I am looking for an efficient way of finding a linear fit $Mx = y$ subject to an inequality constraint:
$\frac{|x_2|}{\sqrt{x_3^2 + x_4^2}} \geq a$, with $a \geq 1$.
The rectangular matrix $M$ is about 1000 x 4 (~1000 observation points), and $x$ is 4-dimensional. If I square the constraint, it looks like a quadratically constrained quadratic programming (QCQP) problem:
$min_x \; x^T (M^TM) x - (2y^TM) x$ s.t. $x Q x \leq 0$
The problem is that Q is diagonal with diagonal entries (0, -1, $a^2$, $a^2$), and because of that -1 eigenvalue is not positive-definite. That, in my understanding, implies that the QCQP problem is nonconvex.
Is there a way to possibly reformulate the problem and solve it more efficiently than with a global constrained minimization? Perhaps something to do with having only one constraint as opposed to nonconvex QCQP cases in the literature that deal with multiple ones? This is to be solved thousands of times in my application, so any hints for a more efficient, case-specific implementation (using R or any other open-source library, or an algorithm reference to code after it) would help.