  In the present work we study a problem of finding a global minimum of a
piecewise affine function. We employ optimality conditions for the problem in
terms of coexhausters and use them to state and prove necessary and sufficient
conditions for a piecewise affine function to be bounded from below. We
construct a simple method based on these conditions which allows one to get the
minimum value of a studied function and the corresponding set of all its global
minimizers. These results are built via coexhauster notion. This notion was
introduced by V. F. Demyanov. Coexhausters are families of convex compact sets
that allow one to represent the approximation of the increment of the studied
function at a considered point in the form of minmax or maxmin of affine
functions. We take these representations as a definition of a piecewise affine
function and show that they correspond with the definitions for piecewise
affine function given by other researchers. All the conditions and methods were
obtained by means of coexhausters theory. In the paper we give some necessary
facts from this theory. A lot of illustrative numerical examples are provided
throughout the paper.
