Which constraints of a numerical problem cause ill-conditioning?

Many numerical problems with input \({x}\) $x$ and output \({y}\) $y$ can be formulated as a system of equations \(F({x}, {y}) = 0\) $F(x,y)=0$ where the goal is to solve for \({y}\) $y$ . The condition number measures the change of \({y}\) $y$ for small perturbations to \({x}\) $x$ . From this numerical problem, one can derive a (typically underdetermined) relaxation by omitting any number of equations from F. We propose a condition number for underdetermined systems that relates the condition number of a numerical problem to those of its relaxations, thereby detecting the ill-conditioned constraints. We illustrate the use of our technique by computing the condition of two problems that do not have a finite condition number in the classic sense: two-factor matrix decompositions and Tucker decompositions.