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Solving optimization problem when the optimization objective is implicit
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I would like to solve an optimization problem which has the following format:
$$max_x (y + f(x)), y in mathbbR, x in mathbbR^n$$ and $y$ can only be solved through an implicit function with $x, g(x,y)=c$.
More specifically, given vectors $a in mathbbR^n, b in mathbbR^n, c in mathbbR^n$, we have $f(x)=fracsum_i x_i a_i c_isum_i x_i a_i$
we have the implicit function $g$
$$sum_i=1^i=n x_i a_i = sum_i=1^i=n [sum_k=1^k=b[i] fracc_i x_iy^k]$$
So how should I proceed to solve this optimization? should I use approximation to solve $y=g'(x)$ ?
Any help appreciated !! Thanks !!!
optimization nonlinear-optimization implicit-differentiation
edited Jul 27 at 14:07
Michael Grant
14.6k11743
asked Jul 25 at 14:11
Yue X
12
 |Â
show 6 more comments
up vote
0
down vote
favorite
I would like to solve an optimization problem which has the following format:
$$max_x (y + f(x)), y in mathbbR, x in mathbbR^n$$ and $y$ can only be solved through an implicit function with $x, g(x,y)=c$.
More specifically, given vectors $a in mathbbR^n, b in mathbbR^n, c in mathbbR^n$, we have $f(x)=fracsum_i x_i a_i c_isum_i x_i a_i$
we have the implicit function $g$
$$sum_i=1^i=n x_i a_i = sum_i=1^i=n [sum_k=1^k=b[i] fracc_i x_iy^k]$$
So how should I proceed to solve this optimization? should I use approximation to solve $y=g'(x)$ ?
Any help appreciated !! Thanks !!!
optimization nonlinear-optimization implicit-differentiation
edited Jul 27 at 14:07
Michael Grant
14.6k11743
asked Jul 25 at 14:11
Yue X
12
can you share $f$ and $g$ with us?
– LinAlg
Jul 25 at 14:15
Are you able to differentiate $g(x,y)$ to determine the rate at which changing $x$ changes $y$? Or conversely, since you make no mention of constraints, why not treat $x$ and $f(x)$ as depending on the choice of $y$? You've told us nothing about $g(x,y)$, so it's guesswork how one solves $g(x,y)=c$ in either direction.
– hardmath
Jul 25 at 14:18
@LinAlg I have edited the post thanks!
– Yue X
Jul 25 at 14:33
@hardmath I have edited the post thanks!
– Yue X
Jul 25 at 14:33
2
There is nothing special about this. This is an equality constrained nonlinear optimization problem in $x$ and $y$.
– Johan Löfberg
Jul 26 at 15:43
 |Â
show 6 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I would like to solve an optimization problem which has the following format:
$$max_x (y + f(x)), y in mathbbR, x in mathbbR^n$$ and $y$ can only be solved through an implicit function with $x, g(x,y)=c$.
More specifically, given vectors $a in mathbbR^n, b in mathbbR^n, c in mathbbR^n$, we have $f(x)=fracsum_i x_i a_i c_isum_i x_i a_i$
we have the implicit function $g$
$$sum_i=1^i=n x_i a_i = sum_i=1^i=n [sum_k=1^k=b[i] fracc_i x_iy^k]$$
So how should I proceed to solve this optimization? should I use approximation to solve $y=g'(x)$ ?
Any help appreciated !! Thanks !!!
optimization nonlinear-optimization implicit-differentiation
edited Jul 27 at 14:07
Michael Grant
14.6k11743
asked Jul 25 at 14:11
Yue X
12
I would like to solve an optimization problem which has the following format:
$$max_x (y + f(x)), y in mathbbR, x in mathbbR^n$$ and $y$ can only be solved through an implicit function with $x, g(x,y)=c$.
More specifically, given vectors $a in mathbbR^n, b in mathbbR^n, c in mathbbR^n$, we have $f(x)=fracsum_i x_i a_i c_isum_i x_i a_i$
we have the implicit function $g$
$$sum_i=1^i=n x_i a_i = sum_i=1^i=n [sum_k=1^k=b[i] fracc_i x_iy^k]$$
So how should I proceed to solve this optimization? should I use approximation to solve $y=g'(x)$ ?
Any help appreciated !! Thanks !!!
optimization nonlinear-optimization implicit-differentiation
edited Jul 27 at 14:07
Michael Grant
14.6k11743
asked Jul 25 at 14:11
Yue X
12
edited Jul 27 at 14:07
Michael Grant
14.6k11743
edited Jul 27 at 14:07
Michael Grant
14.6k11743
edited Jul 27 at 14:07
Michael Grant
14.6k11743
14.6k11743
asked Jul 25 at 14:11
Yue X
12
asked Jul 25 at 14:11
Yue X
12
asked Jul 25 at 14:11
Yue X
12
12
can you share $f$ and $g$ with us?
– LinAlg
Jul 25 at 14:15
Are you able to differentiate $g(x,y)$ to determine the rate at which changing $x$ changes $y$? Or conversely, since you make no mention of constraints, why not treat $x$ and $f(x)$ as depending on the choice of $y$? You've told us nothing about $g(x,y)$, so it's guesswork how one solves $g(x,y)=c$ in either direction.
– hardmath
Jul 25 at 14:18
@LinAlg I have edited the post thanks!
– Yue X
Jul 25 at 14:33
@hardmath I have edited the post thanks!
– Yue X
Jul 25 at 14:33
2
There is nothing special about this. This is an equality constrained nonlinear optimization problem in $x$ and $y$.
– Johan Löfberg
Jul 26 at 15:43
 |Â
show 6 more comments
can you share $f$ and $g$ with us?
– LinAlg
Jul 25 at 14:15
Are you able to differentiate $g(x,y)$ to determine the rate at which changing $x$ changes $y$? Or conversely, since you make no mention of constraints, why not treat $x$ and $f(x)$ as depending on the choice of $y$? You've told us nothing about $g(x,y)$, so it's guesswork how one solves $g(x,y)=c$ in either direction.
– hardmath
Jul 25 at 14:18
@LinAlg I have edited the post thanks!
– Yue X
Jul 25 at 14:33
@hardmath I have edited the post thanks!
– Yue X
Jul 25 at 14:33
2
There is nothing special about this. This is an equality constrained nonlinear optimization problem in $x$ and $y$.
– Johan Löfberg
Jul 26 at 15:43
can you share $f$ and $g$ with us?
– LinAlg
Jul 25 at 14:15
can you share $f$ and $g$ with us?
– LinAlg
Jul 25 at 14:15
Are you able to differentiate $g(x,y)$ to determine the rate at which changing $x$ changes $y$? Or conversely, since you make no mention of constraints, why not treat $x$ and $f(x)$ as depending on the choice of $y$? You've told us nothing about $g(x,y)$, so it's guesswork how one solves $g(x,y)=c$ in either direction.
– hardmath
Jul 25 at 14:18
Are you able to differentiate $g(x,y)$ to determine the rate at which changing $x$ changes $y$? Or conversely, since you make no mention of constraints, why not treat $x$ and $f(x)$ as depending on the choice of $y$? You've told us nothing about $g(x,y)$, so it's guesswork how one solves $g(x,y)=c$ in either direction.
– hardmath
Jul 25 at 14:18
@LinAlg I have edited the post thanks!
– Yue X
Jul 25 at 14:33
@LinAlg I have edited the post thanks!
– Yue X
Jul 25 at 14:33
@hardmath I have edited the post thanks!
– Yue X
Jul 25 at 14:33
@hardmath I have edited the post thanks!
– Yue X
Jul 25 at 14:33
2
2
There is nothing special about this. This is an equality constrained nonlinear optimization problem in $x$ and $y$.
– Johan Löfberg
Jul 26 at 15:43
There is nothing special about this. This is an equality constrained nonlinear optimization problem in $x$ and $y$.
– Johan Löfberg
Jul 26 at 15:43
 |Â
show 6 more comments
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can you share $f$ and $g$ with us?
– LinAlg
Jul 25 at 14:15
Are you able to differentiate $g(x,y)$ to determine the rate at which changing $x$ changes $y$? Or conversely, since you make no mention of constraints, why not treat $x$ and $f(x)$ as depending on the choice of $y$? You've told us nothing about $g(x,y)$, so it's guesswork how one solves $g(x,y)=c$ in either direction.
– hardmath
Jul 25 at 14:18
@LinAlg I have edited the post thanks!
– Yue X
Jul 25 at 14:33
@hardmath I have edited the post thanks!
– Yue X
Jul 25 at 14:33
2
There is nothing special about this. This is an equality constrained nonlinear optimization problem in $x$ and $y$.
– Johan Löfberg
Jul 26 at 15:43