Mixing
Placing $n$ linear functions so that it is best fit to another function in integral norm sense?
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Say we want to build a function which is piecewise linear $$f(x) = sum_forall k (H(x-x_k)-H(x-x_k+1))l_k(x)\l_k(x) = c_k1x+c_k2$$
And also so that it fits best possibly some function $xto g(x)$: $$l_k,x_k=min_l_k,x_kleftf(x)-g(x)$$
Please note that the line end point coordinates $x_k$ we can decide for ourselves.
I've made some numerical approaches which seem promising on this, but how can one approach it algebraically/analytically?
calculus real-analysis optimization linear-approximation
asked Jul 24 at 18:44
mathreadler
13.6k71857
 |Â
show 2 more comments
up vote
-1
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Say we want to build a function which is piecewise linear $$f(x) = sum_forall k (H(x-x_k)-H(x-x_k+1))l_k(x)\l_k(x) = c_k1x+c_k2$$
And also so that it fits best possibly some function $xto g(x)$: $$l_k,x_k=min_l_k,x_kleftf(x)-g(x)$$
Please note that the line end point coordinates $x_k$ we can decide for ourselves.
I've made some numerical approaches which seem promising on this, but how can one approach it algebraically/analytically?
calculus real-analysis optimization linear-approximation
asked Jul 24 at 18:44
mathreadler
13.6k71857
1
If $H(x_k)$ represents the unit step function translated to $x_k$ the notation would be better $f(x) = sum_forall kleft(H(x-x_k)-H(x-x_k+1)right)l_k(x)$
– Cesareo
Jul 24 at 20:26
Yes it is intended to be "active between $x_k$ and $x_k+1$". Feel free to edit if you want.
– mathreadler
Jul 24 at 21:20
This kind of problem can be successfully handled with evolutionary programming -Evolution Strategies. See papers with those authors: Thomas Back, Gunter Rudolph, Hans-Paul Schwefel,
– Cesareo
Jul 25 at 12:59
@Cesareo sounds interesting. Are they related to genetic algorithms?
– mathreadler
Jul 25 at 18:27
Evolution Strategies have an Evolutionary paradigm but they use for each parameter real numbers instead a fractionary representation (chromosome). There are very efficient algorithms (CMA-ES) which converge in difficult non-convex situations.
– Cesareo
Jul 25 at 18:58
 |Â
show 2 more comments
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
Say we want to build a function which is piecewise linear $$f(x) = sum_forall k (H(x-x_k)-H(x-x_k+1))l_k(x)\l_k(x) = c_k1x+c_k2$$
And also so that it fits best possibly some function $xto g(x)$: $$l_k,x_k=min_l_k,x_kleftf(x)-g(x)$$
Please note that the line end point coordinates $x_k$ we can decide for ourselves.
I've made some numerical approaches which seem promising on this, but how can one approach it algebraically/analytically?
calculus real-analysis optimization linear-approximation
asked Jul 24 at 18:44
mathreadler
13.6k71857
Say we want to build a function which is piecewise linear $$f(x) = sum_forall k (H(x-x_k)-H(x-x_k+1))l_k(x)\l_k(x) = c_k1x+c_k2$$
And also so that it fits best possibly some function $xto g(x)$: $$l_k,x_k=min_l_k,x_kleftf(x)-g(x)$$
Please note that the line end point coordinates $x_k$ we can decide for ourselves.
I've made some numerical approaches which seem promising on this, but how can one approach it algebraically/analytically?
calculus real-analysis optimization linear-approximation
asked Jul 24 at 18:44
mathreadler
13.6k71857
edited Jul 24 at 21:22
asked Jul 24 at 18:44
mathreadler
13.6k71857
asked Jul 24 at 18:44
mathreadler
13.6k71857
asked Jul 24 at 18:44
mathreadler
13.6k71857
13.6k71857
1
If $H(x_k)$ represents the unit step function translated to $x_k$ the notation would be better $f(x) = sum_forall kleft(H(x-x_k)-H(x-x_k+1)right)l_k(x)$
– Cesareo
Jul 24 at 20:26
Yes it is intended to be "active between $x_k$ and $x_k+1$". Feel free to edit if you want.
– mathreadler
Jul 24 at 21:20
This kind of problem can be successfully handled with evolutionary programming -Evolution Strategies. See papers with those authors: Thomas Back, Gunter Rudolph, Hans-Paul Schwefel,
– Cesareo
Jul 25 at 12:59
@Cesareo sounds interesting. Are they related to genetic algorithms?
– mathreadler
Jul 25 at 18:27
Evolution Strategies have an Evolutionary paradigm but they use for each parameter real numbers instead a fractionary representation (chromosome). There are very efficient algorithms (CMA-ES) which converge in difficult non-convex situations.
– Cesareo
Jul 25 at 18:58
 |Â
show 2 more comments
1
If $H(x_k)$ represents the unit step function translated to $x_k$ the notation would be better $f(x) = sum_forall kleft(H(x-x_k)-H(x-x_k+1)right)l_k(x)$
– Cesareo
Jul 24 at 20:26
Yes it is intended to be "active between $x_k$ and $x_k+1$". Feel free to edit if you want.
– mathreadler
Jul 24 at 21:20
This kind of problem can be successfully handled with evolutionary programming -Evolution Strategies. See papers with those authors: Thomas Back, Gunter Rudolph, Hans-Paul Schwefel,
– Cesareo
Jul 25 at 12:59
@Cesareo sounds interesting. Are they related to genetic algorithms?
– mathreadler
Jul 25 at 18:27
Evolution Strategies have an Evolutionary paradigm but they use for each parameter real numbers instead a fractionary representation (chromosome). There are very efficient algorithms (CMA-ES) which converge in difficult non-convex situations.
– Cesareo
Jul 25 at 18:58
1
1
If $H(x_k)$ represents the unit step function translated to $x_k$ the notation would be better $f(x) = sum_forall kleft(H(x-x_k)-H(x-x_k+1)right)l_k(x)$
– Cesareo
Jul 24 at 20:26
If $H(x_k)$ represents the unit step function translated to $x_k$ the notation would be better $f(x) = sum_forall kleft(H(x-x_k)-H(x-x_k+1)right)l_k(x)$
– Cesareo
Jul 24 at 20:26
Yes it is intended to be "active between $x_k$ and $x_k+1$". Feel free to edit if you want.
– mathreadler
Jul 24 at 21:20
Yes it is intended to be "active between $x_k$ and $x_k+1$". Feel free to edit if you want.
– mathreadler
Jul 24 at 21:20
This kind of problem can be successfully handled with evolutionary programming -Evolution Strategies. See papers with those authors: Thomas Back, Gunter Rudolph, Hans-Paul Schwefel,
– Cesareo
Jul 25 at 12:59
This kind of problem can be successfully handled with evolutionary programming -Evolution Strategies. See papers with those authors: Thomas Back, Gunter Rudolph, Hans-Paul Schwefel,
– Cesareo
Jul 25 at 12:59
@Cesareo sounds interesting. Are they related to genetic algorithms?
– mathreadler
Jul 25 at 18:27
@Cesareo sounds interesting. Are they related to genetic algorithms?
– mathreadler
Jul 25 at 18:27
Evolution Strategies have an Evolutionary paradigm but they use for each parameter real numbers instead a fractionary representation (chromosome). There are very efficient algorithms (CMA-ES) which converge in difficult non-convex situations.
– Cesareo
Jul 25 at 18:58
Evolution Strategies have an Evolutionary paradigm but they use for each parameter real numbers instead a fractionary representation (chromosome). There are very efficient algorithms (CMA-ES) which converge in difficult non-convex situations.
– Cesareo
Jul 25 at 18:58
 |Â
show 2 more comments
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1
If $H(x_k)$ represents the unit step function translated to $x_k$ the notation would be better $f(x) = sum_forall kleft(H(x-x_k)-H(x-x_k+1)right)l_k(x)$
– Cesareo
Jul 24 at 20:26
Yes it is intended to be "active between $x_k$ and $x_k+1$". Feel free to edit if you want.
– mathreadler
Jul 24 at 21:20
This kind of problem can be successfully handled with evolutionary programming -Evolution Strategies. See papers with those authors: Thomas Back, Gunter Rudolph, Hans-Paul Schwefel,
– Cesareo
Jul 25 at 12:59
@Cesareo sounds interesting. Are they related to genetic algorithms?
– mathreadler
Jul 25 at 18:27
Evolution Strategies have an Evolutionary paradigm but they use for each parameter real numbers instead a fractionary representation (chromosome). There are very efficient algorithms (CMA-ES) which converge in difficult non-convex situations.
– Cesareo
Jul 25 at 18:58