Optimization problem over integration sublevel set in $mathbbR^n$
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Suppose $phi:mathbb R^ntomathbb R^n$ is a fixed smooth vector field; if it's useful, we can assume $|phi(x)|_2leq1$ for all $xinmathbb R^n$. Consider the following optimization problem parameterized by $t$:
$$
f_phi(t):=
sup_lambdainmathbb Rleft[
lambda t - int_Omegamax(0,lambda-nablacdotphi(x)),dx
right].
$$
Can we write an explicit expression for $f_phi(t)$ in terms of the function $phi$ and scalar $t$?
I assume the best way to approach this problem is using the Reynolds Transport Theorem, but the computation seemed to get messy quickly! In a sense, the integral term can also be understood as the integral of $lambda-nablacdotphi$ over the set $x:lambdageqnablacdotphi(x)$.
integration multivariable-calculus derivatives optimization convex-optimization
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up vote
0
down vote
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Suppose $phi:mathbb R^ntomathbb R^n$ is a fixed smooth vector field; if it's useful, we can assume $|phi(x)|_2leq1$ for all $xinmathbb R^n$. Consider the following optimization problem parameterized by $t$:
$$
f_phi(t):=
sup_lambdainmathbb Rleft[
lambda t - int_Omegamax(0,lambda-nablacdotphi(x)),dx
right].
$$
Can we write an explicit expression for $f_phi(t)$ in terms of the function $phi$ and scalar $t$?
I assume the best way to approach this problem is using the Reynolds Transport Theorem, but the computation seemed to get messy quickly! In a sense, the integral term can also be understood as the integral of $lambda-nablacdotphi$ over the set $x:lambdageqnablacdotphi(x)$.
integration multivariable-calculus derivatives optimization convex-optimization
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Suppose $phi:mathbb R^ntomathbb R^n$ is a fixed smooth vector field; if it's useful, we can assume $|phi(x)|_2leq1$ for all $xinmathbb R^n$. Consider the following optimization problem parameterized by $t$:
$$
f_phi(t):=
sup_lambdainmathbb Rleft[
lambda t - int_Omegamax(0,lambda-nablacdotphi(x)),dx
right].
$$
Can we write an explicit expression for $f_phi(t)$ in terms of the function $phi$ and scalar $t$?
I assume the best way to approach this problem is using the Reynolds Transport Theorem, but the computation seemed to get messy quickly! In a sense, the integral term can also be understood as the integral of $lambda-nablacdotphi$ over the set $x:lambdageqnablacdotphi(x)$.
integration multivariable-calculus derivatives optimization convex-optimization
Suppose $phi:mathbb R^ntomathbb R^n$ is a fixed smooth vector field; if it's useful, we can assume $|phi(x)|_2leq1$ for all $xinmathbb R^n$. Consider the following optimization problem parameterized by $t$:
$$
f_phi(t):=
sup_lambdainmathbb Rleft[
lambda t - int_Omegamax(0,lambda-nablacdotphi(x)),dx
right].
$$
Can we write an explicit expression for $f_phi(t)$ in terms of the function $phi$ and scalar $t$?
I assume the best way to approach this problem is using the Reynolds Transport Theorem, but the computation seemed to get messy quickly! In a sense, the integral term can also be understood as the integral of $lambda-nablacdotphi$ over the set $x:lambdageqnablacdotphi(x)$.
integration multivariable-calculus derivatives optimization convex-optimization
edited Aug 12 at 2:16
David G. Stork
8,05421131
8,05421131
asked Aug 12 at 2:00
Justin Solomon
176113
176113
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