With the ultimate goal of iteratively solving piecewise smooth (PS) systems,we consider the solution of piecewise linear (PL) equations. PL models can bederived in the fashion of automatic or algorithmic differentiation as localapproximations of PS functions with a second order error in the distance to agiven reference point. The resulting PL functions are obtained quite naturallyin what we call the abs-normal form, a variant of the state representationproposed by Bokhoven in his dissertation. Apart from the tradition of PLmodeling by electrical engineers, which dates back to the Master thesis ofThomas Stern in 1956, we take into account more recent results on linearcomplementarity problems and semi-smooth equations originating in theoptimization community. We analyze simultaneously the original PL problem (OPL)in abs-normal form and a corresponding complementary system (CPL), which isclosely related to the absolute value equation (AVE) studied by Mangasarian etal and a corresponding linear complementarity problem (LCP). We show that theCPL, like KKT conditions and other simply switched systems, cannot be openwithout being injective. Hence some of the intriguing PL structure described byScholtes is lost in the transformation from OPL to CPL. To both problems onemay apply Newton variants with appropriate generalized Jacobians directlycomputable from the abs-normal representation. Alternatively, the CPL can besolved by Bokhoven's modulus method and related fixed point iterations. Wecompile the properties of the various schemes and highlight the connection tothe properties of the Schur complement matrix, in particular its signed realspectral radius as analyzed by Rump. Numerical experiments and suitablecombinations of the fixed point solvers and stabilized generalized Newtonvariants remain to be realized.
展开▼
