I assume you mean norm rather than absolute value because absolute value would give a vector result? But yes, but this is generally unhelpful since in general

I assume you mean norm rather than absolute value because absolute value would give a vector result? But yes, but this is generally unhelpful since in general the behavior at a given point would give you almost no information regarding behavior at a nearby point. It's a bit like doing the reverse by turning the objective into a feasibility test. e.g. min f(x) is like min 0 : {f(x) <= f(x') for all x'}... you can do it, but it's not really helpful.