Coexhausters are families of convex compact sets that allow one to represent the approximation of the increment of a function at a given point in the form of minmax or maxmin of affine functions. We demonstrate that this representation can be used to define a piecewise affine function and therefore coexhausters are a natural technique for studying the problem of finding a global minimum of piecewise affine functions. All the conditions and methods in the current study were obtained by means of coexhausters theory. Firstly, we apply coexhauster based conditions to state and prove necessary and sufficient conditions for a piecewise affine function to be bounded from below. Secondly, we use coexhausters to construct a simple method which allows one to get the minimum value of the studied function and the corresponding set of all its global minimizers. Illustrative numerical examples are provided throughout the paper.