Piecewise affine bijections of \(\mathbb R^n\), and the equation \(Sx^ +- Tx^ -=y\)
We prove two criteria for bijectivity of piecewise affine maps \(f\) of \(\mathbb R^n\). It is assumed that \(\det f\) has the same nonzero sign everywhere. The first criterion is based solely on the branching properties of the partition of \(\mathbb R^n\) into domains of linearity of \(f\). The second one concerns the map itself and depends on the images under \(f\) of certain two-dimensional angles. This generalizes and improves results of \textit{R. Schramm} [Math. Oper. Res. 5, 510--522 (1980; Zbl 0477.55003)]. The methods are topological. Our results imply \textit{K. G. Murty}'s solution of the linear complementarity problem [Linear Algebra Appl. 5, 65--108 (1972; Zbl 0241.90046)] and some other partly known facts, which we show are equivalent to it. For example, the map \(x\to Sx^+-Tx^-\) is bijective iff all principal minors of \(S^{-1}T\) are strictly positive.
scientific article; zbMATH DE number 4016627
scientific article