
optimization algorithms floor-function
asked Nov 20 at 10:58
Alfred M.
65
We are looking for an algorithm solving the following problem.
Given a sequence $ 0 < x_1< dots < x_n $ find a sequence $0 < y_1 < dots < y_n$ such that $forall j in {2, dots, n-1}, i in {1, dots, j-1}, y in [y_j, y_{j+1}[ quad leftlfloor frac{y}{y_i} rightrfloor= leftlfloor frac{y_{j+1}}{y_i} rightrfloor ,$
while minimizing $sum_{i=1}^n a_i |y_i - x_i| $ with $a_1,dots,a_n in mathbb{R}^+$.
The distance may be replaced by another of the same spirit if it allows for a nice solution.
optimization algorithms floor-function
optimization algorithms floor-function
asked Nov 20 at 10:58
Alfred M.
65
asked Nov 20 at 10:58
Alfred M.
65
edited Nov 27 at 14:21
asked Nov 20 at 10:58
Alfred M.
65
asked Nov 20 at 10:58
Alfred M.
65
asked Nov 20 at 10:58
Alfred M.
65
65
2
What do you mean by "almost surely"? I don't see how the formal meaning would apply here.
– Todor Markov
Nov 22 at 17:04
what is $n$, typically?
– LinAlg
Nov 25 at 20:03
@LinAlg n is typically between 5 and 20
– Alfred M.
Nov 26 at 9:18
@TodorMarkov: True, this was ambiguous. I changed the formulation.
– Alfred M.
Nov 26 at 9:19
Thank for you significantly modifying the question four days after a comment. The new formulation does not make much sense to me since you do not need the index $j$ as you require $leftlfloor frac{y}{y_i} rightrfloor= leftlfloor frac{y_{i+1}}{y_i} rightrfloor$ for all $i$ and for all $y$ in $[y_1,y_i)$
– LinAlg
Nov 26 at 13:48
|
show 2 more comments
2
What do you mean by "almost surely"? I don't see how the formal meaning would apply here.
– Todor Markov
Nov 22 at 17:04
what is $n$, typically?
– LinAlg
Nov 25 at 20:03
@LinAlg n is typically between 5 and 20
– Alfred M.
Nov 26 at 9:18
@TodorMarkov: True, this was ambiguous. I changed the formulation.
– Alfred M.
Nov 26 at 9:19
Thank for you significantly modifying the question four days after a comment. The new formulation does not make much sense to me since you do not need the index $j$ as you require $leftlfloor frac{y}{y_i} rightrfloor= leftlfloor frac{y_{i+1}}{y_i} rightrfloor$ for all $i$ and for all $y$ in $[y_1,y_i)$
– LinAlg
Nov 26 at 13:48
2
2
What do you mean by "almost surely"? I don't see how the formal meaning would apply here.
– Todor Markov
Nov 22 at 17:04
What do you mean by "almost surely"? I don't see how the formal meaning would apply here.
– Todor Markov
Nov 22 at 17:04
what is $n$, typically?
– LinAlg
Nov 25 at 20:03
what is $n$, typically?
– LinAlg
Nov 25 at 20:03
@LinAlg n is typically between 5 and 20
– Alfred M.
Nov 26 at 9:18
@LinAlg n is typically between 5 and 20
– Alfred M.
Nov 26 at 9:18
@TodorMarkov: True, this was ambiguous. I changed the formulation.
– Alfred M.
Nov 26 at 9:19
@TodorMarkov: True, this was ambiguous. I changed the formulation.
– Alfred M.
Nov 26 at 9:19
Thank for you significantly modifying the question four days after a comment. The new formulation does not make much sense to me since you do not need the index $j$ as you require $leftlfloor frac{y}{y_i} rightrfloor= leftlfloor frac{y_{i+1}}{y_i} rightrfloor$ for all $i$ and for all $y$ in $[y_1,y_i)$
– LinAlg
Nov 26 at 13:48
Thank for you significantly modifying the question four days after a comment. The new formulation does not make much sense to me since you do not need the index $j$ as you require $leftlfloor frac{y}{y_i} rightrfloor= leftlfloor frac{y_{i+1}}{y_i} rightrfloor$ for all $i$ and for all $y$ in $[y_1,y_i)$
– LinAlg
Nov 26 at 13:48
|
show 2 more comments
1 Answer
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0
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Here is a solution, likely sub-optimal.
$ forall i in {1, dots, n-2}, ; text{let} ; f_i: x mapsto (leftlfloor {x}/{x_i} rightrfloor + 1) ; x_i$.
- Set $ y_1 := x_1 $ and $ y_2 := x_2 $.
$ forall k in {3, dots, n} $ set $ y_k := min left[ x_k, min_i f_i(x_k) right]. $
Another solution is to correct $ x_2 $:
- Set $ y_1 := x_1 $.
- Set $ y_2 := max (x_2, (leftlfloor x_3/x_1 rightrfloor) ; x_1 )$
$ forall k in {4, dots, n} $ set $ y_k := min left[ x_k, min_i f_i(x_k) right]. $
answered Nov 20 at 11:07
Alfred M.
65
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
Here is a solution, likely sub-optimal.
$ forall i in {1, dots, n-2}, ; text{let} ; f_i: x mapsto (leftlfloor {x}/{x_i} rightrfloor + 1) ; x_i$.
- Set $ y_1 := x_1 $ and $ y_2 := x_2 $.
$ forall k in {3, dots, n} $ set $ y_k := min left[ x_k, min_i f_i(x_k) right]. $
Another solution is to correct $ x_2 $:
- Set $ y_1 := x_1 $.
- Set $ y_2 := max (x_2, (leftlfloor x_3/x_1 rightrfloor) ; x_1 )$
$ forall k in {4, dots, n} $ set $ y_k := min left[ x_k, min_i f_i(x_k) right]. $
answered Nov 20 at 11:07
Alfred M.
65
add a comment |
up vote
0
down vote
Here is a solution, likely sub-optimal.
$ forall i in {1, dots, n-2}, ; text{let} ; f_i: x mapsto (leftlfloor {x}/{x_i} rightrfloor + 1) ; x_i$.
- Set $ y_1 := x_1 $ and $ y_2 := x_2 $.
$ forall k in {3, dots, n} $ set $ y_k := min left[ x_k, min_i f_i(x_k) right]. $
Another solution is to correct $ x_2 $:
- Set $ y_1 := x_1 $.
- Set $ y_2 := max (x_2, (leftlfloor x_3/x_1 rightrfloor) ; x_1 )$
$ forall k in {4, dots, n} $ set $ y_k := min left[ x_k, min_i f_i(x_k) right]. $
answered Nov 20 at 11:07
Alfred M.
65
add a comment |
up vote
0
down vote
up vote
0
down vote
Here is a solution, likely sub-optimal.
$ forall i in {1, dots, n-2}, ; text{let} ; f_i: x mapsto (leftlfloor {x}/{x_i} rightrfloor + 1) ; x_i$.
- Set $ y_1 := x_1 $ and $ y_2 := x_2 $.
$ forall k in {3, dots, n} $ set $ y_k := min left[ x_k, min_i f_i(x_k) right]. $
Another solution is to correct $ x_2 $:
- Set $ y_1 := x_1 $.
- Set $ y_2 := max (x_2, (leftlfloor x_3/x_1 rightrfloor) ; x_1 )$
$ forall k in {4, dots, n} $ set $ y_k := min left[ x_k, min_i f_i(x_k) right]. $
answered Nov 20 at 11:07
Alfred M.
65
Here is a solution, likely sub-optimal.
$ forall i in {1, dots, n-2}, ; text{let} ; f_i: x mapsto (leftlfloor {x}/{x_i} rightrfloor + 1) ; x_i$.
- Set $ y_1 := x_1 $ and $ y_2 := x_2 $.
$ forall k in {3, dots, n} $ set $ y_k := min left[ x_k, min_i f_i(x_k) right]. $
Another solution is to correct $ x_2 $:
- Set $ y_1 := x_1 $.
- Set $ y_2 := max (x_2, (leftlfloor x_3/x_1 rightrfloor) ; x_1 )$
$ forall k in {4, dots, n} $ set $ y_k := min left[ x_k, min_i f_i(x_k) right]. $
answered Nov 20 at 11:07
Alfred M.
65
edited Nov 22 at 14:18
answered Nov 20 at 11:07
Alfred M.
65
answered Nov 20 at 11:07
Alfred M.
65
answered Nov 20 at 11:07
Alfred M.
65
65
add a comment |
add a comment |
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2
What do you mean by "almost surely"? I don't see how the formal meaning would apply here.
– Todor Markov
Nov 22 at 17:04
what is $n$, typically?
– LinAlg
Nov 25 at 20:03
@LinAlg n is typically between 5 and 20
– Alfred M.
Nov 26 at 9:18
@TodorMarkov: True, this was ambiguous. I changed the formulation.
– Alfred M.
Nov 26 at 9:19
Thank for you significantly modifying the question four days after a comment. The new formulation does not make much sense to me since you do not need the index $j$ as you require $leftlfloor frac{y}{y_i} rightrfloor= leftlfloor frac{y_{i+1}}{y_i} rightrfloor$ for all $i$ and for all $y$ in $[y_1,y_i)$
– LinAlg
Nov 26 at 13:48