Quadratic Problem with 2 constraints

Question

Could someone help me to solve the following:

$\min x^Tx$

s.t.

$x^T a=1$

$x^T b=0$

where $x$,$a$ and $b$ are $(N\times1)$ vectors and $1$ and $0$ scalars.

Thank you!

Answer 1

The stated problem is finding the minimum norm point in an underdetermined set of equations, see http://www.math.usm.edu/lambers/mat419/lecture15.pdf

This problem is a special case of the least squares problem with equality constraints, see http://inst.eecs.berkeley.edu/~ee127a/book/login/l_ols_variants.html

Let me solve the problem: Let $C = [a,b]^T$ and $c = [1,0]^T$ such that your problem becomes

$$\min_{x} \|x\|^2 ~~~~~{\rm s.t. }~~~ Cx =c $$

then, the solution is given by

$x^\star = C^T(C C^T)^{-1}c$.