I want to minimize a multivariable function with the following form: $$ f(\vec{x}) = g(\vec{x}) + | h (\vec{x}) | , $$ this is a piecewise function depending on the sign of $h (\vec{x})$, and I assume that these three functions are all differentiable. For the moment I don't think it's too relevant but I also know that this function is convex.
When $h (\vec{x}) \neq 0$ it should be enough to look for the gradient of the two pieces: $$ \frac{\partial}{\partial {\vec{x}} } \left( g(\vec{x}) + h(\vec{x}) \right) = 0 \quad \text{if} \quad h(\vec{x}) > 0\\ \frac{\partial}{\partial {\vec{x}} } \left( g(\vec{x}) - h(\vec{x}) \right) = 0 \quad \text{if} \quad h(\vec{x}) < 0 $$
But what about the surface $\mathcal{Y} = \left\{ \vec{y} \;\; \text{s.t.} \;\; h(\vec{y})=0 \right\} $?
Is studying the gradient of the first term restricted on this surface enough? I mean: $$ \frac{\partial}{\partial {\vec{y}} } g(\vec{y}) = 0 \quad \text{with} \quad h(\vec{y}) = 0 $$
Thanks!
The minimization of $g$ restricted to the surface $h=0$ can be done using a Lagrange multiplier.