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  • $\vec{a}\cdot\vec{b} \approx c$
  • $\vec{\alpha} \cdot \vec{\beta} = c$
  • $\vec{\alpha}$ is close to $\vec{a}$ and $\vec{\beta}$ is close to $\vec{b}$

Given $\vec{a}$, $\vec{b}$ and c, how to find $\vec{\alpha}$, $\vec{\beta}$ quickly?

If necessary, I can assume distance between $\vec{\alpha}$ and $\vec{a}$ is usually much smaller than distance between $\vec{b}$ and $\vec{\beta}$.

Each vector has about 10 - 15 elements but I have to do this for about 10^4 vectors.

If probability should sum to 1 is the constraint, I can normalize the probabilty after each (or several) optimization step to control the numerical error. But I can't do the same thing for this constraint.

Lagrange multipliers and Linearization of the constraints give:

$ \vec{b} \cdot \vec{\alpha} + \vec{a} \cdot \vec{\beta} = \vec{\alpha} \cdot \vec{\beta} + \vec{a} \cdot \vec{b}$

$\vec {\alpha} + \lambda \vec{\beta} = \vec{a} $

$\vec {\beta} + \lambda \vec{\alpha} = \vec{b} $

This is still not a linear system because of the $\lambda \vec{\alpha}$ terms.

How to do this faster?

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  • $\begingroup$ If I use lagrange multipliers, I get a non-linear (quadratic) system. $\endgroup$ – R zu Oct 24 '18 at 17:17
  • $\begingroup$ Linearizing the constraint around $\vec a$ and $\vec b$ leads to 2N + 1 equations with 2N + 1 unknowns. $\endgroup$ – R zu Oct 24 '18 at 17:36
  • $\begingroup$ Maybe I can do two Newton iterations. Fix beta and find alpha and lambda. Then fix alpha and find beta and lambda. Alternate till converge. $\endgroup$ – R zu Oct 24 '18 at 18:13
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    $\begingroup$ This is an underdetermined problem so there are many many solutions. For example, just set $\alpha=a,\beta=bc/(a\cdot b)$. Since $a\cdot b\approx c$ you still get $\beta$ close to $b$. $\endgroup$ – Rahul Oct 24 '18 at 20:20
  • $\begingroup$ If close means least square, then the problem becomes the last 3 equations I wrote. It is a non-linear system with n equations and n unknowns. hmm. I might try your normalization. Any one of the solutions would work. $\endgroup$ – R zu Oct 24 '18 at 20:31
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There are of course infinitely many vectors $\vec \alpha,\vec \beta$ that satisfy $\vec \alpha\cdot\vec \beta=c$. So if you want to have a particular pair of vectors, you will have to be precise when stating what you mean that $\vec \alpha,\vec \beta$ should be "close" to $\vec a,\vec b$.

One pair that satisfies this is $$ \vec \alpha = \vec a, \qquad \qquad \vec \beta = \frac{c}{\vec a \cdot \vec b}\;\vec b. $$

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  • $\begingroup$ I should have read the comments first -- this is of course exactly @Rahul's solution. $\endgroup$ – Wolfgang Bangerth Oct 24 '18 at 22:53

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