Converting function to Quadratic form

[scg4d]

1. Hi everyone. I'm looking for help with converting this function to quadratic form.



2. The function is f(x1,x2)=(x2-x1)^4 + (12*x1*x2) - x1 + x2 - 3.

The quadratic form I need to convert to is: f(x)=(1/2)x'Qx - x'b + h

where x is a vector=[x1 x2]', '=transpose, Q and b are vectors, and h is the constant. Also, Q is symmetric and positive definite (Q=Q' > 0).



3. The trouble I'm running into is that f(x1,x2) is 4th order, and the examples I have convert only 2nd order functions to the f(x) quadratic objective format above.

I hit a wall when I expand the quadratic to:
-b1 x1 + 0.5 a x1^2 - b2 x2 + 0.5 b x1 x2 + 0.5 c x1 x2 + 0.5 d x2^2
where Q=[a, b; c, d] and b=[b1; b2] and try to figure out the variables.

In case your interested, I'm looking for this info to solve a steepest descent problem, where the varying ak value is ak=g(k)'g((k))/( g(k)'Qg(k) ) where g(k)=Qx(k) - b.

Thanks for your help!

 

[Mark44]

1. Hi everyone. I'm looking for help with converting this function to quadratic form.



2. The function is f(x1,x2)=(x2-x1)^4 + (12*x1*x2) - x1 + x2 - 3.

The quadratic form I need to convert to is: f(x)=(1/2)x'Qx - x'b + h

where x is a vector=[x1 x2]', '=transpose, Q and b are vectors, and h is the constant. Also, Q is symmetric and positive definite (Q=Q' > 0).



3. The trouble I'm running into is that f(x1,x2) is 4th order, and the examples I have convert only 2nd order functions to the f(x) quadratic objective format above.

I hit a wall when I expand the quadratic to:
-b1 x1 + 0.5 a x1^2 - b2 x2 + 0.5 b x1 x2 + 0.5 c x1 x2 + 0.5 d x2^2
where Q=[a, b; c, d] and b=[b1; b2] and try to figure out the variables.

In case your interested, I'm looking for this info to solve a steepest descent problem, where the varying ak value is ak=g(k)'g((k))/( g(k)'Qg(k) ) where g(k)=Qx(k) - b.

Thanks for your help!

Here's something that might be helpful, although it has been many years since I did anything with quadratic forms.

You have f(x₁,x₂)=(x₂ - x₁)⁴ + 12x₁x₂ - x₁ + x₂ - 3.

Let u = (x₂ - x₁)², and let v = x₁x₂.
Then your function can be written as
u² + u + 12v - 3
= u² + u + 1/4 + 12 v - 13/4
= (u + 1/2)² + 12v - 13/4.
This could be viewed as quadratic in u and v, but with a coefficient of 0 on the v² term.

Hope this helps.