Optimization problem with constraint is another optimization problem
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I would like to know how to solve a problem like this:
$$min f(x) s.t.; g(x)leq 0$$.
However, we do not have explicit form of $g(x)$. What we know is that $g(x)$ is another solution of an optimization problem, say $g(x)=inf_{beta} h(x,beta)$. Here, $h(x,beta)$ is known, but assuming that we cannot solve the optimization analytically. I would like to know what kind of such optimization called and ways to solve this kind of problem.
optimization convex-optimization nonlinear-optimization
asked Dec 28 '18 at 10:25
will_cheukwill_cheuk
1278
$endgroup$
add a comment |
$begingroup$
I would like to know how to solve a problem like this:
$$min f(x) s.t.; g(x)leq 0$$.
However, we do not have explicit form of $g(x)$. What we know is that $g(x)$ is another solution of an optimization problem, say $g(x)=inf_{beta} h(x,beta)$. Here, $h(x,beta)$ is known, but assuming that we cannot solve the optimization analytically. I would like to know what kind of such optimization called and ways to solve this kind of problem.
optimization convex-optimization nonlinear-optimization
asked Dec 28 '18 at 10:25
will_cheukwill_cheuk
1278
$endgroup$
$begingroup$
Denote $z=(x,beta)$ and definte $F(z)=f(x)$. Then your problem is basically $min_z F(z)$ subject to $h(z)le 0$.
$endgroup$
– A.Γ.
Dec 28 '18 at 19:18
1
$begingroup$
These are called "bilevel optimization problems." There's quite a bit of literature on them.
$endgroup$
– Brian Borchers
Dec 28 '18 at 20:57
add a comment |
$begingroup$
I would like to know how to solve a problem like this:
$$min f(x) s.t.; g(x)leq 0$$.
However, we do not have explicit form of $g(x)$. What we know is that $g(x)$ is another solution of an optimization problem, say $g(x)=inf_{beta} h(x,beta)$. Here, $h(x,beta)$ is known, but assuming that we cannot solve the optimization analytically. I would like to know what kind of such optimization called and ways to solve this kind of problem.
optimization convex-optimization nonlinear-optimization
asked Dec 28 '18 at 10:25
will_cheukwill_cheuk
1278
$endgroup$
I would like to know how to solve a problem like this:
$$min f(x) s.t.; g(x)leq 0$$.
However, we do not have explicit form of $g(x)$. What we know is that $g(x)$ is another solution of an optimization problem, say $g(x)=inf_{beta} h(x,beta)$. Here, $h(x,beta)$ is known, but assuming that we cannot solve the optimization analytically. I would like to know what kind of such optimization called and ways to solve this kind of problem.
optimization convex-optimization nonlinear-optimization
optimization convex-optimization nonlinear-optimization
asked Dec 28 '18 at 10:25
will_cheukwill_cheuk
1278
asked Dec 28 '18 at 10:25
will_cheukwill_cheuk
1278
asked Dec 28 '18 at 10:25
will_cheukwill_cheuk
1278
asked Dec 28 '18 at 10:25
will_cheukwill_cheuk
1278
asked Dec 28 '18 at 10:25
will_cheukwill_cheuk
1278
1278
$begingroup$
Denote $z=(x,beta)$ and definte $F(z)=f(x)$. Then your problem is basically $min_z F(z)$ subject to $h(z)le 0$.
$endgroup$
– A.Γ.
Dec 28 '18 at 19:18
1
$begingroup$
These are called "bilevel optimization problems." There's quite a bit of literature on them.
$endgroup$
– Brian Borchers
Dec 28 '18 at 20:57
add a comment |
$begingroup$
Denote $z=(x,beta)$ and definte $F(z)=f(x)$. Then your problem is basically $min_z F(z)$ subject to $h(z)le 0$.
$endgroup$
– A.Γ.
Dec 28 '18 at 19:18
1
$begingroup$
These are called "bilevel optimization problems." There's quite a bit of literature on them.
$endgroup$
– Brian Borchers
Dec 28 '18 at 20:57
$begingroup$
Denote $z=(x,beta)$ and definte $F(z)=f(x)$. Then your problem is basically $min_z F(z)$ subject to $h(z)le 0$.
$endgroup$
– A.Γ.
Dec 28 '18 at 19:18
$begingroup$
Denote $z=(x,beta)$ and definte $F(z)=f(x)$. Then your problem is basically $min_z F(z)$ subject to $h(z)le 0$.
$endgroup$
– A.Γ.
Dec 28 '18 at 19:18
1
1
$begingroup$
These are called "bilevel optimization problems." There's quite a bit of literature on them.
$endgroup$
– Brian Borchers
Dec 28 '18 at 20:57
$begingroup$
These are called "bilevel optimization problems." There's quite a bit of literature on them.
$endgroup$
– Brian Borchers
Dec 28 '18 at 20:57
add a comment |
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$begingroup$
Denote $z=(x,beta)$ and definte $F(z)=f(x)$. Then your problem is basically $min_z F(z)$ subject to $h(z)le 0$.
$endgroup$
– A.Γ.
Dec 28 '18 at 19:18
1
$begingroup$
These are called "bilevel optimization problems." There's quite a bit of literature on them.
$endgroup$
– Brian Borchers
Dec 28 '18 at 20:57