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How to minimize the following polynomial over a four-dimensional sphere: $(c_0x_0^2+c_1x_1^2+c_2x_2^2+c_3x_3^2)^2+sum_nmc_nm x_nx_m $
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I need to find the global minimum of the following polynomial over a four-dimensional sphere.
$$f(x_0, x_1, x_2, x_3) = left( sumlimits_j=0^3 c_j x_j^2 right)^2 + sumlimits_n,m=0^3 c_nm x_n x_m $$
Here $x_0,x_1,x_2,x_3$ are the variables and are constrained to be on the four-dimensional sphere, i.e.,
$$sum_n=0^3 x_n^2=1$$
and $c_j (j = 0, 1, 2, 3)$ and $c_nm ( n, m = 0,1,2,3)$ are problem constants that can be positive or negative or zero.
Well, I am aware of the general method; find the derivatives calculate the roots etc... however as you can see the problem has some sort of simple structure (quartic part can be diagonalized) so I am wondering if there is a simpler solution. It is also possible to diagonalize the quadratic part and leave the quartic part complicated.
polynomials optimization groebner-basis
edited yesterday
David G. Stork
7,3102728
asked yesterday
Buddha_the_Scientist
115
 |Â
show 2 more comments
up vote
0
down vote
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I need to find the global minimum of the following polynomial over a four-dimensional sphere.
$$f(x_0, x_1, x_2, x_3) = left( sumlimits_j=0^3 c_j x_j^2 right)^2 + sumlimits_n,m=0^3 c_nm x_n x_m $$
Here $x_0,x_1,x_2,x_3$ are the variables and are constrained to be on the four-dimensional sphere, i.e.,
$$sum_n=0^3 x_n^2=1$$
and $c_j (j = 0, 1, 2, 3)$ and $c_nm ( n, m = 0,1,2,3)$ are problem constants that can be positive or negative or zero.
Well, I am aware of the general method; find the derivatives calculate the roots etc... however as you can see the problem has some sort of simple structure (quartic part can be diagonalized) so I am wondering if there is a simpler solution. It is also possible to diagonalize the quadratic part and leave the quartic part complicated.
polynomials optimization groebner-basis
edited yesterday
David G. Stork
7,3102728
asked yesterday
Buddha_the_Scientist
115
Try Lagrange optimization (i.e., a mento for optimization given a constraint). But your problem is unclear: the unit sphere has just $x_1$, $x_2$ and $x_3$. What is $x_0$? Can it actually be a variable? What does it have to do with a sphere?
– David G. Stork
yesterday
Does the range of summation include $0,1,2,3$?
– copper.hat
yesterday
Yes I have edited now. @David G. Stork Yes, thanks. Does the fact that the quartic part can be diagonalized leads to some simplification here?
– Buddha_the_Scientist
yesterday
Are the $c_0,...,c_3$ positive?
– copper.hat
yesterday
@copper.hat No actually, they can be negative.
– Buddha_the_Scientist
yesterday
 |Â
show 2 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I need to find the global minimum of the following polynomial over a four-dimensional sphere.
$$f(x_0, x_1, x_2, x_3) = left( sumlimits_j=0^3 c_j x_j^2 right)^2 + sumlimits_n,m=0^3 c_nm x_n x_m $$
Here $x_0,x_1,x_2,x_3$ are the variables and are constrained to be on the four-dimensional sphere, i.e.,
$$sum_n=0^3 x_n^2=1$$
and $c_j (j = 0, 1, 2, 3)$ and $c_nm ( n, m = 0,1,2,3)$ are problem constants that can be positive or negative or zero.
Well, I am aware of the general method; find the derivatives calculate the roots etc... however as you can see the problem has some sort of simple structure (quartic part can be diagonalized) so I am wondering if there is a simpler solution. It is also possible to diagonalize the quadratic part and leave the quartic part complicated.
polynomials optimization groebner-basis
edited yesterday
David G. Stork
7,3102728
asked yesterday
Buddha_the_Scientist
115
I need to find the global minimum of the following polynomial over a four-dimensional sphere.
$$f(x_0, x_1, x_2, x_3) = left( sumlimits_j=0^3 c_j x_j^2 right)^2 + sumlimits_n,m=0^3 c_nm x_n x_m $$
Here $x_0,x_1,x_2,x_3$ are the variables and are constrained to be on the four-dimensional sphere, i.e.,
$$sum_n=0^3 x_n^2=1$$
and $c_j (j = 0, 1, 2, 3)$ and $c_nm ( n, m = 0,1,2,3)$ are problem constants that can be positive or negative or zero.
Well, I am aware of the general method; find the derivatives calculate the roots etc... however as you can see the problem has some sort of simple structure (quartic part can be diagonalized) so I am wondering if there is a simpler solution. It is also possible to diagonalize the quadratic part and leave the quartic part complicated.
polynomials optimization groebner-basis
edited yesterday
David G. Stork
7,3102728
asked yesterday
Buddha_the_Scientist
115
edited yesterday
David G. Stork
7,3102728
edited yesterday
David G. Stork
7,3102728
edited yesterday
David G. Stork
7,3102728
7,3102728
asked yesterday
Buddha_the_Scientist
115
asked yesterday
Buddha_the_Scientist
115
asked yesterday
Buddha_the_Scientist
115
115
Try Lagrange optimization (i.e., a mento for optimization given a constraint). But your problem is unclear: the unit sphere has just $x_1$, $x_2$ and $x_3$. What is $x_0$? Can it actually be a variable? What does it have to do with a sphere?
– David G. Stork
yesterday
Does the range of summation include $0,1,2,3$?
– copper.hat
yesterday
Yes I have edited now. @David G. Stork Yes, thanks. Does the fact that the quartic part can be diagonalized leads to some simplification here?
– Buddha_the_Scientist
yesterday
Are the $c_0,...,c_3$ positive?
– copper.hat
yesterday
@copper.hat No actually, they can be negative.
– Buddha_the_Scientist
yesterday
 |Â
show 2 more comments
Try Lagrange optimization (i.e., a mento for optimization given a constraint). But your problem is unclear: the unit sphere has just $x_1$, $x_2$ and $x_3$. What is $x_0$? Can it actually be a variable? What does it have to do with a sphere?
– David G. Stork
yesterday
Does the range of summation include $0,1,2,3$?
– copper.hat
yesterday
Yes I have edited now. @David G. Stork Yes, thanks. Does the fact that the quartic part can be diagonalized leads to some simplification here?
– Buddha_the_Scientist
yesterday
Are the $c_0,...,c_3$ positive?
– copper.hat
yesterday
@copper.hat No actually, they can be negative.
– Buddha_the_Scientist
yesterday
Try Lagrange optimization (i.e., a mento for optimization given a constraint). But your problem is unclear: the unit sphere has just $x_1$, $x_2$ and $x_3$. What is $x_0$? Can it actually be a variable? What does it have to do with a sphere?
– David G. Stork
yesterday
Try Lagrange optimization (i.e., a mento for optimization given a constraint). But your problem is unclear: the unit sphere has just $x_1$, $x_2$ and $x_3$. What is $x_0$? Can it actually be a variable? What does it have to do with a sphere?
– David G. Stork
yesterday
Does the range of summation include $0,1,2,3$?
– copper.hat
yesterday
Does the range of summation include $0,1,2,3$?
– copper.hat
yesterday
Yes I have edited now. @David G. Stork Yes, thanks. Does the fact that the quartic part can be diagonalized leads to some simplification here?
– Buddha_the_Scientist
yesterday
Yes I have edited now. @David G. Stork Yes, thanks. Does the fact that the quartic part can be diagonalized leads to some simplification here?
– Buddha_the_Scientist
yesterday
Are the $c_0,...,c_3$ positive?
– copper.hat
yesterday
Are the $c_0,...,c_3$ positive?
– copper.hat
yesterday
@copper.hat No actually, they can be negative.
– Buddha_the_Scientist
yesterday
@copper.hat No actually, they can be negative.
– Buddha_the_Scientist
yesterday
 |Â
show 2 more comments
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Try Lagrange optimization (i.e., a mento for optimization given a constraint). But your problem is unclear: the unit sphere has just $x_1$, $x_2$ and $x_3$. What is $x_0$? Can it actually be a variable? What does it have to do with a sphere?
– David G. Stork
yesterday
Does the range of summation include $0,1,2,3$?
– copper.hat
yesterday
Yes I have edited now. @David G. Stork Yes, thanks. Does the fact that the quartic part can be diagonalized leads to some simplification here?
– Buddha_the_Scientist
yesterday
Are the $c_0,...,c_3$ positive?
– copper.hat
yesterday
@copper.hat No actually, they can be negative.
– Buddha_the_Scientist
yesterday