Equivalence of two optimization problem
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Suppose I have a set of angles $theta_k, 1 leq k leq n$, all between $[-pi,pi]$. Would the two minimization problems
$$
f_1(theta) = frac{1}{2n} sum_{j}(theta-theta_j)^2
$$
and
$$
f_2(theta) = frac{1}{2n} sum_j (1 - cos(theta-theta_j))^2
$$
Lead to the same solution?
The closed form of the first one is the mean of all $theta_j$'s, the second one I think it can only be solved with gradient descent or similar.
I don't think they're the same problem, since the second one can be transformed into the first one if all theta's are close to each other.
Is there formal way to prove/disprove they're the same problem?
Thank you
calculus optimization
asked Dec 3 '18 at 21:19
user8469759user8469759
1,5381618
$endgroup$
add a comment |
$begingroup$
Suppose I have a set of angles $theta_k, 1 leq k leq n$, all between $[-pi,pi]$. Would the two minimization problems
$$
f_1(theta) = frac{1}{2n} sum_{j}(theta-theta_j)^2
$$
and
$$
f_2(theta) = frac{1}{2n} sum_j (1 - cos(theta-theta_j))^2
$$
Lead to the same solution?
The closed form of the first one is the mean of all $theta_j$'s, the second one I think it can only be solved with gradient descent or similar.
I don't think they're the same problem, since the second one can be transformed into the first one if all theta's are close to each other.
Is there formal way to prove/disprove they're the same problem?
Thank you
calculus optimization
asked Dec 3 '18 at 21:19
user8469759user8469759
1,5381618
$endgroup$
$begingroup$
The results will be closed, but are likely to be different in the general case. The simplest way should be to show a counter example. At least $n=3$ seems necessary, with a non uniform distribution ...
$endgroup$
– Damien
Dec 3 '18 at 22:44
$begingroup$
What seems necessary?
$endgroup$
– user8469759
Dec 4 '18 at 7:16
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With $n=2$, the mid angle will be optimal in both cases. Same with 3 angles, one of them being the average of the two others
$endgroup$
– Damien
Dec 4 '18 at 7:22
$begingroup$
It is a condition to find a counter example. Sorry if it is not clear
$endgroup$
– Damien
Dec 4 '18 at 7:23
$begingroup$
Could you please give these counter examples in an answer?
$endgroup$
– user8469759
Dec 4 '18 at 8:42
add a comment |
$begingroup$
Suppose I have a set of angles $theta_k, 1 leq k leq n$, all between $[-pi,pi]$. Would the two minimization problems
$$
f_1(theta) = frac{1}{2n} sum_{j}(theta-theta_j)^2
$$
and
$$
f_2(theta) = frac{1}{2n} sum_j (1 - cos(theta-theta_j))^2
$$
Lead to the same solution?
The closed form of the first one is the mean of all $theta_j$'s, the second one I think it can only be solved with gradient descent or similar.
I don't think they're the same problem, since the second one can be transformed into the first one if all theta's are close to each other.
Is there formal way to prove/disprove they're the same problem?
Thank you
calculus optimization
asked Dec 3 '18 at 21:19
user8469759user8469759
1,5381618
$endgroup$
Suppose I have a set of angles $theta_k, 1 leq k leq n$, all between $[-pi,pi]$. Would the two minimization problems
$$
f_1(theta) = frac{1}{2n} sum_{j}(theta-theta_j)^2
$$
and
$$
f_2(theta) = frac{1}{2n} sum_j (1 - cos(theta-theta_j))^2
$$
Lead to the same solution?
The closed form of the first one is the mean of all $theta_j$'s, the second one I think it can only be solved with gradient descent or similar.
I don't think they're the same problem, since the second one can be transformed into the first one if all theta's are close to each other.
Is there formal way to prove/disprove they're the same problem?
Thank you
calculus optimization
calculus optimization
asked Dec 3 '18 at 21:19
user8469759user8469759
1,5381618
asked Dec 3 '18 at 21:19
user8469759user8469759
1,5381618
asked Dec 3 '18 at 21:19
user8469759user8469759
1,5381618
asked Dec 3 '18 at 21:19
user8469759user8469759
1,5381618
asked Dec 3 '18 at 21:19
user8469759user8469759
1,5381618
1,5381618
$begingroup$
The results will be closed, but are likely to be different in the general case. The simplest way should be to show a counter example. At least $n=3$ seems necessary, with a non uniform distribution ...
$endgroup$
– Damien
Dec 3 '18 at 22:44
$begingroup$
What seems necessary?
$endgroup$
– user8469759
Dec 4 '18 at 7:16
$begingroup$
With $n=2$, the mid angle will be optimal in both cases. Same with 3 angles, one of them being the average of the two others
$endgroup$
– Damien
Dec 4 '18 at 7:22
$begingroup$
It is a condition to find a counter example. Sorry if it is not clear
$endgroup$
– Damien
Dec 4 '18 at 7:23
$begingroup$
Could you please give these counter examples in an answer?
$endgroup$
– user8469759
Dec 4 '18 at 8:42
add a comment |
$begingroup$
The results will be closed, but are likely to be different in the general case. The simplest way should be to show a counter example. At least $n=3$ seems necessary, with a non uniform distribution ...
$endgroup$
– Damien
Dec 3 '18 at 22:44
$begingroup$
What seems necessary?
$endgroup$
– user8469759
Dec 4 '18 at 7:16
$begingroup$
With $n=2$, the mid angle will be optimal in both cases. Same with 3 angles, one of them being the average of the two others
$endgroup$
– Damien
Dec 4 '18 at 7:22
$begingroup$
It is a condition to find a counter example. Sorry if it is not clear
$endgroup$
– Damien
Dec 4 '18 at 7:23
$begingroup$
Could you please give these counter examples in an answer?
$endgroup$
– user8469759
Dec 4 '18 at 8:42
$begingroup$
The results will be closed, but are likely to be different in the general case. The simplest way should be to show a counter example. At least $n=3$ seems necessary, with a non uniform distribution ...
$endgroup$
– Damien
Dec 3 '18 at 22:44
$begingroup$
The results will be closed, but are likely to be different in the general case. The simplest way should be to show a counter example. At least $n=3$ seems necessary, with a non uniform distribution ...
$endgroup$
– Damien
Dec 3 '18 at 22:44
$begingroup$
What seems necessary?
$endgroup$
– user8469759
Dec 4 '18 at 7:16
$begingroup$
What seems necessary?
$endgroup$
– user8469759
Dec 4 '18 at 7:16
$begingroup$
With $n=2$, the mid angle will be optimal in both cases. Same with 3 angles, one of them being the average of the two others
$endgroup$
– Damien
Dec 4 '18 at 7:22
$begingroup$
With $n=2$, the mid angle will be optimal in both cases. Same with 3 angles, one of them being the average of the two others
$endgroup$
– Damien
Dec 4 '18 at 7:22
$begingroup$
It is a condition to find a counter example. Sorry if it is not clear
$endgroup$
– Damien
Dec 4 '18 at 7:23
$begingroup$
It is a condition to find a counter example. Sorry if it is not clear
$endgroup$
– Damien
Dec 4 '18 at 7:23
$begingroup$
Could you please give these counter examples in an answer?
$endgroup$
– user8469759
Dec 4 '18 at 8:42
$begingroup$
Could you please give these counter examples in an answer?
$endgroup$
– user8469759
Dec 4 '18 at 8:42
add a comment |
1 Answer
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For $n=2$ they are equal. For $n=3$ they are not, since $theta_1=theta_2=0$, $theta_3=pi$ is a counterexample. We have $$f_2(theta) = frac{1}{6} left(2left(1-cos(theta)right)^2 + left(1-cos(theta-pi)right)^2right),$$ whose minimum does not occur at $pi/3 approx 1.05$ as can be seen in this graph.
answered Dec 4 '18 at 19:20
LinAlgLinAlg
9,7291521
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add a comment |
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$begingroup$
For $n=2$ they are equal. For $n=3$ they are not, since $theta_1=theta_2=0$, $theta_3=pi$ is a counterexample. We have $$f_2(theta) = frac{1}{6} left(2left(1-cos(theta)right)^2 + left(1-cos(theta-pi)right)^2right),$$ whose minimum does not occur at $pi/3 approx 1.05$ as can be seen in this graph.
answered Dec 4 '18 at 19:20
LinAlgLinAlg
9,7291521
$endgroup$
add a comment |
$begingroup$
For $n=2$ they are equal. For $n=3$ they are not, since $theta_1=theta_2=0$, $theta_3=pi$ is a counterexample. We have $$f_2(theta) = frac{1}{6} left(2left(1-cos(theta)right)^2 + left(1-cos(theta-pi)right)^2right),$$ whose minimum does not occur at $pi/3 approx 1.05$ as can be seen in this graph.
answered Dec 4 '18 at 19:20
LinAlgLinAlg
9,7291521
$endgroup$
add a comment |
$begingroup$
For $n=2$ they are equal. For $n=3$ they are not, since $theta_1=theta_2=0$, $theta_3=pi$ is a counterexample. We have $$f_2(theta) = frac{1}{6} left(2left(1-cos(theta)right)^2 + left(1-cos(theta-pi)right)^2right),$$ whose minimum does not occur at $pi/3 approx 1.05$ as can be seen in this graph.
answered Dec 4 '18 at 19:20
LinAlgLinAlg
9,7291521
$endgroup$
For $n=2$ they are equal. For $n=3$ they are not, since $theta_1=theta_2=0$, $theta_3=pi$ is a counterexample. We have $$f_2(theta) = frac{1}{6} left(2left(1-cos(theta)right)^2 + left(1-cos(theta-pi)right)^2right),$$ whose minimum does not occur at $pi/3 approx 1.05$ as can be seen in this graph.
answered Dec 4 '18 at 19:20
LinAlgLinAlg
9,7291521
answered Dec 4 '18 at 19:20
LinAlgLinAlg
9,7291521
answered Dec 4 '18 at 19:20
LinAlgLinAlg
9,7291521
answered Dec 4 '18 at 19:20
LinAlgLinAlg
9,7291521
9,7291521
add a comment |
add a comment |
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$begingroup$
The results will be closed, but are likely to be different in the general case. The simplest way should be to show a counter example. At least $n=3$ seems necessary, with a non uniform distribution ...
$endgroup$
– Damien
Dec 3 '18 at 22:44
$begingroup$
What seems necessary?
$endgroup$
– user8469759
Dec 4 '18 at 7:16
$begingroup$
With $n=2$, the mid angle will be optimal in both cases. Same with 3 angles, one of them being the average of the two others
$endgroup$
– Damien
Dec 4 '18 at 7:22
$begingroup$
It is a condition to find a counter example. Sorry if it is not clear
$endgroup$
– Damien
Dec 4 '18 at 7:23
$begingroup$
Could you please give these counter examples in an answer?
$endgroup$
– user8469759
Dec 4 '18 at 8:42