# Optimization with Tricky First derivative condition

I have a function $f$ such that the first derivative consists of $f'_i=\alpha_i*\beta_i$, where if $\beta_i=0$ then I get linear dependence in my solution (which is not allowed). So for the first order condition to be satisfied, I must have $f'_i=0$ whilst $\beta_i \ne 0$ for all $i$. I believe this reduces to requiring that $\alpha_i=0$ but I don't think that I can make that a constraint (as it seems awefully weird to have part of the first derivative equal to zero and the other part non-zero such that the product is zero).... Well, its what I really need but am not sure practically how to do this.

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What is the domain of $f$? What does $*$ mean? Are the $\alpha_i$ and the $\beta_i$ constants or expressions? You should tell us more about your problem. –  Christian Blatter Aug 3 '12 at 13:18
Right! Thanks..... * is just the dot product. And αi is the change of the position of a vector (f is a function that takes the sum of differences of vector functions dotted to themselves). (i.e. f = sum_i,j NormSquaredOfDistanceBetweenVectors). The idea is that the vectors are transformed through linear combinations of the basis vectors. –  Squirtle Aug 3 '12 at 15:09
βi however is of a form that (when zero) makes the vectors linearly depen. (that don't start off this way). The way that we are defining the new vectors is through linear combinations of the old vectors (so its odd that they "become" linearly dependent). But they really don't start dependent, .... , geometrically what is happening is that all the vectors are crashing to a point. Also, αi (in the technical language) is the derivative of the position of a given vector i. In other words, the function should have deriv' equal 2 zero when the vectors quit moving αi=0 (not b/c dependence βi=0) –  Squirtle Aug 3 '12 at 15:17