I need to compute a suberivaqtive for the following function $$f\big((x, y) \big) = |x-2y+1| + |x-4y-3| + |2x-y+2|.$$
I also need to find its global extrema.
I know that a subderivarive of function $f$ in $x_0$ is the set $\partial f(x_o)$ of all subgradients which are defined as follows
Vector $\xi \in \mathbb{R^n}$ is a subgradient of a function $f$ in the point $x_0$ if $f(x) \ge f(x_0) + \xi^{T}(x-x_0)$ for all $x$.
It is easy to compute subgradients for functions $f: \mathbb{R} \mapsto \mathbb{R}$ but I have no idea how to deal with that problem for higher dimensions. I would appreciate any hints or tips.
Let $x-4y-3=a$ and $2x-y+2=b$.
Thus, $$x=-\frac{11}{7}+\frac{4b-a}{7},$$ $$y=-\frac{8}{7}+\frac{b-2a}{7}$$ and $$|x-2y+1| + |x-4y-3| + |2x-y+2|=\left|\frac{12}{7}+\frac{3a+2b}{7}\right|+|a|+|b|\geq$$ $$\geq\left|\frac{12}{7}+\frac{3a+2b}{7}\right|+\frac{3}{7}|-a|+\frac{2}{7}|-b|\geq \left|\frac{12}{7}+\frac{3a+2b}{7}-\frac{3}{7}a-\frac{2}{7}b\right|=\frac{12}{7}.$$ The equality occurs for $a=b=0,$ id est, for $$(x,y)=\left(-\frac{11}{7},-\frac{8}{7}\right),$$ which says that we got a minimal value.
The maximal value does not exist, of course.