Solving piecewise linear equations in abs-normal form

Abstract

With the ultimate goal of iteratively solving piecewise smooth (PS) systems, we consider the solution of piecewise linear (PL) equations. PL models can be derived in the fashion of automatic or algorithmic differentiation as local approximations of PS functions with a second order error in the distance to a given reference point. The resulting PL functions are obtained quite naturally in what we call the abs-normal form, a variant of the state representation proposed by Bokhoven in his dissertation. Apart from the tradition of PL modeling by electrical engineers, which dates back to the Master thesis of Thomas Stern in 1956, we take into account more recent results on linear complementarity problems and semi-smooth equations originating in the optimization community. We analyze simultaneously the original PL problem (OPL) in abs-normal form and a corresponding complementary system (CPL), which is closely related to the absolute value equation (AVE) studied by Mangasarian et al and a corresponding linear complementarity problem (LCP). We show that the CPL, like KKT conditions and other simply switched systems, cannot be open without being injective. Hence some of the intriguing PL structure described by Scholtes is lost in the transformation from OPL to CPL. To both problems one may apply Newton variants with appropriate generalized Jacobians directly computable from the abs-normal representation. Alternatively, the CPL can be solved by Bokhoven's modulus method and related fixed point iterations. We compile the properties of the various schemes and highlight the connection to the properties of the Schur complement matrix, in particular its signed real spectral radius as analyzed by Rump. Numerical experiments and suitable combinations of the fixed point solvers and stabilized generalized Newton variants remain to be realized.