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How to maximize the fraction of the square of the sum of some sine and cosine functions
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I have a problem with maximizing the fraction of the square of the sum of some sine and cosine functions. Denote $mathbf{x} triangleq {(x_i)}_{i in {1,2,...,N}}$. The problem is shown below.
$$max_{mathbf{x}} frac{A({(sum_{i = 1}^N cos(x_i - a_i))}^2+{(sum_{i = 1}^N sin(x_i - a_i))}^2)}{B({(sum_{i = 1}^N cos(x_i - b_i))}^2+ {(sum_{i = 1}^N sin(x_i - b_i))}^2)+1}$$
where $A$ and $B$ are constant, and $mathbf{a} triangleq {(a_i)}_{i in {1,2,...,N}}$ and $ mathbf{b} triangleq {(b_i)}_{i in {1,2,...,N}}$ are known.
I tried some variable substitution and found it hard to handle sine and cosine functions decently.
Could anyone give me some hints to handle optimization problem with sine and cosine functions? Thanks!
trigonometry optimization maxima-minima
asked Jan 16 at 13:46
humpbackwhalehumpbackwhale
133
$endgroup$
add a comment |
$begingroup$
I have a problem with maximizing the fraction of the square of the sum of some sine and cosine functions. Denote $mathbf{x} triangleq {(x_i)}_{i in {1,2,...,N}}$. The problem is shown below.
$$max_{mathbf{x}} frac{A({(sum_{i = 1}^N cos(x_i - a_i))}^2+{(sum_{i = 1}^N sin(x_i - a_i))}^2)}{B({(sum_{i = 1}^N cos(x_i - b_i))}^2+ {(sum_{i = 1}^N sin(x_i - b_i))}^2)+1}$$
where $A$ and $B$ are constant, and $mathbf{a} triangleq {(a_i)}_{i in {1,2,...,N}}$ and $ mathbf{b} triangleq {(b_i)}_{i in {1,2,...,N}}$ are known.
I tried some variable substitution and found it hard to handle sine and cosine functions decently.
Could anyone give me some hints to handle optimization problem with sine and cosine functions? Thanks!
trigonometry optimization maxima-minima
asked Jan 16 at 13:46
humpbackwhalehumpbackwhale
133
$endgroup$
$begingroup$
I would try using Euler's formula and see if there is an other way of expressing it, and simplifying some of the terms.
$endgroup$
– P. Quinton
Jan 16 at 13:49
$begingroup$
Thank you for your advice! Do you mean to express sine and cosine functions by $e^{jtheta}$? Actually, the square of the sum of the sine and cosine functions comes from the square of the modulus of a complex number ( and I still don't know how to handle it...)
$endgroup$
– humpbackwhale
Jan 16 at 14:01
$begingroup$
Yes that was what I was suggesting. I think a lot of terms can be factorized if you do so and then it may be easier. But if this is where you come from then you probably already explored that direction.
$endgroup$
– P. Quinton
Jan 16 at 14:03
add a comment |
$begingroup$
I have a problem with maximizing the fraction of the square of the sum of some sine and cosine functions. Denote $mathbf{x} triangleq {(x_i)}_{i in {1,2,...,N}}$. The problem is shown below.
$$max_{mathbf{x}} frac{A({(sum_{i = 1}^N cos(x_i - a_i))}^2+{(sum_{i = 1}^N sin(x_i - a_i))}^2)}{B({(sum_{i = 1}^N cos(x_i - b_i))}^2+ {(sum_{i = 1}^N sin(x_i - b_i))}^2)+1}$$
where $A$ and $B$ are constant, and $mathbf{a} triangleq {(a_i)}_{i in {1,2,...,N}}$ and $ mathbf{b} triangleq {(b_i)}_{i in {1,2,...,N}}$ are known.
I tried some variable substitution and found it hard to handle sine and cosine functions decently.
Could anyone give me some hints to handle optimization problem with sine and cosine functions? Thanks!
trigonometry optimization maxima-minima
asked Jan 16 at 13:46
humpbackwhalehumpbackwhale
133
$endgroup$
I have a problem with maximizing the fraction of the square of the sum of some sine and cosine functions. Denote $mathbf{x} triangleq {(x_i)}_{i in {1,2,...,N}}$. The problem is shown below.
$$max_{mathbf{x}} frac{A({(sum_{i = 1}^N cos(x_i - a_i))}^2+{(sum_{i = 1}^N sin(x_i - a_i))}^2)}{B({(sum_{i = 1}^N cos(x_i - b_i))}^2+ {(sum_{i = 1}^N sin(x_i - b_i))}^2)+1}$$
where $A$ and $B$ are constant, and $mathbf{a} triangleq {(a_i)}_{i in {1,2,...,N}}$ and $ mathbf{b} triangleq {(b_i)}_{i in {1,2,...,N}}$ are known.
I tried some variable substitution and found it hard to handle sine and cosine functions decently.
Could anyone give me some hints to handle optimization problem with sine and cosine functions? Thanks!
trigonometry optimization maxima-minima
trigonometry optimization maxima-minima
asked Jan 16 at 13:46
humpbackwhalehumpbackwhale
133
asked Jan 16 at 13:46
humpbackwhalehumpbackwhale
133
asked Jan 16 at 13:46
humpbackwhalehumpbackwhale
133
asked Jan 16 at 13:46
humpbackwhalehumpbackwhale
133
asked Jan 16 at 13:46
humpbackwhalehumpbackwhale
133
133
$begingroup$
I would try using Euler's formula and see if there is an other way of expressing it, and simplifying some of the terms.
$endgroup$
– P. Quinton
Jan 16 at 13:49
$begingroup$
Thank you for your advice! Do you mean to express sine and cosine functions by $e^{jtheta}$? Actually, the square of the sum of the sine and cosine functions comes from the square of the modulus of a complex number ( and I still don't know how to handle it...)
$endgroup$
– humpbackwhale
Jan 16 at 14:01
$begingroup$
Yes that was what I was suggesting. I think a lot of terms can be factorized if you do so and then it may be easier. But if this is where you come from then you probably already explored that direction.
$endgroup$
– P. Quinton
Jan 16 at 14:03
add a comment |
$begingroup$
I would try using Euler's formula and see if there is an other way of expressing it, and simplifying some of the terms.
$endgroup$
– P. Quinton
Jan 16 at 13:49
$begingroup$
Thank you for your advice! Do you mean to express sine and cosine functions by $e^{jtheta}$? Actually, the square of the sum of the sine and cosine functions comes from the square of the modulus of a complex number ( and I still don't know how to handle it...)
$endgroup$
– humpbackwhale
Jan 16 at 14:01
$begingroup$
Yes that was what I was suggesting. I think a lot of terms can be factorized if you do so and then it may be easier. But if this is where you come from then you probably already explored that direction.
$endgroup$
– P. Quinton
Jan 16 at 14:03
$begingroup$
I would try using Euler's formula and see if there is an other way of expressing it, and simplifying some of the terms.
$endgroup$
– P. Quinton
Jan 16 at 13:49
$begingroup$
I would try using Euler's formula and see if there is an other way of expressing it, and simplifying some of the terms.
$endgroup$
– P. Quinton
Jan 16 at 13:49
$begingroup$
Thank you for your advice! Do you mean to express sine and cosine functions by $e^{jtheta}$? Actually, the square of the sum of the sine and cosine functions comes from the square of the modulus of a complex number ( and I still don't know how to handle it...)
$endgroup$
– humpbackwhale
Jan 16 at 14:01
$begingroup$
Thank you for your advice! Do you mean to express sine and cosine functions by $e^{jtheta}$? Actually, the square of the sum of the sine and cosine functions comes from the square of the modulus of a complex number ( and I still don't know how to handle it...)
$endgroup$
– humpbackwhale
Jan 16 at 14:01
$begingroup$
Yes that was what I was suggesting. I think a lot of terms can be factorized if you do so and then it may be easier. But if this is where you come from then you probably already explored that direction.
$endgroup$
– P. Quinton
Jan 16 at 14:03
$begingroup$
Yes that was what I was suggesting. I think a lot of terms can be factorized if you do so and then it may be easier. But if this is where you come from then you probably already explored that direction.
$endgroup$
– P. Quinton
Jan 16 at 14:03
add a comment |
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$begingroup$
I would try using Euler's formula and see if there is an other way of expressing it, and simplifying some of the terms.
$endgroup$
– P. Quinton
Jan 16 at 13:49
$begingroup$
Thank you for your advice! Do you mean to express sine and cosine functions by $e^{jtheta}$? Actually, the square of the sum of the sine and cosine functions comes from the square of the modulus of a complex number ( and I still don't know how to handle it...)
$endgroup$
– humpbackwhale
Jan 16 at 14:01
$begingroup$
Yes that was what I was suggesting. I think a lot of terms can be factorized if you do so and then it may be easier. But if this is where you come from then you probably already explored that direction.
$endgroup$
– P. Quinton
Jan 16 at 14:03