Function which is convex in a variable cannot achieve maximum?
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This question arose out of studying "nice" conditions on Lagrangians. Let $L : mathbb{R}^n times mathbb{R}^n times mathbb{R} to mathbb{R}$ (let us write $L(x, v, t)$) be convex in $v$ (for any $x, t$). Is it true that any extrema of $L$ must be either global minima, or a saddle point? In particular, $L$ cannot achieve a maximum?
I ask this because the Euler-Lagrange equations are equivalent to being a critical point of the action functional, but generally we want to minimize (at least, as far as I know) the action, and so I'm wondering under what conditions we get critical point $implies$ minimum. I've seen convexity in the $v$-variable as an assumption before. Why in this variable? What about $x$, or $t$?
Many thanks for any help!
analysis dynamical-systems euler-lagrange-equation
asked Dec 21 '18 at 19:04
ChrisChris
1,590413
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add a comment |
$begingroup$
This question arose out of studying "nice" conditions on Lagrangians. Let $L : mathbb{R}^n times mathbb{R}^n times mathbb{R} to mathbb{R}$ (let us write $L(x, v, t)$) be convex in $v$ (for any $x, t$). Is it true that any extrema of $L$ must be either global minima, or a saddle point? In particular, $L$ cannot achieve a maximum?
I ask this because the Euler-Lagrange equations are equivalent to being a critical point of the action functional, but generally we want to minimize (at least, as far as I know) the action, and so I'm wondering under what conditions we get critical point $implies$ minimum. I've seen convexity in the $v$-variable as an assumption before. Why in this variable? What about $x$, or $t$?
Many thanks for any help!
analysis dynamical-systems euler-lagrange-equation
asked Dec 21 '18 at 19:04
ChrisChris
1,590413
$endgroup$
$begingroup$
I slightly updated my answer to account for trivial cases.
$endgroup$
– LinAlg
Dec 21 '18 at 19:35
add a comment |
$begingroup$
This question arose out of studying "nice" conditions on Lagrangians. Let $L : mathbb{R}^n times mathbb{R}^n times mathbb{R} to mathbb{R}$ (let us write $L(x, v, t)$) be convex in $v$ (for any $x, t$). Is it true that any extrema of $L$ must be either global minima, or a saddle point? In particular, $L$ cannot achieve a maximum?
I ask this because the Euler-Lagrange equations are equivalent to being a critical point of the action functional, but generally we want to minimize (at least, as far as I know) the action, and so I'm wondering under what conditions we get critical point $implies$ minimum. I've seen convexity in the $v$-variable as an assumption before. Why in this variable? What about $x$, or $t$?
Many thanks for any help!
analysis dynamical-systems euler-lagrange-equation
asked Dec 21 '18 at 19:04
ChrisChris
1,590413
$endgroup$
This question arose out of studying "nice" conditions on Lagrangians. Let $L : mathbb{R}^n times mathbb{R}^n times mathbb{R} to mathbb{R}$ (let us write $L(x, v, t)$) be convex in $v$ (for any $x, t$). Is it true that any extrema of $L$ must be either global minima, or a saddle point? In particular, $L$ cannot achieve a maximum?
I ask this because the Euler-Lagrange equations are equivalent to being a critical point of the action functional, but generally we want to minimize (at least, as far as I know) the action, and so I'm wondering under what conditions we get critical point $implies$ minimum. I've seen convexity in the $v$-variable as an assumption before. Why in this variable? What about $x$, or $t$?
Many thanks for any help!
analysis dynamical-systems euler-lagrange-equation
analysis dynamical-systems euler-lagrange-equation
asked Dec 21 '18 at 19:04
ChrisChris
1,590413
asked Dec 21 '18 at 19:04
ChrisChris
1,590413
asked Dec 21 '18 at 19:04
ChrisChris
1,590413
asked Dec 21 '18 at 19:04
ChrisChris
1,590413
asked Dec 21 '18 at 19:04
ChrisChris
1,590413
1,590413
$begingroup$
I slightly updated my answer to account for trivial cases.
$endgroup$
– LinAlg
Dec 21 '18 at 19:35
add a comment |
$begingroup$
I slightly updated my answer to account for trivial cases.
$endgroup$
– LinAlg
Dec 21 '18 at 19:35
$begingroup$
I slightly updated my answer to account for trivial cases.
$endgroup$
– LinAlg
Dec 21 '18 at 19:35
$begingroup$
I slightly updated my answer to account for trivial cases.
$endgroup$
– LinAlg
Dec 21 '18 at 19:35
add a comment |
1 Answer
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The answer to your question is negative. If $L(x,v,t)$ has a maximum at $(x^*,v^*,t^*)$, then the convex function $f(t) = (x^*,v^*,t)$ has a maximum at $t^*$, which a convex function with domain $mathbb{R}$ cannot have (unless the function is constant).
answered Dec 21 '18 at 19:14
LinAlgLinAlg
10.1k1521
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add a comment |
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1 Answer
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1 Answer
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$begingroup$
The answer to your question is negative. If $L(x,v,t)$ has a maximum at $(x^*,v^*,t^*)$, then the convex function $f(t) = (x^*,v^*,t)$ has a maximum at $t^*$, which a convex function with domain $mathbb{R}$ cannot have (unless the function is constant).
answered Dec 21 '18 at 19:14
LinAlgLinAlg
10.1k1521
$endgroup$
add a comment |
$begingroup$
The answer to your question is negative. If $L(x,v,t)$ has a maximum at $(x^*,v^*,t^*)$, then the convex function $f(t) = (x^*,v^*,t)$ has a maximum at $t^*$, which a convex function with domain $mathbb{R}$ cannot have (unless the function is constant).
answered Dec 21 '18 at 19:14
LinAlgLinAlg
10.1k1521
$endgroup$
add a comment |
$begingroup$
The answer to your question is negative. If $L(x,v,t)$ has a maximum at $(x^*,v^*,t^*)$, then the convex function $f(t) = (x^*,v^*,t)$ has a maximum at $t^*$, which a convex function with domain $mathbb{R}$ cannot have (unless the function is constant).
answered Dec 21 '18 at 19:14
LinAlgLinAlg
10.1k1521
$endgroup$
The answer to your question is negative. If $L(x,v,t)$ has a maximum at $(x^*,v^*,t^*)$, then the convex function $f(t) = (x^*,v^*,t)$ has a maximum at $t^*$, which a convex function with domain $mathbb{R}$ cannot have (unless the function is constant).
answered Dec 21 '18 at 19:14
LinAlgLinAlg
10.1k1521
edited Dec 21 '18 at 19:34
answered Dec 21 '18 at 19:14
LinAlgLinAlg
10.1k1521
answered Dec 21 '18 at 19:14
LinAlgLinAlg
10.1k1521
answered Dec 21 '18 at 19:14
LinAlgLinAlg
10.1k1521
10.1k1521
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$begingroup$
I slightly updated my answer to account for trivial cases.
$endgroup$
– LinAlg
Dec 21 '18 at 19:35