Global minimum of a many-variables function on a specified domain

Question

My target is to compute the global minimum of the real function $f$. $$ f(x_1,x_2,x_3,x_4): D\subset \mathbb{R}^4\quad \to \quad \mathbb{R} $$ Of course the minimum can either be a stationary point of $f$, i.e. a point where $\nabla f=(0,0,0,0)$ or a point that belongs to the domain's frontier. I can easily compute the stationary points of $f$ and check if they are minima, maxima etc, but I am still trying to figure how to exaustively explore the frontier of the domain. The domain $D$ is such that: $$ 0\le x_1 \le 1 \quad \wedge \quad 0 \le x_2 \le 1 \quad \wedge \quad x_2 \le 1-x_1 $$ $$ 0\le x_3 \le 1 \quad \wedge \quad 0\le x_4 \le 1 \quad \wedge \quad x_4 \le 1-x_3 $$ So, if I am correct, the domain of $f$ is the direct product of 2 triangular regions. Any idea about how to find the global minimum of $f$ in its domain, i.e. about how to explore the domain's frontier of $f$?