How to optimize a non-convex function with nonlinear equality constraints

Question

\begin{equation} \min \|X-UV\| \end{equation} \begin{equation} \textbf{s.t.} \|Uz_i\|_{2}=1 \end{equation} with $V_{n\times m}=(z_1,z_2,...,z_m), z_i=(v_{i1},v_{i2},...,v_{in})^{\top}$

Answer 1

The constraint $\|Uz_i\|_2=1$ is the same as $\|Uz_i\|_2^2=1$. If for the norm in the function $\|X-UV\|$ there exists a $p\geq 1$ such that $\|X-UV\|^p$ is differentiable, then the problem is equivalent to minimize $\|X-UV\|^p$ subject to $\|Uz_i\|_2^2=1$. To do this use Karush-Kuhn-Tucker Theorem.